# Inflation vs. Ekpyrosis — comparing stability in general non-minimal theory

Debottam Nandi<EMAIL_ADDRESS>Manjeet Kaur<EMAIL_ADDRESS>Department of Physics and Astrophysics, University of Delhi, Delhi 110007,
India

###### Abstract

The scalar field is considered to have dominated the early Universe. One
subtle yet crucial requirement of this assumption is that the solution must be
highly stable, i.e., indifferent to any initial conditions because there are
no favored ones. Inflation, which is now the most successful early Universe
paradigm, answers most of the early Universe’s problems, including the fact
that it is mostly stable. In this article, in addition to the inflationary
solution, we systematically investigate every possible early Universe solution
in the presence of a barotropic fluid in the general non-minimal (scalar-
tensor) theory. In doing so, we rely upon the classical perturbative
techniques. We find, to our surprise, that inflation does not always ensure
stability in the Einstein frame, although ekpyrosis can. We also discover
that, contrary to the inflationary paradigm, ekpyrosis always assures
stability in the presence of any fluid with any equation of state in general
non-minimal models. We utilize the conformal transformation to map the
inflationary theory in the minimal frame to the ekpyrotic theory in the non-
minimal frame, and show that the latter is always much more stable than the
former, resulting in a much more preferred model that can even be studied in
different contexts such as late time cosmology.

##  I Introduction

Stable (attractor) solutions are crucial for solving equation in general
relativity because they enable highly nonlinear Einstein equations to
asymptotically converge to a particular solution regardless of the initial
conditions or the presence of additional matter in the system. For this
reason, attractor solutions are consistently favored over non-attractor
solutions. These benefits particularly establish themselves in a highly
critical position during the early Universe because there are no favorable
initial conditions available to us. This means that different initial
conditions lead to different solutions in the early Universe, which is
undesirable unless the result is an attractor.

At the very early stage of the Universe, it is assumed to be dominated by the
scalar field(s) and the most successful theory of it by far is the
inflationary paradigm [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], in which
the Universe expands at an accelerated rate, thus solving early Universe
puzzles like the horizon or the flatness problem. The beauty of inflation is,
not only it satisfies the observational constraints [14, 15], but also acts as
a stable attractor solution [16, 17]. As a result, perturbations, which are
generated quantum mechanically and inherent to the system, do not grow beyond
a specific threshold, rendering the system stable and free of different
instabilities. Anisotropic stress, which at that high energy is always assumed
to be present, is also diminished over this period. Certain concerns, however,
like as the trans-Planckian or the initial singularity problem [18, 19, 20,
21], ruling out of inflationary models [22, 23, 24, 25], etc., plague the
inflationary scenario and compel us to look beyond this paradigm.

On the other hand, well-known alternatives to inflation, such as the bouncing
paradigm [26, 27, 28, 29, 30, 31], in which the Universe initially contracts
to a minimum and then expands, can also address the problems associated with
the early Universe, such as the Horizon and flatness problems without the
trans-Planckian problem or the initial singularity problem. However, the
greatest challenge lies in constructing a viable bouncing model that satisfies
observational restrictions [32, 33, 34, 35, 36, 37, 38]. It also has problems
with ghost and gradient instabilities [39, 40, 41, 42, 43, 44, 45, 46, 47,
48]. However, the issue of stability is of the utmost importance, i.e., in
general, bouncing solutions are unstable, and as a result, intrinsic
perturbations or anisotropies (Belinsky-Khalatnikov-Lifshitz (BKL)
instability) can accumulate and make the system extremely unstable [49, 50,
51, 52, 53, 54]. These issues lead to in search for non-minimal theories, viz.
the Horndeski theories or even beyond the Horndeski theories [55, 56, 57, 53,
39, 40, 41, 42, 43, 44, 45, 46, 39, 35, 36, 58, 59, 60, 61].

