# Emergent Cosmology in 4D Einstein Gauss Bonnet Theory of Gravity

Mrinnoy M. Gohain  0000-0002-1097-2124<EMAIL_ADDRESS>Department of
Physics, Dibrugarh University, Dibrugarh
Assam, India, 786004  Kalyan Bhuyan  0000-0002-8896-7691
<EMAIL_ADDRESS>Department of Physics, Dibrugarh University,
Dibrugarh
Assam, India, 786004  Theoretical Physics Divison, Centre for Atmospheric
Studies, Dibrugarh University, Dibrugarh, Assam, India 786004

###### Abstract

In this paper, in an FLRW background and a perfect fluid equation of state, we
explore the possibility of the realization of an emergent scenario in a 4D
regularized extension of Einstein-Gauss-Bonnet gravity, with the field
equations particularly expressed in terms of scalar-tensor degrees of freedom.
By assuming non-zero spatial curvature ($k=\pm 1$), the stability of the
Einstein static universe (ESU) and its subsequent exit into the standard
inflationary system is tested through different approaches. In terms of
dynamical systems, a spatially closed universe rather than an open universe
shows appealing behaviour to exhibit a graceful transition from the Einstein
static universe to standard cosmological history. We found that under linear
homogeneous perturbations, for some constraints imposed on the model
parameters, the Einstein static universe is stable under those perturbations.
Moreover, it is noted that for a successful graceful transition, the equation
of state $\omega$ must satisfy the conditions $-1<\omega<0$ and $\omega<-1$
for closed and open universes, respectively. Also, under density
perturbations, the Einstein static universe is unstable if the fluid satisfies
the strong energy condition but is stable if it violates it, for both closed
and open universes. Furthermore, the Einstein static universe is seen to be
stable under vector perturbations and tensor perturbations, regardless of
whether the fluid obeys or violates the SEC.

4D-Einstein-Gauss-Bonnet Gravity, Cosmology, Emergent Cosmology, Modified
Gravity, Einstein Static Universe

##  I Introduction

The theory of general relativity suggests that at the genesis of the Universe,
spacetime and matter were compressed to a region of infinitesimally pointlike
singularity (the initial singularity) with infinite density, which in other
words is also known as the Planck’s scale. At Planck’s scale, the known
classical laws of physics break down and quantum mechanics becomes
significantly important. In an attempt to unravel the nature of the initial
singularity, the incompetence of classical general relativity led to the
development of alternative theories, or to be precise theories of quantum
gravity which are based on the application of quantum mechanics in gravity.
Thus quantum gravity (and hence quantum cosmology) has proven to be quite
popular in resolving the problem of the initial singularity. Apart from this,
string theory, ekpyrotic/cyclic, and bouncing universe theories are also some
of the candidates proposed to address the initial singularity problem. In
addition, a relatively new idea to settling the debate of the problem of
initial singularity, Ellis et al. [1, 2] suggested the so-called “ _emergent
Universe_ ” scenario in the framework of general relativity and closed
Friedmann-Lemaitre-Robertson-Walker (FLRW) universe. The emergent Universe is,
therefore, a singularity-free Universe where the Universe is supposed to be
ever-existing in an Einstein static phase and subsequently undergo a phase
transition into the standard inflationary domain. Therefore, in an emergent
Universe, the initial singularity is replaced by the Einstein static Universe.
The original Ellis et al. model [1, 2], which was developed in general
relativity, faced significant fine-tuning problems. The authors suggested
that, rather than a Big Bang singularity, an initially static state known as
the Einstein Static Universe (ESU) existed in the eternal past, following a
closed FLRW cosmological setting with positive spatial curvature, the exit
from which leads to a brief phase of inflationary era. Consequently, a self-
consistent, or stable, departure from the ESU and a graceful transition from
the ESU to the inflationary phase are necessary for the scenario to
successfully describe the fixing of the initial singularity. Notably, the
former and latter are, respectively, sufficient and required criteria for
singularity evasion, indicating that the EU scenario fails if any one of the
two requirements is not satisfied. Ellis et al’s model failed to achieve the
former criteria, posing a significant obstacle to stability. The original EU
scenario failed to successfully resolve the big bang singularity issue given
that Barrow et al [3] discovered that the ESU in GR is not stable, suggesting
that the universe in such an initial static state cannot survive for long
against perturbations in the ES phase. However, in the early universe,
physical situations in particular, like gravity quantization or GR-based
corrections may tip the balance in favour of the EU scenario. In a nutshell,
although the EU scenario collapsed in the context of GR, current modified
gravity theories could potentially able to ameliorate the situation. This idea
has prompted various investigations on the natural improvements of the
original EU setup into modified gravity theories, intending to achieve a few
promising findings in contrast to GR. In the aforementioned context, it seems
that the issues of stability are resolved while working on modified theories
of gravity[4]. With this motive, the Einstein static Universe is extensively
studied in [5, 6, 7], which led to interesting properties that are
substantially dissimilar to those of general relativity, when stability is
concerned. In agreement with this statement, we take into consideration the
possibility of a modified matter-geometry scenario in case of an emergent
Universe scenario with a motive to account for the stability of the Einstein
static Universe. Therefore, it is quite crucial to address the stability of a
model to avoid the collapse of the Einstein static Universe into the initial
singularity. Emergent cosmology is an approach to address the problem of the
big-bang singularity, with a modification of the standard inflationary
Universe, with an ever-existing phase of ESU, with a radius greater than
Planck’s scale to avoid the quantum gravity era. The ESU has been revisited as
a potential foundation for an emergent Universe, which is dedicated to
addressing the avoidance of the initial singularity prevalent within
inflationary cosmology. The ESU exhibit notable features, including the
absence of an initial singularity and mitigation of the issues associated with
quantum gravity. It is important to emphasize that the stability of the ESU
plays a pivotal role in the successful implementation of the emergent Universe
scenario. Extensive studies on the stability of the ESU have been carried out
within various modified gravity theories as well as in theories with various
physically motivated corrections in different theories. These studies can be
found in references [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22,
23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40]

With these motivations at hand, the paper is dedicated to the possibility and
consequences of emergent cosmology in the framework of 4D-Einstein-Gauss-
Bonnet gravity, which is planned as follows: In section II we discuss and
review the basic formulation of $4D$-Einstein Gauss-Bonnet gravity. In section
III we study the stability and graceful exit mechanism of emergent cosmology
using dynamical systems. In sectionIV, we study the stability of the ESU
subject to homogeneous linear perturbations. In section V, we discuss the
consequences of density perturbations to the ESU. In section VI we discuss how
vector and tensor perturbations to the ESU affect its stability and finally in
section VII we summarize and conclude the outcomes of the study.

