similar images based on nearby sources instead of the central galaxy. A query-
by-example is run as described in Sect. 4.1. The embedding of the query image
is the concept embedding. Grad-CAM is then used to highlight regions in the
images closest to the query image. This test is valid since our self-
supervised approach does not account for background sources during training,
such as using data augmentations. Thus, if background sources are indeed
dominating similarity decisions, it can be attributed to the model’s
incapability to be robust to background sources rather than inappropriate
training.

We now discuss Fig. 22. The LEDA662179 query image in the first example shows
bright green-coloured sources surrounding the central galaxy. The closest
image (LEDA83102), using the similarity search, also contains similar
background sources. The Grad-CAM heatmap for LEDA83102 highlights the bright
green-coloured source instead of the central galaxy, which suggests that the
similarity search was affected by the neighbouring sources.

A similar phenomenon is observed in the second example (ESO358-49 query
image). For example, the heatmap for the second-closest image (ESO357-29)
highlights the two closely-spaced sources in its bottom right more prominently
than the central galaxy. This can be attributed to the fact that the query
image (ESO358-49) also contained two similar sources, so the similarity search
focused on the surrounding sources instead of the central galaxy. However, the
subsequent similar images were not severely affected by surrounding sources.
This could be because the surrounding sources in these similar images did not
dominate the image in spatial size or brightness. The third example also shows
that the nearby sources in the top two similar images to ESO35867 (ESO359-16
and ESO418-13) significantly contributed to the high similarity to ESO35867.
However, ESO359-16 and ESO418-13 were considered visual mergers (not labelled
in the figure). Thus, interacting galaxies could pose difficulties in
similarity search, especially if there is a considerable difference in the
spatial size or brightness of the interacting galaxies.

These examples suggest the importance of removing background sources, as deep
learning models are not inherently robust to background sources. Thus, we have
used the galmask package to explicitly remove background sources for the
machine learning application in the main text. Cases where galmask could not
effectively remove background sources were not considered part of our machine
learning dataset. An alternative possibility to solve this issue is to account
for background sources in the data augmentation pipeline, but this was not
experimented in our study.

Figure 22: Demonstration using the Grad-CAM method of how background sources
in the galaxy images can affect similarity search. Three examples are shown.
The similarity search results are shown for each example, and the
corresponding Grad-CAM heatmaps overlayed on the galaxy images are shown in
the bottom row. This figure motivates handling background sources before or
during model training, especially when the background sources have a size or
brightness comparable to the central galaxy under consideration.

##  Appendix E Classification performance on training and testing on different
galaxy clusters

Appendix. B discussed the case where testing was performed on galaxies from
the Fornax galaxy cluster. For completeness, we extend the comparison of
supervised and self-supervised classification results, where testing is
performed on the Antlia and Hydra galaxy clusters.

Fig. 23 shows the macro-averaged precision and recall of the supervised and
self-supervised classifications. In the case of the Antlia galaxy cluster,
self-supervised classification outperforms the supervised case due to higher
precision and recall scores. However, in the Hydra cluster case, the
supervised approach has a higher precision (91%) than the self-supervised
approach (82%). On the other hand, the recall scores are similar for the self-
supervised approach (86%) and the supervised approach (85%). These two
experiments reinforce the fact that SSL provides results competitive to
supervised learning and can even improve classification results compared to
supervised classification.

Figure 23: Macro-averaged precision and recall scores for supervised and self-
supervised approaches when testing on galaxy images from the Antlia and Hydra
clusters.


# Anisotropic Spacetimes in Chiral Scalar Field Cosmology

A. Giacomini<EMAIL_ADDRESS>Instituto de Ciencias Físicas y
Matemáticas, Universidad Austral de Chile, Valdivia 5090000, Chile  P.G.L.
Leach<EMAIL_ADDRESS>Institute of Systems Science, Durban University of
Technology, Durban 4000, South Africa  G. Leon<EMAIL_ADDRESS>Departamento
de Matemáticas, Universidad Católica del Norte, Avda. Angamos 0610, Casilla
1280 Antofagasta, Chile  A. Paliathanasis<EMAIL_ADDRESS>Instituto de
Ciencias Físicas y Matemáticas, Universidad Austral de Chile, Valdivia
5090000, Chile  Institute of Systems Science, Durban University of Technology,
Durban 4000, South Africa

###### Abstract

We study the behaviour and the evolution of the cosmological field equations
in an homogeneous and anisotropic spacetime with two scalar fields coupled in
the kinetic term. Specifically, the kinetic energy for the scalar field
Lagrangian is that of the Chiral model and defines a two-dimensional maximally
symmetric space with negative curvature. For the background space we assume
the locally rotational spacetime which describes the Bianchi I, the Bianchi
III and the Kantowski-Sachs anisotropic spaces. We work on the
$H$-normalization and we investigate the stationary points and their
stability. For the exponential potential we find a new exact solution which
describes an anisotropic inflationary solution. The anisotropic inflation is
always unstable, while future attractors are the scaling inflationary solution
or the hyperbolic inflation. For scalar field potential different from the
exponential, the de Sitter universe exists.

Multifield Cosmology; Chiral Cosmology; Dynamical analysis; Kantowski-Sachs

###### pacs:

98.80.-k, 95.35.+d, 95.36.+x

##  I Introduction

Gravitational models with two or more scalar fields for the description of the
matter part for the gravitational field equations have been widely studied in
the literature during recent years mf0 ; mf1 ; mf2 ; mf3 ; mf4 ; mf5 ; mf6 ;
mf7 ; mf8 ; mf9 ; mf10 ; mf11 ; mf12 . Multi-scalar field cosmological models
have been used as alternative mechanisms for the description of inflation guth
as also as unified dark energy models. Indeed, because of the additional
degrees of freedom provided by the scalar field, the exit from the
inflationary era is different from the single-scalar field theory.
Specifically, it is not necessary the values for the scalar fields to be the
same at the beginning of the inflation and at the end of the inflation. Hence,
the curvature perturbations can be affected by the different number of e-folds
g1 ; g2 . On the other hand, multi-scalar fields provide non-adiabatic field
perturbations which generate observable non-Gaussianities in the power
spectrum g3 ; g4 ; g5 . As far as the late time universe is concerned, multi-
scalar field models provide dark energy models which can cross the phantom
divide line without the presence of ghosts qq2 , as also to describe the dark
matter component of the universe ancqg .