Nevertheless, there appears to be an exception to this rule: a very slowly
contracting bouncing scenario, known as the ekpyrotic bounce [62], can avoid
this issue even more effectively than the inflationary paradigm. Using non-
perturbative simulations, the authors of Ref. [63] demonstrated that the
inflationary expansion begins under very precise conditions, casting doubt on
the claim that inflation is insensitive to initial conditions. In Ref. [64],
the authors showed that under the majority of initial conditions, a small
field potential is insufficient to initiate inflation. In fact, authors
demonstrated in Ref [65] that the ekpyrotic contraction is “super-smooth,”
i.e., it is resistant to a broad variety of initial conditions and avoids
“Kasner/mixmaster” chaos. A similar assertion on any other primordial scenario
could never be substantiated. Also, using classical perturbative analysis, it
is possible to demonstrate that inflation is incapable of suppressing
cosmological constant-like fluids, but ekpyrosis, on the other hand, can
dilute any fluid with any equation of state parameter. In contrast to the
inflationary paradigm (mostly slow-roll inflation), however, ekpyrotic bounce
cannot produce outcomes that are consistent with observations.

Therefore, the motivation of this article is in uncovering options in quest of
a model that may simultaneously suppress the presence of any additional
fluid(s) while satisfying observations. Clearly, finding such a model within
the framework of minimal Einstein’s theory is exceedingly challenging. Once we
depart from this, however, and reach the realm of non-minimal theory, it may
be possible to find such a model. This is because, as demonstrated in Ref.
[66, 67], it is possible to transform any bouncing solution into an attractor
solution via non-minimal coupling. Moreover, we showed that these models are
also capable of satisfying observational constraints [68]. Upon further
investigation, we discovered that conformal transformation is the key, since
it preserves the invariance of the perturbations but, at the same time, can
modify the stability of the system, potentially making it more stable and
hence, preferred than the original model. Using these insights, in Ref. [69,
70, 71], we constructed bouncing models in the non-minimal frame and have
shown that they satisfy the observations and are free of instabilities.
However, the question is always finding models with the best apparent
stability and observational viability to date.

To find the answers of the above questions, in this work, we extensively study
the stability in the context of the early Universe. In this regard, we
consider general non-minimally coupled (with scalar field) gravity model in
the presence of an additional fluid. Here, for simplicity, we consider the
fluid to be barotropic in nature, i.e., the pressure is proportional to the
energy density of the system. In doing so, we choose a wide range of equation
of state parameter $w_{m}$, i.e., the ratio of pressure to the energy density
from $w_{m}=-1$ to $w_{m}=1$, and study the stability analysis. One could
wonder why there is such a large selection of $w_{m}$. This is because, as far
as we are aware, fields in our universe — whether they be associated with dark
energy ($w_{m}=-1$), dark matter ($w_{m}=0$), radiation ($w_{m}=1/3$) or the
stiff matter ($w_{m}=1$) — contain such equations of state parameters.

Therefore, this work has a dual purpose. The first is to compare the stability
of two non-minimal theories that are related by a conformal transformation,
and the second is to build up the non-minimal frame in such a way that we can
discover a more preferred model in terms of stability while maintaining the
observations to be invariant. Conformally connected theories, i.e., two
theories associated by the remapping of the metric tensors, are generally
assumed to be indistinguishable (equivalent) as long as perturbations remain
invariant. In the first half of this paper, we attempt to show a similar
relation (equivalence) in the context of stability. In doing so, we first
examine the stability in the non-minimal frame without the presence of an
additional fluid, and we establish the relationship between the stability
criteria in various conformal frames, which has been thoroughly demonstrated
in a different way in Ref. [67]. Then, the conformal invariance is
deliberately disrupted by introducing the additional fluid, and the dynamical
analysis is performed once more. As expected, we discover that the conformal
stability criterion is now violated, and we compare the stability in various
non-minimal frames once more. We find that, regardless of the presence of non-
minimal couplings, the ekpyrotic scenario is always favored because it can
dilute any presence of the additional fluid. It can even diminish the effect
of the cosmological constant, which may provide a solution to the age-old
conundrum of dark energy’s fine-tuning. Therefore, an ideal illustration of
such a situation would be the following: it is possible to design a non-
minimal model that conforms to any viable inflationary theory and change it
into an ekpyrotic scenario, making it significantly more desirable than the
original inflationary theory. There may be other countless benefits, and it
may also be possible to study it in the context of the late universe. However,
this is outside the scope of the present project, so we will save it for
future initiatives.