##  II Brief Review of 4DEGB Gravity

Fernandes et al.[41] have proposed an innovative regularisation method for
Einstein Gauss-Bonnet gravity, resulting in a set of field equations that can
be stated in closed form in 4D. Their approach demands the inclusion of a
counter-term to the action term and is unaffected by the embedding and
compactification of any higher-dimensional space. This counter-term eliminates
the divergence in the action that would otherwise occur. For detailed
derivation see Ref. [41, 42, 43, 44, 45] .Moreover, from an observational
perspective Clifton et al [46]studied the observational constraints on the
4DEGB gravity and Toniato et al [47] developed a complete post-Newtonian
analysis of 4DEGB theories, improving some observational constraints and also
discussed extensions to the PPN formalism accordingly. The action resulting
from the addition of the Gauss-Bonnet term in the Einstein-Hilbert action in
$D$ dimensions with the Glavan-Lin rescaling $\alpha\to\frac{\alpha}{D-4}$ is
[41]

$S=\int d^{D}x\sqrt{-g}\left(R+\frac{\alpha}{D-4}\mathcal{G}\right)$ (1)

where $\mathcal{G}$ is the Gauss-Bonnet scalar given by

$\mathcal{G}=R^{2}-4R_{\mu\nu}R^{\mu\nu}+R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma},$
(2)

It may be noted that the rescaling of the Gauss-Bonnet coupling constant, is a
way of accounting for the so-called _conformal_ or _trace_ anomaly in quantum
field theory [43]. Following [41] directly, one may write the resulting action
after $4D$-regularization first reported by Lu and Pang [44]

$\displaystyle
S=\int_{\mathcal{M}}d^{D}x\sqrt{-g}\left[R+\alpha(D-4)\left(4(D-3)G^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi\right.\right.$
(3)
$\displaystyle\left.\left.-\phi\mathcal{G}-4(D-5)(D-3)\square\phi(\nabla\phi)^{2}-(D-5)(D-3)\right.\right.$
$\displaystyle\left.\left.\hskip
128.0374pt(D-2)(\nabla\phi)^{4}\right)\right]+S_{m}$

In the $4-D$ limit equation (3) reduces to

$\displaystyle S$
$\displaystyle=\int_{\mathcal{M}}\mathrm{d}^{4}x\sqrt{-g}\left[R+\alpha\left(4G^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi-\phi\mathcal{G}\right.\right.$
(4) $\displaystyle\left.\left.\hskip
85.35826pt+4\square\phi(\nabla\phi)^{2}+2(\nabla\phi)^{4}\right)\right]+S_{M},$

which is a $4-D$ action free of divergences. One may also observe that this
action belongs to a class of Horndeski theories with the functions
$G_{2}=8\alpha X^{2}$, $G_{3}=8\alpha X$, $G_{4}=1+4\alpha X$ and
$G_{5}=4\alpha\ln X$ with $X=-\frac{1}{2}\nabla_{\mu}\phi\nabla^{\mu}\phi$.
Moreover, the scalar field action in this action exhibits a shift symmetry
under the transformation $\phi\to\phi+\mathcal{C}$, where $\mathcal{C}$ is an
arbitrary constant.

Varying the above action (4), one may obtain the field equations

$G_{\mu\nu}+\alpha\mathcal{H}_{\mu\nu}=T_{\mu\nu},$ (5)

where

$\displaystyle\mathcal{H}_{\mu\nu}$
$\displaystyle=2G_{\mu\nu}(\nabla\phi)^{2}+4P_{\mu\alpha\nu\beta}\left(\nabla^{\alpha}\phi\nabla^{\beta}\phi-\nabla^{\beta}\nabla^{\alpha}\phi\right)$
(6)
$\displaystyle+4\left(\nabla_{\mu}\phi\nabla_{\nu}\phi-\nabla_{\nu}\nabla_{\mu}\phi\right)\square\phi+4\left(\nabla_{\alpha}\phi\nabla_{\mu}\phi-\nabla_{\alpha}\nabla_{\mu}\phi\right)$
$\displaystyle\left(\nabla^{\alpha}\phi\nabla_{\nu}\phi-\nabla^{\alpha}\nabla_{\nu}\phi\right)+g_{\mu\nu}\left(2(\square\phi)^{2}-(\nabla\phi)^{4}\right.$
$\displaystyle\left.\hskip
56.9055pt+2\nabla_{\beta}\nabla_{\alpha}\phi\left(2\nabla^{\alpha}\phi\nabla^{\beta}\phi-\nabla^{\beta}\nabla^{\alpha}\phi\right)\right),$

with

$\displaystyle P_{\alpha\beta\mu\nu}$
$\displaystyle\equiv*R*_{\alpha\beta\mu\nu}=-R_{\alpha\beta\mu\nu}-g_{\alpha\nu}R_{\beta\mu}+g_{\alpha\mu}R_{\beta\nu}$
(7) $\displaystyle-
g_{\beta\mu}R_{\alpha\nu}+g_{\beta\nu}R_{\alpha\mu}-\frac{1}{2}\left(g_{\alpha\mu}g_{\beta\nu}+g_{\alpha\nu}g_{\beta\mu}\right)R$

which is the double dual of the Riemann tensor.

Let us now consider the homogeneous and isotropic FLRW metric in $D$
dimensions

$ds^{2}=-dt^{2}+a^{2}(t)\left[d\chi^{2}+S_{k}^{2}(\chi)d\Omega^{2}\right],$
(8)

where $a(t)$ is the scale-factor,

$S_{k}(\chi)=\left\\{\begin{matrix}\chi,\hskip 51.21504ptk=0\\\
\sin(\chi),\hskip 34.14322ptk=1\\\ \hskip 8.5359pt\sinh(\chi),\hskip
28.45274ptk=-1\end{matrix}\right.$

and $d\Omega^{2}$ represents the line-element for a $D-2$ sphere. We assume a
perfect fluid energy-momentum tensor $T^{\mu}_{\nu}=\text{diag}(-\rho,p,p,p)$
and the scalar-tensor version of 4DEGB gravity [42, 43] for this particular
work. By using (8) the scalar field equation can be solved which results in
the following field equation [43]

$\alpha\left(k+a^{2}(H+\dot{\phi})^{2}\right)(\ddot{\phi}+\dot{H}+H(\dot{\phi}+H))=0,$
(9)

In this theory, the Friedmann equation is

$H^{2}+\frac{k}{a^{2}}=\frac{8\pi
G}{3}\rho+\alpha\dot{\phi}(2H+\dot{\phi})\left(2\left(H^{2}+\frac{k}{a^{2}}\right)+2H\dot{\phi}+\dot{\phi}^{2}\right),$
(10)

Now, from (9) one obtains

$\dot{\phi}=-H+\frac{K}{a},$ (11)

which, after substitution in the in equation (10), results in

$H^{2}+\frac{k}{a^{2}}+\alpha\left(H^{2}-\frac{K^{2}}{a^{2}}\right)\left(H^{2}+\frac{K^{2}+2k}{a^{2}}\right)=\frac{8\pi
G}{3}\rho.$