In this study we focus upon the asymptotic dynamics for the field equations in
a two-scalar field theory known as the Chiral model within an homogeneous and
anisotropic background space atr6 ; vr91 ; vr92 ; ts1 . The kinetic energy of
the two scalar field lies on a two-dimensional maximally symmetric space of
negative curvature, hyperbolic space. This multi-scalar field model provides
the so-called hyperbolic inflation vr91 ; s1 . However, there are various
applications of this model and in other areas of the cosmic evolution ts1 ;
ts2 ; ts3 ; ts4 ; ts5 ; ts6 ; ts7 ; ts8 ; ts9 ; ts10 .

On the other hand, homogeneous and anisotropic are mainly expressed by the
Bianchi class of spatially homogeneous spacetimes. Bianchi spacetimes have
been mainly used for the discussion of anisotropies in the very early universe
Mis69 ; jacobs2 ; collins . The presence of a cosmological constant in Bianchi
spacetimes leads to isotropic universe as a future solution wald . For the
physical space in this study we assume the generic line element nilson

$ds^{2}=-dt^{2}+e^{2\alpha\left(t\right)}\left(e^{2\beta\left(t\right)}dx^{2}+e^{-\beta\left(t\right)}\left(dy^{2}+f^{2}\left(y\right)dz^{2}\right)\right),$
(1)

where the function $f\left(y\right)$ has one of the following forms,
$f_{A}\left(y\right)=1$,
$f_{B}\left(y\right)=\sinh\left(\sqrt{\left|K\right|}y\right)$ and
$f_{C}\left(y\right)=\sin\left(\sqrt{\left|K\right|}y\right)$. The line
element (1) corresponds to homogeneous locally rotational spacetimes (LRS)
induced with four isometries. For $f_{A}\left(y\right)$ the spacetime is that
of Bianchi I, for $f_{B}\left(y\right)$ is that of the Bianchi III metric
while for $f_{C}\left(y\right)$ the line element reduces to that of Kantowski-
Sachs. These three different families of spacetimes reduce to the flat, closed
and open Friedmann–Lemaître–Robertson–Walker (FLRW) spacetimes when the
parameter, $\beta\left(t\right)$, becomes constant. Indeed, the parameter,
$\beta\left(t\right)$, is the anisotropic parameter while
$\alpha\left(t\right)$ is the scale factor for the three-dimensional
hypersurface. These spacetimes play an important role on the description of
the very early universe and specifically during the pre-inflationary era kas2
; kas3 ; kas4 ; kas5 ; kas6 ; kas7 ; kas8 ; kas9 ; kas10 .

In the following we investigate the asymptotic dynamics and the evolution for
the field equations by investigate the stationary points for the field
equations aco . Every stationary point describes a specific era for the
evolution of the field equations sc1 . The analysis of the stability
properties for the stationary points is essential in order to construct the
cosmological history sc2 ; sc3 . Such an analysis provides important
information for the viability of a given gravitational theory sc4 . In
addition this analysis provides important information about the initial
condition problem. Such analysis has been widely studied in various
gravitational models sc5 ; sc6 ; sc7  while some studies in anisotropic
universes can be found in sc8 ; sc9 ; sc10 ; sc11 ; sc12 ; sc13 ; sc14 . The
plan of the paper is as follows.

In Section II we present the field equations for the Chiral theory with
anisotropic background space described by the anisotropic line element (1).
Section III includes the new results of this analysis in which we study the
general evolution and the asymptotic behaviour for the field equations for the
Chiral theory for the potential function of the hyperbolic inflation vr91 . In
Section IV we investigate the dynamics for a scalar field potential beyond the
exponential. Finally, in Section V we summarize the results and we draw our
conclusions.

##  II Field equations

We assume the gravitational model in a Riemann manifold
$g_{\mu\nu}\left(x^{\kappa}\right)$ and Ricci scalar
$R\left(x^{\kappa}\right)~{}$with two-scalar fields minimally coupled to the
gravity, which is described by the following Action Integral.

$S=\int\sqrt{-g}dx^{4}\left(\frac{R}{2}+L_{C}\left(\phi,\nabla_{\mu}\phi,\psi,\nabla_{\mu}\psi\right)\right).$
(2)

Lagrangian $L_{C}\left(\phi,\nabla_{\mu}\phi,\psi,\nabla_{\mu}\psi\right)$ is
assumed to be that of the Chiral model, that is

$L_{C}\left(\phi,\nabla_{\mu}\phi,\psi,\nabla_{\mu}\psi\right)=-\frac{1}{2}g^{\mu\nu}\left(\nabla_{\mu}\phi\nabla_{\nu}\phi+e^{-2\kappa\phi}\nabla_{\mu}\psi\nabla_{\nu}\psi\right)+V(\phi).$
(3)

Consequently, the two scalar fields, $\phi$ and $\psi$, are defined on a two-
dimensional space of constant and negative curvature, while their evolution is
defined on the physical space with metric $g_{\mu\nu}.$ In the following we
assume that $\kappa\neq 0$ and the scalar field potential is that of the
hyperbolic inflation, that is, $V\left(\phi\right)=V_{0}e^{-\lambda\phi}$.