The article is written in the following way. In the next section II, we set up
the equations needed for dynamical analysis in the most generalized manner. In
Sec. III, we introduce the conformal transformation and show how different
non-minimal theories are connected in the new set up established for the
dynamical analysis. Then, in Sec. IV, we study the dynamical analysis in
different non-minimally coupled models without the presence of the barotropic
fluid and we find the equivalence of stability in these theories. In the
following section IV, we again study the dynamical analysis, now in the
presence of the barotropic fluid and compare our results in different non-
minimally coupled theories. We find the best result lies in ekpyrosis
scenario. At the end, we summarize and conclude the final remarks in Sec. VI.

In this work, we work with natural units of $\hbar=c=1$ and we define the
Planck mass to be $M_{{}_{\rm Pl}}\equiv(8\,\pi\,G)^{-1/2}=1$. We adopt the
metric signature of $(-,+,+,+)$. The partial derivatives are expressed as
$\partial$. The overdot $(\dot{\,\,})$ and the subscript $N\equiv\log(a)$
denote derivative with respect to cosmic time $t$ and the e-fold variable $N$,
respectively, associated with the Friedmann-Lema$\hat{i}$tre-Robertson-Walker
(FLRW) line element, respectively with $a$ being the scale factor and $H$
being used as the Hubble parameter, defined as
$\left(\frac{\dot{a}}{a}\right).$ We have normalized all the scalar fields in
this work with $M_{{}_{\rm Pl}}$.

##  II Dynamical equations in general non-minimal theory

Let us now discuss the general equations required to perform such analysis. In
this work, we focus on the theories where the action consisting of a scalar
field non-minimally coupled to gravity with the presence of an additional
barotropic fluid as

$\displaystyle\mathcal{S}$ $\displaystyle=$ $\displaystyle\frac{1}{2}\int{\rm
d}^{4}{\rm\bf
x}\sqrt{-g}\left\\{f^{2}(\phi)\,R-g^{\mu\nu}\,\omega(\phi)\,\partial_{\mu}\phi\partial_{\nu}\phi\right.$
(1) $\displaystyle\hskip
75.0pt\left.-2\,V(\phi)\right\\}+S_{m}(g_{\mu\nu},\Psi_{m}),$

where, $g_{\mu\nu}$ is the metric tensor, $R$ is the Ricci scalar, $f(\phi)$
is the non-minimal scalar coupling function, $\omega(\phi)$ is the derivative
coupling function of $\phi$, $V(\phi)$ is the scalar potential function, and
$S_{m}(g_{\mu\nu},\Psi_{m})$ is the action associated with the additional
barotropic fluid. The corresponding equations of motion can be written as

$\displaystyle
f^{2}\left(R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R\right)-2\nabla_{\mu}f\,\nabla_{\nu}f-2f\,\nabla_{\mu\nu}f+$
$\displaystyle 2g_{\mu\nu}\,\left(\nabla^{\lambda}f\,\nabla_{\lambda}f+f\,\Box
f\right)\ =\omega(\phi)\left(\nabla_{\mu}\phi\nabla_{\nu}{\phi}-\right.$
$\displaystyle\hskip
60.0pt\left.\frac{1}{2}g_{\mu\nu}\nabla^{\lambda}\phi\nabla_{\lambda}\phi\right)-g_{\mu\nu}V+T^{(m)}_{\mu\nu},$
(2)
$\displaystyle\Box\phi+\frac{1}{2\omega(\phi)}\left(\omega_{,\phi}\,\nabla^{\lambda}\phi\nabla_{\lambda}\phi-2V_{,\phi}+2\,f\,f_{,\phi}\,R\right)=0,\qquad$
(3)