When $k=0$, the equation reduces to

$H^{2}+\frac{k}{a^{2}}+\alpha\left(H^{2}+\frac{k}{a^{2}}\right)^{2}=\frac{8\pi
G}{3}\rho,$ (12)

which is the Friedmann equation in the limit $D\to 4$. For a flat Universe,
$K=0$ and for a closed Universe $K=\pm i$. For any other value parametrized by
$K=-k+C$, where $C$ is any free parameter leads to a Friedmann equation with
an extra dark radiation term given as [43]

$H^{2}+\frac{k}{a^{2}}+\alpha\left(H^{2}+\frac{k}{a^{2}}\right)^{2}=\frac{8\pi
G}{3}\rho+\frac{\alpha C^{2}}{a^{4}}.$ (13)

Very recently, Zanoletti et al [48] obtained detailed cosmological aspects of
the 4DEGB theory that places empirical constraints on the basis of CMB data on
the dark radiation parameter $C$, and also computed the perturbed equations of
motion for all values of the curvature parameter $k$. It will be interesting
to explore, the feasibility of non-singular Einstein static behaviour of the
early Universe, which is a crucial element of emergent cosmology. Therefore,
the Friedmann equation (13) shall be our primary interest from which the
dynamics of emergent cosmology shall be studied. The other counterpart to this
equation is the Raychaudhuri equation, which can be obtained by
differentiating equation (13) and using the continuity equation
$\dot{\rho}+3H(\rho+p)=0$, which gives

$\left(\dot{H}-\frac{k}{a^{2}}\right)\left[1+2\alpha\left(H^{2}+\frac{k}{a^{2}}\right)\right]+\frac{2\alpha
C^{2}}{a^{4}}=-4\pi G(\rho+p),$ (14)

The crucial point to be noted here is, in the case of $k=0$ (flat Universe),
from the Friedmann equation (13) one may see that $\rho_{s}=-\frac{\alpha
C^{2}}{a_{s}^{4}}$, ($\rho_{s}$ and $a_{s}$ are the energy density and
Einstein static radius respectively), is negative for $\alpha>0$, which is
unphysical. However, $\alpha<0$ leads to positive energy density, but
$\alpha<0$ is inconvenient for consistent cosmology in [43]. Therefore, in
this work, the cases of closed and open Universes ($k=1$ and $k=-1$) will be
investigated.

The stability analysis of the ESU has been extensively used in Gauss-Bonnet
gravity by different authors. For instance, Huang et al performed a stability
analysis of the ESU in modified Gauss-Bonnet gravity under scalar
perturbations to the Newtonian Gauge through harmonic decomposition of the
potentials associated with it [25]. They found that a closed Universe admits
stable ES solutions subject to homogeneous perturbation but are unstable in
terms of inhomogeneous perturbations, whereas an open Universe is unstable
subject either to homogeneous or inhomogeneous perturbations. More recently,
Li et al [49] found stable ES solutions to scalar perturbations in a 4D Gauss-
Bonnet gravity by rescaling the Gauss-Bonnet coupling constant and performing
the analysis in the $D\to 4$ limit. Böhmer and Lobo [50] studied the stability
of the Einstein static universe in the context of linear homogeneous
perturbations within modified Gauss-Bonnet gravity, by assuming a general form
of the Gauss-Bonnet function characterized by a linear equation of state and
the second derivative of the Gauss-Bonnet term. To the best of our knowledge,
we find that the graceful exit dynamics, which is a requirement of emergent
cosmology has not been explored in the aforementioned works or any other
literature, as far as 4DEGB is concerned. Therefore in our work, we consider a
4D-regularized scalar-tensor version of the Gauss-Bonnet field equations and
perform the stability analysis in various contexts like dynamical stability,
homogeneous scalar perturbation to scale factor and energy density,
inhomogeneous density perturbation, vector and tensor perturbations. As stated
before, given the requirement of a successful emergent cosmology, we shall
also try to address the possibility of graceful exit through the method of
dynamical systems and scalar homogeneous perturbations.

##  III Graceful Exit Mechanism and Stability Analysis

Emergent cosmology is based on the assumption that the initial singularity is
replaced by a stable Einstein static Universe. This follows from the basic
criteria to be fulfilled for a successful emergent cosmology: _existence of
stable and sustained Einstein static Universe_ and _a graceful exit from the
stable Einstein static Universe to the standard cosmology._ In this section,
we shall address both the mechanism of phase transition and stability based on
dynamical system analysis. The stability of a dynamical system is performed
based on the linearised system $\dot{x}_{i}=J_{ij}(x_{j}-x_{j0})$ around the
equilibrium point or critical point $(x_{10},x_{20})=(a_{s},0)$, where $a_{s}$
is the Einstein static radius in our context. $J_{ij}$ represents the elements
of the Jacobian $J$ defined as

$J=\left(\frac{\partial\mathcal{X}_{i}}{\partial
x_{j}}\right)_{(a_{ES},0)}=\begin{pmatrix}0&1\\\
\frac{\partial\mathcal{X}_{2}}{\partial
x_{1}}\left.\right|_{(a_{ES},0)}&0\end{pmatrix}.$ (15)

Using Lyapunov’s method, the stability of the critical point
$(x_{1},x_{2})=(a_{ES},0)$ is determined by the eigenvalues $\lambda$ of the
$J$-matrix (15). We may now construct the dynamical system for the
Raychaudhuri equation (14).

Let us consider $x_{1}=a$, $x_{2}=\dot{x_{1}}=\dot{a}$. Thus the dynamical
system of equations becomes

$\displaystyle\dot{x_{1}}$
$\displaystyle=x_{2}\equiv\mathcal{X}_{1}(x_{1},x_{2}),$ (16)
$\displaystyle\dot{x_{2}}$ $\displaystyle=\frac{-\frac{2\alpha
C^{2}}{x_{1}^{3}}-4\pi G\rho x_{1}(\omega+1)+\frac{2\alpha
k^{2}}{x_{1}^{3}}+\frac{2\alpha kx_{2}^{2}}{x_{1}^{2}}+\frac{2\alpha
kx_{2}^{2}}{x_{1}^{3}}+\frac{k}{x_{1}}+\frac{2\alpha
x_{2}^{4}}{x_{1}^{4}}+x_{2}}{\frac{2\alpha k}{x_{1}^{2}}+\frac{2\alpha
x_{2}^{2}}{x_{1}^{2}}+1}\equiv\mathcal{X}_{2}(x_{1},x_{2})$

###  III.1 Model 1: $k=1$

Let us first consider the case of a closed Universe. By setting $k=1$ in the
Raychaudhuri equation (14) and considering $x_{1}=a$ and
$x_{2}=\dot{x}_{1}=\dot{a}$, we can construct the dynamical system in this
model as