For the line element (1) we derive

$R\left(\alpha,\dot{\alpha},\beta,\dot{\beta}\right)=6\ddot{\alpha}+12\dot{\alpha}^{2}+\frac{3}{2}\dot{\beta}^{2}-2e^{\beta-2\alpha}K$
(4)

and $\sqrt{-g}=e^{3\alpha}$, where the overdot means total derivative with
respect the independent parameter $t$, i.e. $\dot{\alpha}=\frac{d\alpha}{dt}$.

Hence, by substituting into (2) and assuming that the scalar fields inherit
the symmetries of the background space, we obtain the following system of
second-order differential equations anuni1

$2\ddot{\alpha}+3\dot{\alpha}^{2}+\frac{3}{4}\dot{\beta}^{2}+\frac{1}{2}\left(\dot{\phi}^{2}+e^{-2\kappa\phi}\dot{\psi}\right)-V\left(\phi\right)-\frac{1}{3}e^{-2\alpha-\beta}K=0,$
(5) $\ddot{\beta}+3\dot{\alpha}\dot{\beta}+\frac{2}{3}e^{-2\alpha-\beta}K=0,$
(6) $\ddot{\phi}+\kappa
e^{-2\kappa\phi}\dot{\psi}^{2}+3\dot{\alpha}\dot{\phi}+V_{,\phi}=0,$ (7)
$\ddot{\psi}-2\kappa\dot{\phi}\dot{\psi}+3\dot{\alpha}\dot{\psi}=0$ (8)

and the constraint equation

$e^{3\alpha}\left(3\dot{\alpha}^{2}-\frac{3}{4}\dot{\beta}^{2}-\frac{1}{2}\left(\dot{\phi}^{2}+e^{-2\kappa\phi}\dot{\psi}^{2}\right)-V\left(\phi\right)\right)-e^{\alpha-\beta}K=0.$
(9)

The parameter $K$ denotes the spatial curvature of the three-dimensional
hypersurface part for (1). Indeed, for Bianchi I space $K=0$, for Bianchi III
space is positive $K>0$ while for the Kantowski-Sachs space, $K<0$.

###  II.1 Dimensionless variables

In order to study the global evolution of the field equations we define the
new set of variables

$\Sigma=\frac{\dot{\beta}}{2H}~{},~{}x=\frac{\dot{\phi}}{\sqrt{6}H}~{},~{}y^{2}=\frac{V\left(\phi\right)}{3H^{2}}~{},$
(10)
$z=\frac{e^{-\kappa\phi}\dot{\psi}}{\sqrt{6}H}~{},~{}\omega_{R}=\frac{R^{\left(3\right)}}{3H^{2}}~{},$
(11)

where $R^{\left(3\right)}=e^{\alpha-\beta}K$ and
$H\left(t\right)=\dot{\alpha}$ is the expansion rate.

Moreover, we select the new independent variable to be
$dt=d\tau~{},~{}\tau=\alpha$. Thus, in the new variables the field equations
(5)-(8) read

$\Sigma^{\prime}=-y^{2}\left(1+\Sigma\right)+\left(2\Sigma-1\right)\left(x^{2}+z^{2}+\Sigma^{2}-1\right)~{},$
(12)
$x^{\prime}=2x^{3}+\frac{\sqrt{6}}{2}\left(y^{2}\lambda-2z^{2}\kappa\right)-x\left(y^{2}-2\left(z^{2}+\Sigma^{2}-1\right)\right),$
(13)
$y^{\prime}=\frac{1}{2}y\left(2\left(1-y^{2}\right)+4\left(x^{2}+z^{2}+\Sigma^{2}\right)-\sqrt{6}x\lambda\right)$
(14)

and

$z^{\prime}=z\left(\sqrt{6}x\kappa+2\left(x^{2}+z^{2}+\Sigma^{2}-1\right)-y^{2}\right)~{},$
(15)

where $V\left(\phi\right)=V_{0}e^{-\lambda\phi}$,
$\lambda=\frac{V_{,\phi}}{V}$ and $\Sigma^{\prime}=\frac{d\Sigma}{d\tau}$.
Furthermore, the constraint equation (9) reduces to the following algebraic
equation

$\omega_{R}=1-\left(\Sigma^{2}+x^{2}+y^{2}+z^{2}\right).$ (16)

By definition, the parameter $y$ is positive, while the field equations remain
invariant under the discrete transformation $z\rightarrow-z$. Hence we select
to work with $z>0$.

Moreover, we define the deceleration parameter
$q=-1-\frac{\ddot{a}}{\dot{a}^{2}}$, which with the use of the dimensionless
variables is

$q\left(\Sigma,x,y,z\right)=2\left(x^{2}+z^{2}+\Sigma^{2}\right)-y^{2}\text{.}$
(17)

##  III Asymptotic dynamics

We continue our analysis with the study of the dynamics provided by the
dynamical system (12)-(16). Specifically, we determine the stationary points
and we investigate their stability. Every stationary point corresponds to a
specific era in the evolution of the cosmological history.

We summarize the stationary points,
$P=\left(\Sigma\left(P\right),x\left(P\right),y\left(P\right),z\left(P\right)\right)$,
in three categories, (A) stationary points of General Relativity; (B)
stationary points of quintessence and (C) stationary points with two scalar
fields. The first family of stationary points describes exact solutions
without any matter source. Thus, the exact solutions described by these points
are these of General Relativity in the vacuum
$\left(x,y,z\right)=\left(0,0,0\right)$. For the family (B) of points, only
the scalar field $\phi$ contributes in the cosmological solution, that is
$z=0,~{}\left(x,y\right)\neq\left(0,0\right)$, while for the third family of
points, both the scalar fields contribute, i.e. $\dot{\phi}\dot{\psi}\neq 0$.
It is important to mention that the stability properties of the points on the
families (A) and (B) depend upon the existence of the second field, that is,
of the dynamical variable $z$. Thus we should perform a detailed analysis on
the stability conditions. Moreover, stationary points with $\Sigma=0$,
correspond to isotopic background space, while stationary points with
$\eta=0$, indicate that the exact solution is a static solution. Furthermore,
the background space in a asymptotic solution is that of Bianchi I or
spatially flat FLRW metric when $\omega_{R}=0,$ of Bianchi III or closed FLRW
metric when $\omega_{R}>0$, or Kantowski-Sachs or open FLRW universe when
$\omega_{R}<0$.