where, $A_{,\phi}\equiv\left(\partial A/\partial\phi\right),$ $\Box\equiv
g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}$, $R_{\mu\nu}$ is the Ricci tensor,
$R\equiv g^{\mu\nu}R_{\mu\nu}$ is the Ricci scalar and $T^{(m)}_{\mu\nu}$ is
the energy-momentum tensor associated with the barotropic fluid satisfying the
continuity equation

$\displaystyle\nabla^{\mu}T^{(m)}_{\mu\nu}=0.$ (4)

Using the Friedmann-Lemaître-Robertson-Walker (FLRW) line element, describing
the homogeneous and isotropic Universe

$\displaystyle{\rm d}s^{2}=-{\rm d}t^{2}+a^{2}(t)\,{\rm d}{\bf
x}^{2}=a^{2}(\eta)\left(-{\rm d}\eta^{2}+{\rm d}{\bf x}^{2}\right),$ (5)

where, $a(t)$ is the scale factor, the above equations take the following
form:

$\displaystyle
3f^{2}\,H^{2}=-6H\,f\,f_{,\phi}\dot{\phi}+\left(\frac{1}{2}\omega\,\dot{\phi}^{2}+V+\rho_{M}\right),$
(6) $\displaystyle
2f^{2}\dot{H}=-2\,\left(f_{,\phi}^{2}+\,ff_{,\phi\phi}\right)\dot{\phi}^{2}-\,2ff_{,\phi}\ddot{\phi}+\,2H\,ff_{,\phi}\dot{\phi}$
$\displaystyle\hskip 40.0pt-\,(\omega\dot{\phi}^{2}+\,(1+w_{m})\rho_{m})$ (7)
$\displaystyle\ddot{\phi}+3H\,\dot{\phi}+\frac{1}{2\omega}\left(\omega_{,\phi}\dot{\phi}^{2}+2V_{,\phi}-2ff_{,\phi}R\right)=0,$
(8) $\displaystyle\dot{\rho}_{m}+3H\left(\rho_{m}+P_{m}\right)=0,$ (9)

where, $\rho_{m}$ and $P_{m}$ are the energy density and pressure of the
additional fluid, i.e.,

$T^{(m)}{}^{0}_{~{}0}\equiv-\rho_{m},\quad T^{(m)}{}^{i}_{~{}j}\equiv
P_{m}\,\delta^{i}_{j},$

and the equation of state of the barotropic fluid is

$w_{m}\equiv\frac{P_{m}}{\rho_{m}}=\mbox{Constant.}$

For example, if the fluid is dust-matter, the corresponding equation of state
is $w_{m}=0$ and, by using Eq. (9), one can obtain the energy density of the
additional fluid as $\rho_{m}\propto a^{-3}.$ Similarly, $w_{m}=1/3$ signifies
the additional fluid is radiation-like with $\rho_{m}\propto a^{-4}.$ Most
importantly, stiff anisotropic fluid represents $w_{m}=1$ and $\rho_{m}\propto
a^{-6}$.

Let us now define the parameters on which we will analyze dynamical
properties. These are

$\displaystyle\gamma\equiv\frac{Vf_{,\phi}}{fV_{,\phi}},\quad\mu\equiv\frac{V\sqrt{\omega}}{fV_{,\phi}}.$
(10)

These are scalar field functions in general. In our case, we treat them as
constants because, as we’ll see later, they make it easier to find fixed point
solutions, which are otherwise quite difficult to obtain. In the case of slow-
roll, for example, these functions vary, but the change in the slow-roll limit
is insignificant, therefore they can again be considered to be constants.