$\displaystyle\dot{x}_{1}$
$\displaystyle=x_{2}\equiv\mathcal{X}_{1}(x_{1},x_{2}),$ (17)
$\displaystyle\dot{x}_{2}$ $\displaystyle=\frac{2\alpha
x_{1}\left(-C^{2}+x_{2}^{2}+1\right)-4\pi G\rho x_{1}^{5}(\omega+1)+2\alpha
x_{2}^{2}x_{1}^{2}+2\alpha
x_{2}^{4}+x_{2}x_{1}^{4}+x_{1}^{3}}{2\alpha\left(x_{2}^{2}+1\right)x_{1}^{2}+x_{1}^{4}}\equiv\mathcal{X}_{2}(x_{1},x_{2}).$

The eigenvalue $\lambda$ for this system about the critical point
$(x_{1},x_{2})=(a_{s},0)$ is

$\displaystyle\lambda^{2}$ $\displaystyle=\frac{1}{2\left(2\alpha
a_{s}+a_{s}^{3}\right)^{2}}\left[4\alpha\left(3C^{2}-2\right)a_{s}^{2}-\right.$
(18)
$\displaystyle\left.\left((\omega+1)a_{s}^{4}\rho_{s}\left(a_{s}^{2}+6\alpha\right)\right)-2a_{s}^{4}+8\alpha^{2}\left(C^{2}-1\right)\right]$

Since the ESR $a_{s}$ is arbitrary and positive, let us assume $a_{s}=1$
111Note that the assumed value for $a_{s}$ chosen in such a way it does not
violate the requirement of the classical notion of emergent cosmology. Since
we are working with a natural unit system, $c=\hbar=1$ and $G=1/8\pi$, this
gives the Planck length $\sqrt{1/8\pi}$. To avoid the quantum gravity era, the
ES radius must be greater than the Planck length. Also, on a similar footing,
the Planck energy density must be chosen such that $\rho_{s}<64\pi^{2}$. This
gives

$\lambda^{2}=-\frac{8\alpha(\alpha+1)-4\alpha(2\alpha+3)C^{2}+(6\alpha+1)(\omega+1)\rho_{s}+2}{2(2\alpha+1)^{2}}$
(19)

This form of eigenvalues squared is useful as it relates the energy density of
the ESU with the model parameters $\alpha$ and $C$ along with the EoS
$\omega$. From this relation, we may obtain the stability regions
corresponding to the requirement of the model parameters. The sign of the
eigenvalue squared ($\lambda^{2}$) determines the type of stability of the
dynamical system. When $\lambda^{2}<0$, the critical point obtained for the
ESU is a centre equilibrium point and has circular stability. In other words,
any small perturbation from the critical point will lead to indefinite
oscillations about the point rather than an exponential deviation from it. By
simultaneously setting $\lambda^{2}<0$ required for stable solutions and
$\rho_{s}>0$, $\alpha>0$ and $C>0$, one may obtain the constraints on $\omega$
as

$\omega>\frac{-8\alpha^{2}-8\alpha+8\alpha^{2}C^{2}+12\alpha
C^{2}-6\alpha\rho_{s}-\rho_{s}-2}{6\alpha\rho_{s}+\rho_{s}}$ (20)

This constraint on $\omega$ is however not entirely informative about the
explicit existence regions as it involves the unknown parameters in a coupled
form. To reduce it further to a convincingly simpler form, we may need to
eliminate $\rho_{s}$ and $\alpha$ (or $C$) from the expression. Also, note
that the constraint on $\omega$ is not changed by the sign of $C$, which keeps
the inequality (20) unchanged irrespective of the sign of $C$. Now solving for
$\rho_{s}$ from equation (13) we get

$\rho_{s}=3\left(1+\alpha-\alpha C^{2}\right)$ (21)

Again setting $\rho_{s}>0,\alpha>0$ the constraint on $C$ is obtained as

$-\sqrt{\frac{\alpha+1}{\alpha}}<C<\sqrt{\frac{\alpha+1}{\alpha}}$ (22)

This relation gives the possible ranges of $C$ given the values of $\alpha$.
For $k=1,$ with $\rho=\rho_{s},a=a_{s}=1,\dot{a}=\ddot{a}=0$, the Raychaudhuri
equation can be expressed in a simplified form for the ESU

$\frac{3}{2}(\omega+1)\left(\alpha-\alpha C^{2}+1\right)+2\alpha
C^{2}-(1+2\alpha)=0,$ (23)

Solving for $\alpha$ gives

$\alpha=\frac{3\omega+1}{\left(C^{2}-1\right)(3\omega-1)},$ (24)

Keeping in mind the statement addressed earlier at the end of section I, we
set $\alpha>0$. Also, let us choose $C>0$ 222Setting $C>0$ or $C<0$ would not
have any effect on the stability regions as observed from (24). We may choose
$C>0$ conventionally., from which we find the following constraints on the EoS
parameter from the $\alpha>0$ perspective for different ranges of $C$

$\displaystyle\text{For }0<C<1,-\frac{1}{3}<\omega<\frac{1}{3},$ (25)
$\displaystyle\text{and for }C>1,\omega<-\frac{1}{3}\text{ or
}\omega>\frac{1}{3}$

With the substitution of equations (21) and (24) into the inequality (20) we
find the following existence regions for different ranges of $C$

$\displaystyle\left\\{(C,\omega)\mid 0<C<\frac{1}{\sqrt{2}}\ \text{and}\
\left(\frac{-3+C^{2}}{3+3C^{2}}<\omega<-\frac{5}{9}\ \text{or}\
\omega>\frac{-7+C^{2}}{15+3C^{2}}\right)\right\\}$ (26)
$\displaystyle\cup\left\\{(C,\omega)\mid C=\frac{1}{\sqrt{2}}\ \text{and}\
\omega>-\frac{13}{33}\right\\}$
$\displaystyle\cup\left\\{(C,\omega)\mid\frac{1}{\sqrt{2}}<C\leq 1\
\text{and}\ \left(-\frac{5}{9}<\omega<\frac{-3+C^{2}}{3+3C^{2}}\ \text{or}\
\omega>\frac{-7+C^{2}}{15+3C^{2}}\right)\right\\}$
$\displaystyle\cup\left\\{(C,\omega)\mid C>1\ \text{and}\
\left(-\frac{5}{9}<\omega<\frac{-7+C^{2}}{15+3C^{2}}\
\omega>\frac{-3+C^{2}}{3+3C^{2}}\right)\right\\}$

Note that these stability ranges are free from the dependence on $\rho_{s}$
and $\alpha$. Thus, the admitted parameter values for $\lambda^{2}<0$ lead to
a centre equilibrium point and refer to a stable ESU.