We determine the stationary points for the dynamical system (12)-(16) for
values of the dynamical variables in the finite regime.

###  III.1 Stationary points of family A

The stationary points which belong to the family A are

$A_{1}^{\pm}=\left(\pm
1,0,0,0\right)~{},~{}A_{2}=\left(\frac{1}{2},0,0,0\right).$

For each of the stationary points we calculate
$\omega_{R}\left(A_{1}^{\pm}\right)=0~{}$ and
$\omega_{R}\left(A_{2}^{\pm}\right)=\frac{3}{4}$.

We continue with the discussion of the physical properties for the asymptotic
solutions at the stationary points while we investigate the stability
properties of the points.

Points $A_{1}^{\pm}$ describe anisotropic spacetime with zero spatial
curvature, that is, the asymptotic solution at the points correspond to Kasner
universes. The eigenvalues of the linearized system near the asymptotic points
are $e_{1}\left(A_{1}^{+}\right)=6~{},~{}e_{2}\left(A_{1}^{+}\right)=3$,
$e_{3}\left(A_{1}^{+}\right)=0$ and $e_{4}\left(A_{1}^{+}\right)=0~{}$;
$e_{1}\left(A_{1}^{-}\right)=3~{},~{}e_{2}\left(A_{1}^{-}\right)=2$,
$e_{3}\left(A_{1}^{-}\right)=0$ and $e_{4}\left(A_{1}^{-}\right)=0$. Thus,
points $A_{1}^{\pm}$ are sources and the asymptotic solutions are always
unstable.

Point $A_{2}$ describes an anisotropic exact solution with nonzero spatial
curvature. The eigenvalues are
$e_{1}\left(A_{3}^{\pm}\right)=\frac{3}{2}~{},~{}e_{2}\left(A_{3}^{\pm}\right)=-\frac{3}{2}$,
$e_{3}\left(A_{3}^{\pm}\right)=-\frac{3}{2}$ and
$e_{4}\left(A_{3}^{\pm}\right)=-\frac{3}{2}$ from which we conclude that the
vacuum anisotropic solution is unstable, while point $A_{2}$ is a saddle point
for the dynamical system

For all the stationary points we derive positive value for the deceleration
parameter, $q\left(A_{1}^{\pm}\right)=2$, $q\left(A_{2}\right)=\frac{1}{2}$.

###  III.2 Stationary points of family B

The family B for the stationary points of the dynamical system (12)-(16) is
consists of the points,

$B_{1}^{\pm}=\left(\pm\sqrt{1-x^{2}},x,0,0\right)~{},~{}B_{2}=\left(0,\frac{\lambda}{\sqrt{6}},\sqrt{1-\frac{\lambda^{2}}{6}},0\right)$
$B_{3}=\left(\frac{1}{2}\left(1-\frac{3}{2\left(1+\lambda^{2}\right)}\right),\frac{\sqrt{6}\lambda}{2\left(1+\lambda^{2}\right)},\frac{\sqrt{6\left(2+\lambda^{2}\right)}}{2\left(1+\lambda^{2}\right)},0\right)$

with $\omega_{R}\left(B_{1}^{\pm}\right)=0$,
$\omega_{R}\left(B_{2}^{\pm}\right)=0$, and
$\omega_{R}\left(B_{3}^{\pm}\right)=\frac{3}{4}\frac{\left(\lambda^{2}-4\right)}{\left(1+\lambda^{2}\right)^{2}}$.

Points $B_{1}^{\pm},$ exist when $x^{2}\leq 1$, for $x^{2}<1$. They describe
families of points where the asymptotic solutions are Kasner universes,
respectively. Moreover, for $x^{2}=1$, the solution is that of spatially flat
FLRW universe dominated by a stiff fluid. For the asymptotic solutions at the
stationary points the decelerating parameter is derived to be $q=2$. Hence,
the points do not describe acceleration. The eigenvalues are derived to be
$e_{1}\left(B_{1}^{\pm}\right)=2\left(2\mp\sqrt{1-x^{2}}\right)$,
$e_{2}\left(B_{1}^{\pm}\right)=\sqrt{6}\kappa x$ ,
$e_{3}\left(B_{1}^{\pm}\right)=\frac{1}{2}\left(6-\sqrt{6}\kappa\lambda\right)$
and$~{}e_{4}\left(B_{1}^{\pm}\right)=0$. Consequently, points $B_{1}^{\pm}$
are sources or saddle points. Specifically, points $B_{2}^{+}$ are sources
when $\left\\{\kappa<0,-1\leq x<0,\lambda>\frac{\sqrt{6}}{x}\right\\}$ or
$\left\\{\kappa>0,0<x\leq 1,\lambda<\frac{\sqrt{6}}{x}\right\\}$. Otherwise
they are saddle points.

Point $B_{2}$ is real and physical accepted when $\lambda^{2}\leq 6$ and
describes the scaling solution for the quintessence field with the exponential
scalar field potential in a spatially flat FLRW background space. The
deceleration parameter is calculated to be
$q\left(B_{2}\right)=\frac{\lambda^{2}-2}{2}$, which means that for
$\lambda^{2}<2$ the asymptotic solution describes an accelerating universe.
The eigenvalues of the linearized system are
$e_{1}\left(B_{2}\right)=\frac{\lambda^{2}+2\kappa\lambda-6}{2}~{},~{}e_{2}\left(B_{2}\right)=\frac{\lambda^{2}-6}{2}$
,$~{}e_{3}\left(B_{2}\right)=\frac{\lambda^{2}-6}{2}$,$~{}e_{4}\left(B_{2}\right)=\left(\lambda^{2}-2\right)$.
Hence, the asymptotic solution at the point $B_{2}$ is stable and the point
$B_{2}$ is an attractor when
$\left\\{-\sqrt{2}<\lambda<0,\kappa>\frac{6-\lambda^{2}}{2\lambda}\right\\}$
or
$\left\\{0<\lambda<\sqrt{2},\kappa<\frac{6-\lambda^{2}}{2\lambda}\right\\}$.
The region where point $B_{2}$ is an attractor is presented in Fig. 1.