We can further simplify background equations by defining two dimensionless
quantities as:

$\displaystyle x\equiv\sqrt{\frac{\omega}{6}}\frac{\dot{\phi}}{Hf},\quad
y\equiv\frac{\sqrt{V}}{\sqrt{3}fH}.$ (11)

Using the above definitions of $x$ and $y,$ along with $\gamma$ and $\mu,$ the
energy equation (6) can now be written as

$\displaystyle\Omega_{m}\equiv\frac{\rho_{m}}{3f^{2}H^{2}}=1+2\,\frac{\sqrt{6}}{\mu}\gamma\,x-y^{2}-x^{2},$
(12)

where, $\Omega_{m}$ is the fractional energy density of the additional fluid.
One can also express the (first) slow-roll parameter in terms of $x$ and $y$
as

$\displaystyle\epsilon\equiv-\frac{\dot{H}}{H^{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{2\left(6\gamma^{2}+\mu^{2}\right)}\left(3\mu^{2}\left(-w_{m}x^{2}-(w_{m}+1)y^{2}\right.\right.$
(13)
$\displaystyle\left.\left.+w_{m}+x^{2}+1\right)+2\sqrt{6}\gamma\mu(3w_{m}-1)x\right.$
$\displaystyle\left.+6\gamma\left(2\gamma\left(x^{2}+2\right)-y^{2}\right)\right),$

as, for constant $\epsilon$, it defines the scale factor solution, i.e.,
$a(\eta)\propto(-\eta)^{1/(\epsilon-1)}.$ Finally, the effective equation of
state of the Universe in terms of the slow-roll parameter can then be written
as

$\displaystyle w_{\rm eff}=-1+\frac{2}{3}\,\epsilon,$ (14)

which essentially signifies how the effective energy density of the Universe
depends on the scale factor, i.e.,

$\displaystyle\rho_{\rm eff}\propto a^{-3(1+w_{\rm eff})}.$ (15)

As the system contain two scalar degrees of freedom ($\phi$ and the barotropic
fluid), the above two dimensionless variables can be used as dynamical
quantities, and the equations of motion of $x$ and $y$ can be obtained as

$\displaystyle\frac{{\rm d}x}{{\rm d}N}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{H}\frac{{\rm d}x}{{\rm
d}t}=-\frac{1}{2\left(6\gamma^{2}\mu+\mu^{3}\right)}\left(-3x^{3}\left(4\gamma^{2}\mu-\mu^{3}\right.\right.$
(16) $\displaystyle\left.\left.(w_{m}-1)\right)+\sqrt{6}\gamma
x^{2}\left(24\gamma^{2}+\mu^{2}(7-9w_{m})\right)\right.$
$\displaystyle\left.+6\gamma\mu x\left(6\gamma
w_{m}+y^{2}\right)+3\mu^{3}x\left(w_{m}\left(y^{2}-1\right)\right.\right.$
$\displaystyle\left.\left.+y^{2}+1\right)+\sqrt{6}\mu^{2}\left(\gamma(3w_{m}-1)\right.\right.$
$\displaystyle\left.\left.+y^{2}(1-3\gamma(w_{m}+1))\right)\right),$
$\displaystyle\frac{{\rm d}y}{{\rm d}N}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{H}\frac{{\rm d}y}{{\rm
d}t}=\frac{y}{2\left(6\gamma^{2}\mu+\mu^{3}\right)}\left(3x^{2}\left(4\gamma^{2}\mu-\right.\right.$
(17)
$\displaystyle\left.\left.\mu^{3}(w_{m}-1)\right)+\sqrt{6}x\left(6(1-2\gamma)\gamma^{2}+\right.\right.$
$\displaystyle\left.\left.\mu^{2}(\gamma(6w_{m}-4)+1)\right)-3\mu^{3}(w_{m}+1)\left(y^{2}-1\right)\right.$
$\displaystyle\left.-6\gamma\mu\left(y^{2}-4\gamma\right)\right),$