###  III.2 Model 2: $k=-1$

For the case of an open Universe, setting $k=-1$ in the Raychaudhuri equation
(14), and again considering $x_{1}=a$ and $x_{2}=\dot{x}_{1}=\dot{a}$, the
dynamical system for this case becomes

$\displaystyle\dot{x}_{1}$ $\displaystyle=x_{2},$ (27)
$\displaystyle\dot{x}_{2}$ $\displaystyle=-\frac{2\alpha
x_{1}\left(C^{2}+x_{2}^{2}-1\right)+4\pi G\rho x_{1}^{5}(\omega+1)+2\alpha
x_{2}^{2}x_{1}^{2}-2\alpha
x_{2}^{4}-x_{2}x_{1}^{4}+x_{1}^{3}}{2\alpha\left(x_{2}^{2}-1\right)x_{1}^{2}+x_{1}^{4}}\equiv\mathcal{X}(x_{1},x_{2}).$

The eigenvalue $\lambda$ for this system around the critical point
$(x_{1},x_{2})=(a_{s},0)$ is

$\displaystyle\lambda^{2}$ $\displaystyle=\frac{1}{2\left(a_{s}^{3}-2\alpha
a_{s}\right)^{2}}\left[4\alpha\left(3C^{2}-2\right)a_{s}^{2}+a_{s}^{4}\right.$
(28)
$\displaystyle\left.\left(6\alpha(\omega+1)\rho_{s}+2\right)-\left((\omega+1)a_{s}^{6}\rho_{s}\right)-8\alpha^{2}\left(C^{2}-1\right)\right]$

Setting $a_{s}=1$ (also setting $G=1/8\pi$ simultaneously) the eigenvalue
squared from equation (28) becomes,

$\lambda^{2}=\frac{\begin{split}-8\alpha^{2}\left(C^{2}-1\right)&+4\alpha\left(3C^{2}-2\right)+(6\alpha-1)\\\
&\hskip 56.9055pt(\omega+1)\rho_{s}+2\end{split}}{2(1-2\alpha)^{2}}$ (29)

In this case, the energy density of the ESU becomes

$\rho_{s}=3(\alpha-1-\alpha C^{2}),$ (30)

Setting $\rho_{s}>0$ in equation (30) the constraint on $C$ becomes

$-\sqrt{\frac{\alpha-1}{\alpha}}<C<\sqrt{\frac{\alpha-1}{\alpha}},\quad\forall\,\alpha>1.$
(31)

Now, for stable solutions like the previous case the constraints on $\omega$
with $\lambda^{2}<0$ for $\rho_{s}>0,\alpha>1$ and $C>0$ are found as

$\omega<\frac{-8\alpha^{2}+8\alpha+8\alpha^{2}C^{2}-12\alpha
C^{2}-6\alpha\rho_{s}+\rho_{s}-2}{6\alpha\rho_{s}-\rho_{s}}$ (32)

Substuting $\rho=\rho_{s},$ from equation (30) and $\dot{a}=\ddot{a}=0$ in
equation (14) we get

$\frac{3}{2}(\omega+1)\left(\alpha-\alpha C^{2}+1\right)+2\alpha
C^{2}+(1-2\alpha)=0,$ (33)

Solving (33) for $\alpha$ gives,

$\alpha=\frac{3\omega+5}{\left(C^{2}-1\right)(3\omega-1)}$ (34)

It is seen from equation (31), that real values of $C$ must require
$\alpha>1$. Imposing $\alpha>1$ and $C>0$ the constraints on $C$ and $\omega$
are

$\displaystyle\text{For
}0<C<1,\quad\frac{C^{2}+4}{3C^{2}-6}<\omega<\frac{1}{3}\text{ or }$ (35)
$\displaystyle\text{For
}1<C<\sqrt{2},\quad\left(\omega<\frac{C^{2}+4}{3C^{2}-6}\,\cup\,\omega>\frac{1}{3}\right)\text{
or }$ $\displaystyle\text{ For
}\left(C=\sqrt{2},\quad\omega>\frac{1}{3}\right)\text{ or }$
$\displaystyle\text{For
}\left(C>\sqrt{2},\quad\frac{1}{3}<\omega<\frac{C^{2}+4}{3C^{2}-6}\right)$

Substituting equation (30) and (34) into equation (32) eliminates the
dependence of $\omega$ on $\rho_{s}$ and $\alpha$ which gives the existence
regions of stability (shown in appendix A). As mentioned earlier, the sign of
$C$ does not affect the analysis. This stability regions are true for both
positive and negative values of $C$ provided the condition (31) is met.

Figure 1: The evolution of $\lambda^{2}$ with $\omega$ is shown for different
combinations of the parameter values for both $k=1$ (See (i) and (ii)) and
$k=-1$ (See (iii) and (iv)) cases.

Let us now address the graceful exit mechanism of the ESU to the standard
cosmology for both scenarios of $k=1$ and $k=-1$ by illustrating the evolution
of the eigenvalue squared against the EoS parameter $\omega$. For $k=1$, it is
evident that $\lambda^{2}$ evolves linearly with decreasing $\omega$ from the
negative to the positive region of $\lambda^{2}$. The phase transition occurs
in the interval $w=[-1,0]$ for different sets of $(\alpha,C)$ parameter
values, where $C$ is adjusted and $\alpha$ is kept fixed, as shown in Fig. 1
(i). Fig. 1 (ii) shows a similar scenario where $\alpha$ varies but $C$
remains constant. In contrast, when $k=-1$, it is found that as $\omega$
decreases, the variation of $\lambda^{2}$ is from the positive to the negative
regions, regardless of different combinations of $(\alpha,C)$. Negative
$\lambda^{2}$ indicates imaginary eigenvalues, which correspond to a stable
centre where perturbations to the static solutions do not lead to a collapse
or diverging of the solutions, resulting in a perpetual loop of oscillations
around the ES critical point. However, $\lambda^{2}>0$ characterises a saddle-
like point that indicates an unstable critical point. So, at this stage, one
might anticipate finding a mechanism that overcomes these endless oscillations
about the ESU critical point and leads to a subsequent unstable saddle point
indicating the inflationary phase. The transition from negative to positive
$\lambda^{2}$ may be viewed as a graceful exit from the stable ESU to the
standard cosmological scenario, a fundamental requirement for emergent
cosmology. 333One may see that the decrease of $\omega$ from higher to lower
numerical values may be interpreted as the overall growth of cosmic time. This
interpretation is explained rigorously in [22]. In recapitulation, $k=1$
displays promising behaviour while exhibiting a graceful exit from ESU to a
standard inflationary scenario whereas $k=-1$ does not show the correct
graceful exit mechanism. From the analysis, it is also clearly seen that
$\omega$ plays the role of a bifurcation parameter, since depending on these
values the qualitative behaviour of the ESU changes.