Point $B_{3}$ describes the exact solution with anisotropic spacetime, with
positive spatial curvature when $\lambda^{2}>4$, or with negative spatial
curvature when for $\lambda^{2}<4$, while for $\lambda^{2}=4$ it describes a
Bianchi I universe. The deceleration parameter is
$q\left(B_{3}\right)=\frac{\lambda^{2}-2}{2\left(\lambda^{2}+1\right)}$. Hence
there is acceleration for $\lambda^{2}<2$. The eigenvalues of the linearized
system are
$e_{1}\left(B_{3}\right)=-\frac{3\left(2-2\kappa\lambda+\lambda^{2}\right)}{2\left(1+\lambda^{2}\right)}$
,
$e_{2}\left(B_{3}\right)=-\frac{3\left(2+3\lambda^{2}+\lambda^{4}\right)}{2\left(1+\lambda^{2}\right)}$
,
$e_{3}\left(B_{3}\right)=-\frac{3\left(2+3\lambda^{2}+\lambda^{4}+\sqrt{\left(1+\lambda\right)^{2}\left(2+\lambda^{2}\right)\left(7\lambda^{2}-18\right)}\right)}{4\left(1+\lambda^{2}\right)^{2}}$
and
$e_{4}\left(B_{3}\right)=-\frac{3\left(2+3\lambda^{2}+\lambda^{4}\pm\sqrt{\left(1+\lambda\right)^{2}\left(2+\lambda^{2}\right)\left(7\lambda^{2}-18\right)}\right)}{4\left(1+\lambda^{2}\right)^{2}}$.
In Fig. 1 we present the region in the space $\left(\lambda,\kappa\right)$ in
which the point $B_{3}$ is an attractor.

Figure 1: Region plot in the space of the free parameters
$\left(\lambda,\kappa\right)$ where the asymptotic solutions at points
$B_{2}$, $B_{3}$ and $C_{1}$ are stable solutions and points are attractors,
and when point $C_{2}$ is physical accepted and describes anisotropic
inflation.

###  III.3 Stationary points of family C

The third family of stationary points for the dynamical (12)-(16) includes the
points

$C_{1}=\left(0,\frac{\sqrt{6}}{2\kappa+\lambda},\sqrt{\frac{2\kappa}{2\kappa+\lambda}},\frac{\sqrt{\lambda^{2}+2\kappa\lambda-6}}{2\kappa+\lambda}\right)~{},$
$C_{2}=\left(2-\frac{6\kappa}{2\kappa+\lambda},\frac{\sqrt{6}}{2\kappa+\lambda},\frac{\sqrt{6\kappa\left(2\kappa-\lambda\right)}}{2\kappa+\lambda},\frac{\sqrt{6\kappa\lambda-3\left(\lambda^{2}+2\right)}}{2\kappa+\lambda}\right)~{}\text{.}$

The stationary point, $C_{1}$, exists when $2\kappa+\lambda\neq 0$ and
$\left\\{\lambda\leq-\sqrt{6},\kappa<0\right\\}$,
$\left\\{-\sqrt{6}<\lambda<0,\kappa<\frac{6-\lambda^{2}}{2\lambda}\right\\}~{}$,$~{}\left\\{0<\lambda<\sqrt{6},\kappa>\frac{6-\lambda^{2}}{2\lambda}\right\\}~{}$,$~{}\left\\{\lambda\geq\sqrt{6},\kappa>0\right\\}$
, while the spatial curvature for the background space is zero, that is,
$\omega_{R}\left(C_{1}\right)=0$. Hence the background space at the stationary
point is that of spatially flat FLRW universe. The point $C_{1}$ describes
hyperbolic inflation vr91 . The deceleration parameter is written as
$q\left(C_{1}\right)=2-\frac{6\kappa}{2\kappa+\lambda}$. Hence, the asymptotic
solution at $C_{1}$ describes inflationary isotropic inflation when
$\left\\{\lambda\leq-\sqrt{2},\kappa<\lambda\right\\}$,
$\left\\{-\sqrt{2}<\lambda<0,\kappa<\frac{6-\lambda^{2}}{2\lambda}\right\\}~{}$,$~{}\left\\{0<\lambda<\sqrt{2},\kappa>\frac{6-\lambda^{2}}{2\lambda}\right\\}~{}$
and $~{}\left\\{\lambda\geq\sqrt{2},\kappa>\lambda\right\\}$. In Fig. 1 we
present the region in the space $\left(\lambda,\kappa\right)$ in which the
point $C_{1}$ is an attractor. Note that, when $C_{1}$ is an attractor, the
asymptotic solution describes an inflationary universe, i.e.
$q\left(C_{1}\right)<0$.

The point $C_{2}$ describes anisotropic solutions with
$\omega_{R}\left(C_{2}\right)=-\frac{12\kappa\left(\kappa-\lambda\right)}{\left(2\kappa+\lambda\right)^{2}}$.
The points are real and physical acceptable when
$2\kappa\lambda>\left(2+\lambda^{2}\right)^{2}.~{}$ The deceleration parameter
is derived to be
$q\left(C_{2}\right)=2-\frac{6\kappa}{2\kappa+\lambda}.~{}$Consequently, when
the point is real and physical accepted, it describes accelerated anisotropic
inflationary solution in a Kantowski-Sachs background space. We derive the
eigenvalues of the linearized system and we find that the four eigenvalues do
not have real parts with negative values, for the same values of the
parameters, $\lambda$ and $\kappa$. Moreover, they do not have real parts with
positive values for the same values of the parameters, $\lambda$ and $\kappa$.
We conclude that the asymptotic anisotropic solution is unstable and point
$C_{2}$ is a saddle point. In Fig. 1 we present the region in the two-
dimensional space $\left(\lambda,\kappa\right)$ where the point is real and
physical acceptable.