where, instead of the cosmic time, we have expressed the time variable as $N$,
the e-folding number defined as the logarithmic change of scale factor, i.e.,
$N\equiv\ln{(a)}.$ Notice that, only if $\gamma$ and $\mu$ are (almost)
constants can the preceding relations be obtained in terms of $x$ and $y$,
resulting in fixed point (analytically solvable) solutions. One can debate the
legitimacy of such a choice. However, one might counter the notion that
because this is always achievable in a short range of time, they can be deemed
constant.

Since we now have the equations in terms of the dimensionless variables $x$
and $y$, we may solve the system, which will be discussed in the subsequent
sections. However, before we do that, we will explore the conformal
transformation and the relationship between two conformally connected frames
in the following part, which can assist us in establishing the relationships
of the scale factors between these two frames.

##  III Conformal transformation

In this section, given a theory in expression (1), we apply the conformal
transformation on that action, with the goal being to discover connections
between various non-minimal frames, as was previously described. To ensure
that conformal invariance breaks down between the two models and the physics
differ in them, we just transform the scalar part of the action, leaving the
additional fluid unaffected. In the conformal transformation, the metric field
as well as the scale factor are redefined as

$\displaystyle\tilde{g}_{\mu\nu}=\Omega^{2}(\phi)\,g_{\mu\nu}\quad\Rightarrow\quad\tilde{a}(\tilde{t})=\Omega(\phi)\,a(t).$
(18)

$\tilde{g}_{\mu\nu}$ and $\tilde{a}(\tilde{t})$ are the new conformally
transformed metric and the scale factor, respectively, describing a new
Universe, whereas, $g_{\mu\nu}$ and $a(t)$ are the old metric and the scale
factor, respectively. Also, the cosmic time is replaced by the new cosmic time
parameter $\tilde{t}$, where they are related by ${\rm
d}\tilde{t}=\Omega\,{\rm d}t.$ Under such transformation, the non-minimal
theory described in (1) transforms into a new action

$\displaystyle\tilde{\mathcal{S}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int{\rm d}^{4}{\rm\bf
x}\sqrt{-\tilde{g}}\left[\tilde{f}^{2}(\phi)\,\tilde{R}-\tilde{\omega}(\phi)\,\tilde{g}^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi\right.$
(19)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\left.-2\,\tilde{V}(\phi)\right]+S_{m}.$

Again, please note that $S_{m}$ represents the identical action for the
barotropic fluid defined in the original action in (1) as we are conformally
transforming only the scalar sector of the action. The other functions,
$\tilde{f}(\phi),\,\tilde{\omega}(\phi)$ and the potential $\tilde{V}(\phi)$
depend on the coupling functions $f(\phi),~{}\omega(\phi)$ and the potential
$V(\phi)$ as

$\displaystyle\tilde{f}(\phi)$ $\displaystyle=$
$\displaystyle\frac{f}{\Omega},$ (20) $\displaystyle\tilde{\omega}(\phi)$
$\displaystyle=$
$\displaystyle\left(\frac{\omega}{\Omega^{2}}+6\frac{f^{2}}{\Omega^{2}}\left(2\frac{f_{,\phi}}{f}-\frac{\Omega,_{\phi}}{\Omega}\right)\frac{\Omega,_{\phi}}{\Omega}\right),$
(21) $\displaystyle\tilde{V}(\phi)$ $\displaystyle=$
$\displaystyle\frac{V}{\Omega^{4}}.$ (22)

The corresponding equations of motion of this theory is identical to Eqs. (2)
and (3) with the old functions replaced by the functions defined in the new
conformally transformed theory with new time variable $\tilde{t}.$ Note that,
in the new transformed theory, the new $\tilde{\gamma}$ and $\tilde{\mu}$ are
defined, again, as