##  IV Stability under Homogeneous Linear Perturbation

In this section, we aim to understand the stability of the ES Universe under
linear homogeneous perturbations for $k=\pm 1$ Universes. Our motive is to
find a possible influence of such perturbation on the stability of the ESU. A
time-dependent perturbation is introduced into the scale factor and the energy
density up to a linear order, given by

$a(t)=a_{0}(1+\delta a(t)),\quad\rho(t)=\rho_{0}(1+\delta\rho(t)),$ (36)

where $\delta a(t)$ and $\delta\rho(t)$ are infinitesimal linear perturbations
introduced to the scale factor and energy density respectively.

For the ESU in the $k=1$ case, setting $\dot{a}=\ddot{a}=0$ and also
$G=1/8\pi$ in equation (13) and (14) we obtain

$\rho_{s}d\rho=3\left(\frac{4\alpha
C^{2}}{a_{s}^{4}}-\frac{2}{a_{s}^{2}}-\frac{4\alpha}{a_{s}^{4}}\right)\delta
a,$ (37)
$\frac{1}{a_{s}^{2}}+\frac{\alpha}{a_{s}^{4}}-\frac{1}{3}\rho_{s}-\frac{\alpha
C^{2}}{a_{s}^{4}}=0$ (38)

and for $k=-1$ case we get

$\rho_{s}d\rho=3\left(\frac{4\alpha
C^{2}}{a_{s}^{4}}-\frac{2}{a_{s}^{2}}-\frac{4\alpha}{a_{s}^{4}}\right)\delta
a,$ (39)
$-\frac{1}{a_{s}^{2}}+\frac{\alpha}{a_{s}^{4}}-\frac{1}{3}\rho_{s}-\frac{\alpha
C^{2}}{a_{s}^{4}}=0$ (40)

Now, substituting the perturbed scale factor and the energy density in
equation (13) and using the respective $\rho_{s}d\rho$ equations (37) and
(39), we finally obtain

$\delta\ddot{a}+\Omega_{1}\delta a=0,$ (41)

for $k=1$ case and

$\delta\ddot{a}+\Omega_{2}\delta a=0,$ (42)

for $k=-1$ case, where

$\displaystyle\Omega_{1}$
$\displaystyle=\frac{3(1+\omega)}{2(a_{s}^{2}+2\alpha)}\left(\frac{4\alpha
C^{2}}{a_{s}^{2}}-\frac{4\alpha}{a_{s}^{2}}-2\right),$ (43)
$\displaystyle\Omega_{2}$
$\displaystyle=\frac{3(1+\omega)}{2(a_{s}^{2}+2\alpha)}\left(\frac{4\alpha
C^{2}}{a_{s}^{2}}-\frac{4\alpha}{a_{s}^{2}}+2\right),$

For finite oscillating perturbation modes, which admit stable ES solutions,
$\Omega_{1},\Omega_{2}>0$. The stable solutions of the equations (41) and (42)
are then

$\delta a(t)=C_{1}e^{i\Omega_{1}t}+C_{2}e^{-i\Omega_{1}t},$ (44)

and

$\delta a(t)=C_{3}e^{i\Omega_{2}t}+C_{4}e^{-i\Omega_{2}t},$ (45)

where $C_{1}$ and $C_{2}$ are integration constants. Therefore given the
conditions $\Omega_{1},\Omega_{2}>0$, the stability intervals are obtained to
be

$\displaystyle\left\\{(C,\alpha,\omega)\mid 0<C\leq 1\ \text{and}\ \alpha>0\
\text{and}\ \omega<-1\right\\}$
$\displaystyle\cup\left\\{(C,\alpha,\omega)\mid C>1\ \text{and}\
\left(0<\alpha<\frac{1}{-2+2C^{2}}\right.\right.$
$\displaystyle\left.\left.\hskip 56.9055pt\ \text{and}\
\omega<-1\right.\right.$ $\displaystyle\quad\quad\left.\left.\text{or}\
\alpha>\frac{1}{-2+2C^{2}}\ \text{and}\ \omega>-1\right)\right\\}$

for $k=1$ whereas for $k=-1$ we obtain

$\displaystyle\left\\{(C,\alpha,\omega)\mid 0<C<1\ \text{and}\
\left(0<\alpha<-\frac{1}{-2+2C^{2}}\ \text{and}\ \right.\right.$
$\displaystyle\left.\left.\hskip 56.9055pt\omega>-1\right.\right.$
$\displaystyle\quad\quad\left.\left.\text{or}\ \alpha>-\frac{1}{-2+2C^{2}}\
\text{and}\ \omega<-1\right)\right\\}$
$\displaystyle\cup\left\\{(C,\alpha,\omega)\mid C\geq 1\ \text{and}\ \alpha>0\
\text{and}\ \omega>-1\right\\}$

Like the previous analysis, we have also set $a_{s}=1$ for simplicity.

Figure 2: Graceful exit mechanism for $k=1$ and $k=-1$ under the influence of
homogeneous perturbation

Figure 2 shows the graceful exit mechanism from a stable ES phase to the
inflationary phase. In this analysis, we have set $t_{0}=0$ as the transition
point. For $k=1$, setting $\omega=-0.3,\alpha=0.5,C=1.8$ gives the stable
region of the ESU subject to homogeneous perturbation. However, a change in
the value of $C$ from 1.8 to 0.5 breaks the infinite series of oscillations of
the scale factor and leads to the exponential inflationary phase. But, for
$\omega<-1$, the oscillations do not break and a graceful exit does not occur.
Therefore, a phantom-like fluid does not allow a successful graceful exit from
a closed Universe. For $k=-1$ case by setting $\omega=-0.3,\alpha=1.5,C=1.5$,
we find that ESU is stable subject to homogeneous perturbation, but breaks
when $\omega<-1$. This suggests that a successful graceful exit from ESU to
the standard inflationary cosmology demands the requirement of a fluid of
phantom nature ($\omega<-1$) for an open Universe.

##  V Stability under inhomogeneous density perturbation

The dynamical equation of inhomogeneous density perturbation in an FLRW
background is given by [3]

$\displaystyle\ddot{\Delta}+\left(2-6\omega+3c_{\mathrm{s}}^{2}\right)H\dot{\Delta}+\left[12\left(\omega-
c_{\mathrm{s}}^{2}\right)\frac{k}{a^{2}}\right.$ (46)
$\displaystyle\left.\quad+4\pi
G\left(3\omega^{2}+6c_{\mathrm{s}}^{2}-8\omega-1\right)\rho\right]\Delta$
$\displaystyle\quad-
c_{\mathrm{s}}^{2}\mathrm{D}^{2}\Delta-\omega\left(\mathrm{D}^{2}+3\frac{\kappa}{a^{2}}\right)\mathcal{E}=0$

The density perturbations are defined using the 1+3 - covariant gauge-
invariant approach as $\Delta=a^{2}\mathrm{D}^{2}\rho/\rho$, with $\mathrm{D}$
as the covariant spatial Laplacian. Here $c_{s}^{2}=dp/d\rho$ and
$\mathcal{E}=(a^{2}\mathrm{D}^{2}p-\rho c_{s}^{2}\Delta)/p$ are the squared-
sound speed and entropy perturbation respectively.