We summarize the results of this analysis in Table 1. We present the
stationary points, their physical properties as also we summarize their
stability conditions.

Table 1: Stationary points and their stability for the anisotropic Chiral model Point | $\left(\mathbf{\Sigma,x,y,z}\right)$ | $\mathbf{\omega}_{R}$ | $\mathbf{q}$ | Stability
---|---|---|---|---
$A_{1}^{\pm}$ | $\left(\pm 1,0,0,0\right)$ | $0$ | $2$ | Source
$A_{2}$ | $\left(\frac{1}{2},0,0,0\right)$ | $\frac{3}{4}$ | $\frac{1}{2}$ | Saddle
$B_{1}^{\pm}$ | $\left(\pm\sqrt{1-x^{2}},x,0,0\right)$ | $0$ | $2$ | Source/Saddle
$B_{2}$ | $\left(0,\frac{\lambda}{\sqrt{6}},\sqrt{1-\frac{\lambda^{2}}{6}},0\right)$ | $0$ | $\frac{\lambda^{2}-2}{2}$ | Attractor Fig. 1
$B_{3}$ | $\left(\frac{1}{2}\left(1-\frac{3}{2\left(1+\lambda^{2}\right)}\right),\frac{\sqrt{6}\lambda}{2\left(1+\lambda^{2}\right)},\frac{\sqrt{6\left(2+\lambda^{2}\right)}}{2\left(1+\lambda^{2}\right)},0\right)$ | $\frac{3\left(\lambda^{2}-4\right)}{4\left(1+\lambda^{2}\right)^{2}}$ | $\frac{\lambda^{2}-2}{2\left(\lambda^{2}+1\right)}$ | Attractor Fig. 1
$C_{1}$ | $\left(0,\frac{\sqrt{6}}{2\kappa+\lambda},\sqrt{\frac{2\kappa}{2\kappa+\lambda}},\frac{\sqrt{\lambda^{2}+2\kappa\lambda-6}}{2\kappa+\lambda}\right)$ | $0$ | $2-\frac{6\kappa}{2\kappa+\lambda}$ | Attractor Fig. 1
$C_{2}$ | $\left(2-\frac{6\kappa}{2\kappa+\lambda},\frac{\sqrt{6}}{2\kappa+\lambda},\frac{\sqrt{6\kappa\left(2\kappa-\lambda\right)}}{2\kappa+\lambda},\frac{\sqrt{6\kappa\lambda-3\left(\lambda^{2}+2\right)}}{2\kappa+\lambda}\right)$ | $-\frac{12\kappa\left(\kappa-\lambda\right)}{\left(2\kappa+\lambda\right)^{2}}$ | $2-\frac{6\kappa}{2\kappa+\lambda}$ | Saddle

##  IV Beyond the exponential potential

We proceed with our analysis by considering a potential function different
from the exponential potential. In particular we assume the existence of a
cosmological constant term, such that the scalar field potential is

$V\left(\phi\right)=V_{0}\left(e^{-\sigma\phi}-\Lambda\right)\text{.}$ (18)

For this potential function parameter $\lambda=\frac{V_{,\phi}}{V}$ is not a
constant, but it depends upon time variable. Indeed, $\lambda=\frac{\sigma
e^{-\sigma\phi}}{e^{-\sigma\phi}-\Lambda}$, such that
$\phi=-\frac{1}{\sigma}\ln\left(\frac{\lambda\Lambda}{\lambda-\sigma}\right)$.
Consequently, the derivative of $\lambda$ is different from zero, that is,

$\lambda^{\prime}=\sqrt{6}x\lambda\left(\sigma-\lambda\right)\text{.}$ (19)

Therefore, for a non-exponential scalar field potential, the gravitational
field equations have one extra dimension, i.e. equation (19). Because of the
existence of equation (19) new stationary points follow, but the stability
properties of the previous points may change.

As above, we summarize the stationary points
$P=\left(\Sigma\left(P\right),x\left(P\right),y\left(P\right),z\left(P\right),\lambda\left(P\right)\right)$
on three families of points, families $\bar{A}$ , $\bar{B}$ and $\bar{C}$. The
stationary points are categorized according to the contribution of the scalar
field in the cosmological fluid as above.

###  IV.1 Stationary points of the family $\bar{A}$

The family $\bar{A}$ is defined by the stationary points

$\bar{A}_{1}^{\pm}=\left(\pm
1,0,0,0,\lambda\right)~{},~{}\bar{A}_{2}=\left(\frac{1}{2},0,0,0,\lambda\right)~{},~{}\lambda~{}\text{\
arbitrary.}$ (20)

The physical properties of the asymptotic solutions at the stationary points
are similar to those of the exponential potential. The eigenvalues for the
five-dimensional linearized system are derived to be
$e_{1}\left(\bar{A}_{1}^{+}\right)=6$
,$~{}e_{2}\left(\bar{A}_{1}^{+}\right)=3$,$~{}e_{3}\left(\bar{A}_{1}^{+}\right)=0$
, $e_{4}\left(\bar{A}_{1}^{+}\right)=0$,
$e_{5}\left(\bar{A}_{1}^{+}\right)=0~{}$;
$e_{1}\left(\bar{A}_{1}^{-}\right)=3$ , $e_{2}\left(\bar{A}_{1}^{-}\right)=2$,
$e_{3}\left(\bar{A}_{1}^{-}\right)=0$, $e_{4}\left(\bar{A}_{1}^{-}\right)=0$ ,
$e_{5}\left(\bar{A}_{1}^{-}\right)=0$ ;
$e_{1}\left(\bar{A}_{2}\right)=\frac{3}{2}~{},~{}e_{2}\left(\bar{A}_{2}\right)=-\frac{3}{2}$,
$e_{3}\left(\bar{A}_{2}\right)=-\frac{3}{2}$
,$~{}e_{4}\left(\bar{A}_{2}\right)=-\frac{3}{2}$
,$~{}~{}e_{5}\left(\bar{A}_{2}\right)=0$. We observe that the stability
properties do not change for the stationary points. Hence, the points
$\bar{A}_{1}^{\pm}$ are sources, while the point $\bar{A}_{2}$ is a saddle
point.