$\displaystyle\tilde{\gamma}\equiv\frac{\tilde{V}\tilde{f}_{,\phi}}{\tilde{f}\tilde{V}_{,\phi}},\quad\tilde{\mu}\equiv\frac{\tilde{V}\sqrt{\tilde{\omega}}}{\tilde{f}\tilde{V}_{,\phi}}.$
(23)

As a result, the dynamical equations given in Eqs. (16) and (17) remain
identical in the new frame, with all quantities replaced by newly specified
functions in the new frame, including $\tilde{\gamma}$ and $\tilde{\mu}.$ The
above relation shows the link between $\\{\gamma,\mu\\}$ and
$\\{\tilde{\gamma},\tilde{\mu}\\}$, and because dynamics rely on them, we can
now build a direct relationship between two frames, which will be realized as
we progress.

Since the set up is now developed, we now can dynamically analyze the system.
First, in the next section, we will consider the action without the additional
fluid, and later, we will focus on the effect of the additional fluid causing
the breaking of conformal invariance to understand the complete picture.

##  IV With the absence of barotropic fluid

In the absence of the additional fluid, $\rho_{m}=0$, which leads to

$\displaystyle\Omega_{m}=0,\quad\Rightarrow
y^{2}=1+2\,\frac{\sqrt{6}}{\mu}\gamma\,x-x^{2}.$ (24)

It implies that the two variables depend on each other, which essentially
removes one degree of freedom, which is obvious as there is no additional
fluid. Considering $x$ to be the dynamical variable, we can now obtain the
evolution equation of it by replacing $y$ in terms of $x$ in Eq. (16) and it
becomes

$\displaystyle\frac{{\rm d}x}{{\rm d}N}$ $\displaystyle=$
$\displaystyle\frac{1}{2\left(6\gamma^{2}\mu+\mu^{3}\right)}\left(\sqrt{6}(4\gamma-1)\mu^{2}+6\mu
x^{3}\left(2\gamma^{2}\right.\right.$ (25)
$\displaystyle\left.\left.+\gamma+\mu^{2}\right)+\sqrt{6}x^{2}\left(\mu^{2}-4\gamma\left(6\gamma^{2}+3\gamma+4\mu^{2}\right)\right)\right.$
$\displaystyle\left.-6\mu x\left(-6\gamma^{2}+3\gamma+\mu^{2}\right)\right).$

The fixed points can be found by equating the velocity ${\rm d}x/{\rm d}N$ to
zero, which, here, leads to three critical points:

$\displaystyle x_{1}^{*}$ $\displaystyle=$
$\displaystyle\frac{(4\gamma-1)\mu}{\sqrt{6}\left(2\gamma^{2}+\gamma+\mu^{2}\right)},$
(26) $\displaystyle x_{2}^{*}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{6}\gamma-\sqrt{6\gamma^{2}+\mu^{2}}}{\mu},$ (27)
$\displaystyle x_{3}^{*}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{6\gamma^{2}+\mu^{2}}+\sqrt{6}\gamma}{\mu},$ (28)

with the corresponding slow-roll parameters

$\displaystyle\epsilon_{1}$ $\displaystyle=$
$\displaystyle\frac{8\gamma^{2}-6\gamma+1}{2\left(2\gamma^{2}+\gamma+\mu^{2}\right)}$
(29) $\displaystyle\epsilon_{2}$ $\displaystyle=$ $\displaystyle
3+\frac{2\gamma\left(6\gamma-\sqrt{6}\sqrt{6\gamma^{2}+\mu^{2}}\right)}{\mu^{2}}$
(30) $\displaystyle\epsilon_{3}$ $\displaystyle=$ $\displaystyle
3+\frac{2\gamma\left(\sqrt{6}\sqrt{6\gamma^{2}+\mu^{2}}+6\gamma\right)}{\mu^{2}}$
(31)

Notice that, the second and third fixed point correspond to kinetic energy
dominated solutions as these lead to

$\displaystyle y_{2}^{*}=y_{3}^{*}=0,$ (32)

i.e., the potential energy is zero. Instead, we’re interested in solutions
that involve potential energy, therefore we focus on the first fixed point
(26), which is critical to the dynamics of the early Universe. Please bear in
mind that, even if an additional fluid is present, we will always evaluate
only this fixed point because it reflects our desired scalar field dominated
solution in the early Universe.