For $k=1$ case after setting the conditions for ESU ($H=0$) and also the fact
that for an ESU $\mathcal{E}=0$ and without the loss of generality,
conveniently choosing $a_{s}=1$, (46) becomes

$\ddot{\Delta}_{\kappa}+\Theta\Delta_{\kappa}=0,$ (47)

with

$\displaystyle\Theta=\frac{3}{2}\left(\alpha-\alpha C^{2}+1\right)$
$\displaystyle\left(6c_{s}^{2}+3\omega^{2}-8\omega-1\right)+\kappa^{2}c_{s}^{2}$
(48) $\displaystyle\hskip 56.9055pt+12\left(\omega-c_{s}^{2}\right),$

Also $\kappa$ is the comoving index which appears through the substitution
$\mathrm{D}^{2}\to-\frac{\kappa^{2}}{a_{s}^{2}}$.

Similarly, for $k=-1$ case we have

$\displaystyle\Theta=\frac{3}{2}\left(\alpha-\alpha C^{2}-1\right)$
$\displaystyle\left(6c_{s}^{2}+3\omega^{2}-8\omega-1\right)+\kappa^{2}c_{s}^{2}$
(49) $\displaystyle\hskip 56.9055pt-12\left(\omega-c_{s}^{2}\right)$

Now, one may see that (47) is a second-order differential equation that gives
finite sinusoidal solutions provided $\Theta>0$. The stability intervals in
this case are complicated inequalities as there are four parameters of
dependency, and so we have refrained from expressing it explicitly. Instead,
we solve equation (47) numerically for different combinations of the model
parameters.

Figure 3: The evolution of density perturbation is shown for different values
of $\omega$. Here we have assumed $(\alpha,C,c_{s})=(0.5,0.5,0.2)$ (see (i))
and $(\alpha,C,c_{s})=(1.2,0.5,0.2)$ (see (ii)) for $k=1$ and $k=-1$
respectively.

Figure 4: The evolution of tensor perturbation is shown for different values
of $\omega$. Here we have assumed $(\alpha,C)=(0.5,0.5)$ (see (i)) and
$(\alpha,C)=(1.2,0.5)$ (see (ii)) for $k=1$ and $k=-1$ respectively.

From figure 3, it is observed that negative values of $\omega$
($\omega=-0.8,-1.2$) lead to finite and stable oscillatory modes for both
$k=1$ and $k=-1$ cases. Thus the ESU is stable subject to density perturbation
for fluids violating the strong energy condition (SEC). However, for fluids
satisfying the SEC $\omega=0.3$ for instance, the perturbations exponentially
diverge indicating instability of the ESU under density perturbations.

##  VI Stability under Vector and Tensor Perturbations

###  VI.1 Vector Perturbation

In an FLRW background, the vector perturbations of a perfect fluid are given
by the comoving dimensionless vorticity defined as $\varpi_{a}=a\varpi$, with
modes that satisfy the following propagation equation [3]

$\dot{\varpi}_{k}+H(1-3c_{s}^{2})\varpi_{k}=0,$ (50)

with $H$ as the Hubble parameter. In an ESU, as $H=0$ equation (50) becomes

$\dot{\varpi}_{k}=0,$ (51)

From equation (51), it is clear that the initial vector perturbations are
frozen, so regardless of all equations of states and on all possible scales,
the ESU is neutrally stable against vector perturbation.

###  VI.2 Tensor Perturbations

In the case of tensor perturbations, or gravitational wave perturbations of a
perfect fluid with EoS, $p=\omega\rho$, that is defined by the comoving
dimensionless transverse-traceless shear $\Sigma_{ab}=a\sigma_{ab}$, with
modes that satisfy the following evolution equation [3]

$\ddot{\Sigma}_{k}+3H\dot{\Sigma}_{k}+\left[\frac{2}{a^{2}}+\frac{k}{a^{2}}-\frac{(1+3\omega)\rho}{3}\right]\Sigma_{k}=0,$
(52)

For the case $k=1$, setting $a=a_{s}=1$ and $\rho=\rho_{s}$ and using equation
(21), the above evolution equation becomes

$\ddot{\Sigma}_{k}+\Phi\Sigma_{k}=0,$ (53)

where

$\Phi=3-(3\omega+1)\left(\alpha-\alpha C^{2}+1\right)$ (54)

Equation (53) determines the neutral stability of tensor perturbation. Stable
and unstable oscillation modes are specified by the conditions $\Phi>0$ and
$\Phi<0$ respectively. The stability intervals for stable oscillation modes
are

$\displaystyle\left\\{(C,\alpha,\omega)\mid 0<C\leq 1\ \text{and}\
\alpha>0\right.$ (55) $\displaystyle\left.\hskip 113.81102pt\ \text{and}\
\omega<\frac{-2+\alpha-C^{2}\alpha}{-3-3\alpha+3C^{2}\alpha}\right\\}$
$\displaystyle\cup\left\\{(C,\alpha,\omega)\mid C>1\ \text{and}\
\left(\left(0<\alpha<\frac{1}{-1+C^{2}}\right.\right.\right.$
$\displaystyle\left.\left.\left.\hskip 113.81102pt\ \text{and}\
\omega<\frac{-2+\alpha-C^{2}\alpha}{-3-3\alpha+3C^{2}\alpha}\right)\right.\right.$
$\displaystyle\quad\quad\left.\left.\text{or}\ \alpha=\frac{1}{-1+C^{2}}\
\text{or}\ \left(\alpha>\frac{1}{-1+C^{2}}\right.\right.\right.$
$\displaystyle\left.\left.\left.\hskip 85.35826pt\ \text{and}\
\omega>\frac{-2+\alpha-C^{2}\alpha}{-3-3\alpha+3C^{2}\alpha}\right)\right)\right\\}$

Likewise, for $k=-1$ case we have

$\Phi=3-(3\omega+1)\left(\alpha-\alpha C^{2}-1\right),$ (56)

Again for stable modes setting $\Phi>0$ gives

$\displaystyle\left\\{(C,\alpha,\omega)\mid 0<C<1\ \text{and}\
\left(\left(1<\alpha<-\frac{1}{-1+C^{2}}\right.\right.\right.$ (57)
$\displaystyle\left.\left.\left.\hskip 113.81102pt\ \text{and}\
\omega>\frac{-4+\alpha-C^{2}\alpha}{3-3\alpha+3C^{2}\alpha}\right)\right.\right.$
$\displaystyle\quad\quad\left.\left.\text{or}\ \alpha=-\frac{1}{-1+C^{2}}\
\text{or}\ \left(\alpha>-\frac{1}{-1+C^{2}}\right.\right.\right.$
$\displaystyle\left.\left.\left.\hskip 113.81102pt\ \text{and}\
\omega<\frac{-4+\alpha-C^{2}\alpha}{3-3\alpha+3C^{2}\alpha}\right)\right)\right\\}$
$\displaystyle\cup\left\\{(C,\alpha,\omega)\mid C\geq 1\ \text{and}\ \alpha>1\
\text{and}\
\omega>\frac{-4+\alpha-C^{2}\alpha}{3-3\alpha+3C^{2}\alpha}\right\\}$

The evolutionary behaviour of the tensor perturbations is shown in figure 4.
The perturbations for $k=1$ and $k=-1$ remain finite for both positive and
negative $\omega$ values. Thus, regardless of whether the fluid obeys or
violates the SEC, the ESU displays stable behaviour concerning tensor
perturbations.