###  IV.2 Stationary points of family $\bar{B}$

Family $\bar{B}$ is composed of the following points

$\bar{B}_{1}^{\pm}=\left(\pm\sqrt{1-x^{2}},x,0,0,\sigma\right)~{},~{}\bar{B}_{2}=\left(0,\frac{\sigma}{\sqrt{6}},\sqrt{1-\frac{\sigma^{2}}{6}},0,\sigma\right),$

$\bar{B}_{3}=\left(\frac{1}{2}\left(1-\frac{3}{2\left(1+\sigma^{2}\right)}\right),\frac{\sqrt{6}\sigma}{2\left(1+\sigma^{2}\right)},\frac{\sqrt{6\left(2+\sigma^{2}\right)}}{2\left(1+\sigma^{2}\right)},0,\sigma\right)~{},$
$\bar{B}_{4}^{\pm}=\left(\pm\sqrt{1-x^{2}},x,0,0,0\right)~{},~{}\bar{B}_{5}=\left(-1,0,\sqrt{3},0,0\right)~{},~{}\bar{B}_{6}=\left(0,0,1,0,0\right).$

The stationary points, $\bar{B}_{1}^{\pm}$, $\bar{B}_{2}$ and $\bar{B}_{3}$,
have the same physical properties with the corresponding points of family $B$.
Thus we study only their stability properties.

The eigenvalues for the linearized system around $\bar{B}_{1}^{\pm}$ are
$e_{1}\left(\bar{B}_{1}^{\pm}\right)=2\left(2\mp\sqrt{1-x^{2}}\right)$
,$~{}e_{2}\left(\bar{B}_{1}^{\pm}\right)=\sqrt{6}x\kappa$,$~{}e_{3}\left(\bar{B}_{1}^{\pm}\right)=-\sqrt{6}x\sigma$
,
$e_{4}\left(\bar{B}_{1}^{\pm}\right)=\frac{1}{2}\left(6-\sqrt{6}x\sigma\right)$,
$e_{5}\left(\bar{B}_{1}^{\pm}\right)=0$. Therefore, the family of points
$\bar{B}_{1}^{\pm}$ are saddle points. Moreover, the eigenvalues around
$\bar{B}_{2}$ are
$e_{1}\left(\bar{B}_{2}\right)=\frac{\sigma^{2}+2\kappa\sigma-6}{2}$
,$~{}e_{2}\left(\bar{B}_{2}\right)=\frac{\sigma^{2}-6}{2}$,$~{}e_{3}\left(\bar{B}_{2}\right)=-\sqrt{6}x\sigma$
, $e_{4}\left(\bar{B}_{2}\right)=\sigma^{2}-2$,
$e_{5}\left(\bar{B}_{2}\right)=-\sigma^{2}$, that is, point $\bar{B}_{2}$ has
similar stability properties with point $B_{2}$, as presented in Fig. 1.
Moreover, for point $\bar{B}_{3}$ we calculate the eigenvalues
$e_{1}\left(\bar{B}_{3}\right)=-\frac{3\left(2-2\kappa\sigma+\sigma^{2}\right)}{2\left(1+\sigma^{2}\right)}$
,
$e_{2}\left(\bar{B}_{3}\right)=-\frac{3\left(2+3\sigma^{2}+\sigma^{4}\right)}{2\left(1+\sigma^{2}\right)}$
,
$e_{3}\left(\bar{B}_{3}\right)=-\frac{3\left(2+3\sigma^{2}+\sigma^{4}+\sqrt{\left(1+\sigma\right)^{2}\left(2+\sigma^{2}\right)\left(7\sigma^{2}-18\right)}\right)}{4\left(1+\sigma^{2}\right)^{2}}$,
$e_{4}\left(\bar{B}_{3}\right)=-\frac{3\left(2+3\sigma^{2}+\sigma^{4}\pm\sqrt{\left(1+\sigma\right)^{2}\left(2+\sigma^{2}\right)\left(7\sigma^{2}-18\right)}\right)}{4\left(1+\sigma^{2}\right)^{2}}$
and $e_{5}\left(B_{3}\right)=-\frac{3\sigma^{2}}{1+\sigma^{2}}$. The
eigenvalue $e_{5}\left(\bar{B}_{3}\right)$ is always negative, while the rest
are similar to those of point $B_{3}$. Thus, point $\bar{B}_{3}$ is an
attractor in the region presented in Fig. 1, where $\lambda=\sigma$.

The stationary points $\bar{B}_{4}^{\pm}$, $\bar{B}_{5}$, $\bar{B}_{6}$
correspond to asymptotic solutions for which the scalar field potential plays
the role of the cosmological constant, that is $V_{,\phi}=0$, and
$V\left(\phi\right)=const.$ Points $\bar{B}_{4}^{\pm}$ have the same physical
properties with $\bar{B}_{1}^{\pm}$, while the eigenvalues of the linearized
system are the same, for $\sigma=0$.

The point $\bar{B}_{5}$ describes an anisotropic inflationary exact solution
in a Kantowski-Sachs spacetime, $\omega_{R}\left(\bar{B}_{5}\right)=-3$,
$q\left(\bar{B}_{5}\right)=-1$. The eigenvalues are
$e_{1}\left(\bar{B}_{5}\right)=-6$, $e_{2}\left(\bar{B}_{5}\right)=-3$,
$e_{3}\left(\bar{B}_{5}\right)=-3$, $e_{4}\left(\bar{B}_{5}\right)=3$,
$e_{5}\left(\bar{B}_{5}\right)=0$, which means that $\bar{B}_{5}$ is a saddle
point.

Finally, points $\bar{B}_{6}$ describe the de Sitter solution in a spatially
flat FLRW spacetime, $\omega_{R}\left(\bar{B}_{6}\right)=0$,
$q\left(\bar{B}_{6}\right)=-1$. The eigenvalues of the linearized system are
$e_{1}\left(\bar{B}_{6}\right)=-3$, $e_{2}\left(\bar{B}_{6}\right)=-3$,
$e_{3}\left(\bar{B}_{6}\right)=-3$, $e_{4}\left(\bar{B}_{6}\right)=-2$,
$e_{5}\left(\bar{B}_{6}\right)=0$. In Fig. 2 we discuss the stability
properties for the stationary point $\bar{B}_{6}$. We observe that the point
is in general a saddle space, but it has a stable manifold in the subspace
$\left\\{\Sigma,x,y,z\right\\}$ for $\lambda=0$. The exact form for the stable
manifold can be derived with the application of the center manifold theorem.
We select to omit such presentation because it does not contribute in the
physical discussion for the anisotropic model.

Figure 2: Phase-space portraits for the dynamical system around the stationary
point $\bar{B}_{6}$. Left figures are for $\sigma=+1$, while right figures are
for $\sigma=-1$. Figures of the first row are in the plane
$\left\\{x,\lambda\right\\}$, where we observe that $\bar{B}_{6}$ is a saddle
point. However, because the dynamical system at the stationary point has four
eigenvalues with negative real parts, there is a stable submanifold when
$\lambda=0$, as it can be seen from the figures of the second row.

###  IV.3 Stationary points of family $\bar{C}$

The stationary points with $xz\neq 0$ are

$\bar{C}_{1}=\left(0,\frac{\sqrt{6}}{2\kappa+\lambda},\sqrt{\frac{2\kappa}{2\kappa+\lambda}},\frac{\sqrt{\lambda^{2}+2\kappa\lambda-6}}{2\kappa+\lambda},\sigma\right)~{}$

and

$\bar{C}_{2}=\left(2-\frac{6\kappa}{2\kappa+\lambda},\frac{\sqrt{6}}{2\kappa+\lambda},\frac{\sqrt{6\kappa\left(2\kappa-\lambda\right)}}{2\kappa+\lambda},\frac{\sqrt{6\kappa\lambda-3\left(\lambda^{2}+2\right)}}{2\kappa+\lambda},\sigma\right)~{}\text{.}$

Thus the physical properties of the solutions are similar those of points
$C_{1}$ and $C_{2}^{\pm}$, respectively. Moreover, the stability properties
are the same as above. Indeed, point $C_{1}$ is an attractor as presented in
Fig. 1 while point $C_{2}$ is always a saddle point.

Finally, there are no (real valued) stationary points for $\lambda=0$.

We conclude that the consideration of a different potential function different
from the exponential, provides new stationary points only in family $B$, that
is, of the quintessence case, in which the second scalar field does not
contribute in the cosmological fluid, $z=0$. The latter can be easily seen
and, if we consider an arbitrary potential function $V\left(\phi\right)$,
where no new stationary points in the family $C$ follow.

Finally, the new physical solutions are anisotropic inflationary solutions in
a Kantowski-Sachs spacetime with cosmological constant and the de Sitter
universe.

##  V Conclusions

We performed a detailed analysis on the dynamics for the Chiral cosmological
theory in a anisotropic background space. The Chiral model belongs to the
multi scalar field theories, in which the energy-momentum tensor of the theory
is consisted by two interacting scalar fields minimally coupled to the
gravity. The two scalar fields interact in the kinetic part, such that the
scalar fields to lie on the hyperbolic plane.

For this model, and for the generic LRS background space which describes the
Bianchi I, the Bianchi III and the Kantowski-Sachs spacetimes we wrote the
field equations by using dimensionless variables in the $H$-normalization
approach. Because of the large number of the dependent variables, we selected
to work on the $H$-normalization instead of other dimensionless variables. We
determined the stationary points of the field equations and we investigated
their stability properties. Every stationary point corresponds to a specific
exact solution for the field equations which describe a specific asymptotic
behaviour during the cosmological evolution.

The stationary points have been categorized in three families. Points of
family $A$ describe the limit of General Relativity without matter source,
points of family $B$ correspond to the stationary points with a quintessence
matter source, while points of the third family, namely family $C$, describe
exact solutions where the two fields contributes. The points of the third
family are of special interests because they can describe isotropic and
anisotropic inflationary solutions with two scalar fields. In particular we
recovered the isotropic inflationary model known hyperbolic inflation vr91 ,
while the anisotropic inflationary solution can be sees as the analogue of
hyperbolic inflation. Because the anisotropic hyperbolic inflationary solution
is always unstable, point $C_{2}$ is a saddle point, we can say that the
anisotropic inflationary solution played role in the very early universe such
is in the beginning of the inflation.

This work contributes on the anisotropic inflationary models. In a future
study we plan to investigate further the effects of the Chiral model in
anisotropic background spaces.

###### Acknowledgements.

The research of AG was funded by Agencia Nacional de Investigación y
Desarrollo - ANID through the program FONDECYT Regular grant no. 1200293. The
research of AP and GL was funded by Agencia Nacional de Investigación y
Desarrollo - ANID through the program FONDECYT Iniciación grant no. 11180126\.
Additionally, GL was funded by Vicerrectoría de Investigación y Desarrollo
Tecnológico at Universidad Catolica del Norte. This work is based on research
supported in part by the National Research Foundation of South Africa (Grant
Numbers 131604).

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