##  VII Conclusion

In this paper, we have investigated emergent cosmology by studying the
fundamental requirements of an emergent scenario namely _stability of the ESU_
and _graceful exit mechanism_. The stability is analysed based on several
techniques like dynamical systems, homogeneous scalar perturbations, density
perturbations, vector perturbations and tensor perturbations. We assumed that
the total matter content is a perfect fluid described by the constant equation
of state $\omega$ with closed and open spatial geometry. From the dynamical
system point of view, we have found that a closed Universe shows promising
behaviour exhibiting stable ES solutions and providing a successful graceful
exit into standard cosmology. However, a spatially open Universe, although
stable ES solutions can be found does not show a successful graceful exit
mechanism.

As far as linear homogeneous perturbation to the scale factor and energy
density is concerned, the ESU is found to be stable under these perturbations.
To realise standard cosmology, the stability of the ESU should break and exit
into the inflationary era (graceful exit). It is observed that for
$\omega<-1$, the oscillations do not break and a graceful exit does not occur.
Therefore, a phantom-like fluid does not allow a graceful exit from a closed
Universe. It suggests that for the realization of a successful graceful exit,
$-1<\omega<0$. But in contrast, in the case of an open Universe, a successful
graceful exit is only observed when $\omega<-1$. In other words, it requires a
phantom equation of state for the successful realization of standard
inflationary cosmology after a phase transition from the ESU in an open
Universe.

Concerning the stability of the ESU under density perturbations, in the case
of a closed Universe, for fluids satisfying the SEC, the perturbations diverge
thus making the ESU unstable to these perturbations. It is also observed that
a change in the value of the parameter $C$ leads to a phase transition from
the stable ESU to the standard inflationary Universe, which advocates a
successful graceful exit. However, the exact reason why $C$ should be a
dynamic degree of freedom that triggers the inflationary scenario is not very
clear in this paper and is currently beyond the scope of the paper. We plan to
investigate this concern in future work.

Vector perturbations on the other hand are frozen in the ESU, and thus the ESU
is stable under these perturbations at all scales. Tensor perturbations, on
the other hand, it is found that the ESU is stable under these perturbations
regardless the fluid obeys or violates the SEC for both closed and open
Universes.

##  Appendix A Stability Region for $k=-1$

$\displaystyle\left\\{(C,\omega)\mid 0<C<0.98995\ \text{and}\
\left(\omega<-\frac{5}{3}\ \text{or}\
\frac{29+C^{2}}{-21+3C^{2}}<\omega<\frac{19-3C^{2}}{3(-7+C^{2})}-\frac{2}{3}\sqrt{\frac{1-28C^{2}+4C^{4}}{(-7+C^{2})^{2}}}\right.\right.$
(58) $\displaystyle\qquad\left.\left.\text{or}\
\frac{19-3C^{2}}{3(-7+C^{2})}+\frac{2}{3}\sqrt{\frac{1-28C^{2}+4C^{4}}{(-7+C^{2})^{2}}}<\omega<-\frac{2}{3}\right)\right\\}$
$\displaystyle\cup\left\\{(C,\omega)\mid C=0.98995\ \text{and}\
\left(\omega<-\frac{5}{3}\ \text{or}\
\frac{29+C^{2}}{-21+3C^{2}}<\omega<\frac{19-3C^{2}}{3(-7+C^{2})}-\frac{2}{3}\sqrt{\frac{1-28C^{2}+4C^{4}}{(-7+C^{2})^{2}}}\right.\right.$
$\displaystyle\qquad\left.\left.\text{or}\
\frac{19-3C^{2}}{3(-7+C^{2})}-\frac{2}{3}\sqrt{\frac{1-28C^{2}+4C^{4}}{(-7+C^{2})^{2}}}<\omega<-\frac{2}{3}\right)\right\\}$
$\displaystyle\cup\left\\{(C,\omega)\mid 0.98995<C\leq 1\ \text{and}\
\left(\omega<-\frac{5}{3}\ \text{or}\
\frac{29+C^{2}}{-21+3C^{2}}<\omega<-\frac{2}{3}\right)\right\\}$
$\displaystyle\cup\left\\{(C,\omega)\mid 1<C\leq 1.05573\ \text{and}\
\left(\omega<\frac{29+C^{2}}{-21+3C^{2}}\ \text{or}\
-\frac{5}{3}<\omega<-\frac{2}{3}\right)\right\\}$
$\displaystyle\cup\left\\{(C,\omega)\mid 1.05573<C<\sqrt{7}\ \text{and}\
\left(\omega<\frac{29+C^{2}}{-21+3C^{2}}\ \text{or}\
-\frac{5}{3}<\omega<-\frac{2}{3}\right.\right.$
$\displaystyle\qquad\left.\left.\text{or}\
\frac{19-3C^{2}}{3(-7+C^{2})}-\frac{2}{3}\sqrt{\frac{1-28C^{2}+4C^{4}}{(-7+C^{2})^{2}}}<\omega<\frac{19-3C^{2}}{3(-7+C^{2})}+\frac{2}{3}\sqrt{\frac{1-28C^{2}+4C^{4}}{(-7+C^{2})^{2}}}\right)\right\\}$
$\displaystyle\cup\left\\{(C,\omega)\mid C=\sqrt{7}\ \text{and}\
\left(-\frac{5}{3}<\omega<-\frac{2}{3}\ \text{or}\
\omega>\frac{25}{3}\right)\right\\}$ $\displaystyle\cup\left\\{(C,\omega)\mid
C>\sqrt{7}\ \text{and}\
\left(\omega<\frac{19-3C^{2}}{3(-7+C^{2})}-\frac{2}{3}\sqrt{\frac{1-28C^{2}+4C^{4}}{(-7+C^{2})^{2}}}\
\text{or}\ -\frac{5}{3}<\omega<-\frac{2}{3}\right.\right.$
$\displaystyle\qquad\left.\left.\text{or}\
\frac{19-3C^{2}}{3(-7+C^{2})}+\frac{2}{3}\sqrt{\frac{1-28C^{2}+4C^{4}}{(-7+C^{2})^{2}}}<\omega<\frac{29+C^{2}}{-21+3C^{2}}\right)\right\\}$

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