Let us now find out the solution of around the fixed point. To obtain this, we
assume linear perturbation theory, i.e., the deviation from the fixed point
solution is small: $x=x_{*}+\delta x,~{}\delta x\ll x_{*}.$ One can then
immediately find the equation of motion of $\delta x$ to be

$\displaystyle\frac{{\rm d}\delta x}{{\rm d}N}=\frac{\partial A(x)}{\partial
x}\Big{|}_{*}\,\delta x,$ (33)

where, $A(x)$ is the right hand side of Eq. (25) and $|_{*}$ represents the
value evaluated at the corresponding fixed point. Keep in mind that here we
ignore the higher-order terms of $\delta x$ because we are treating $\delta x$
as a perturbed quantity. In a nutshell, we use the classical perturbation
analysis to evaluate the dynamics of the deviation. Following that, it is
simple to assess the solution of the above equation as

$\displaystyle\delta x=\delta x_{0}\exp\left(\lambda
N\right),\quad\lambda\equiv\frac{\partial A(x)}{\partial x}\Big{|}_{*},$ (34)

and for the three fixed points, the eigenvalues, referred to as the Lyapunov
exponents, take the following forms:

$\displaystyle\lambda_{1}$ $\displaystyle=$
$\displaystyle-3+\frac{1-2\gamma(4\gamma+1)}{2\left(2\gamma^{2}+\gamma+\mu^{2}\right)},$
(35) $\displaystyle\lambda_{2}$ $\displaystyle=$ $\displaystyle
6-\frac{6(2\gamma+1)}{\sqrt{6}\sqrt{6\gamma^{2}+\mu^{2}}+6\gamma},$ (36)
$\displaystyle\lambda_{3}$ $\displaystyle=$ $\displaystyle
6+\frac{6(2\gamma+1)}{\sqrt{6}\sqrt{6\gamma^{2}+\mu^{2}}-6\gamma}.$ (37)

Given initial conditions close to the fixed point (i.e., choosing $\delta
x_{0}\ll x_{*}$), the appropriate value of $\lambda$ determines whether the
initial deviation (from the fixed point) decays over time. If initial
conditions are slightly deviated from the fixed point solution, $\lambda$
influences whether the solution grows and moves away from the fixed point
solution, known as a repeller solution, or decays and eventually merges with
the fixed point solution, known as an attractor solution. Thus, for an
expanding universe, $\lambda>0$ indicates that a fixed point is unstable,
whereas $\lambda<0$ indicates that the point is stable. In the next sections,
we will examine the relationship between the solution in the minimal Einstein
frame and the non-minimally coupled frame.

###  IV.1 Minimal Einstein frame

Let us now concentrate on Einstein’s minimal frame. As discussed before, we
are now considering only the first fixed point (26). In this instance, the
coupling function $f(\phi)$ is constant, and hence, $\gamma=0$. Consequently,
Eqs. (26) and (29) become

$\displaystyle
x_{1}^{*}=-\frac{1}{\sqrt{6}\mu},\quad\epsilon_{1}=\frac{1}{2\mu^{2}},$ (38)

and the corresponding eigenvalue in Eq. (35) take the form

$\displaystyle\lambda_{1}=-3+\frac{1}{2\mu^{2}}.$ (39)

Using Eq. (38), it is convenient to express the above relation in terms of the
slow-roll parameter as

$\displaystyle\lambda_{1}=\epsilon_{1}-3\quad\Rightarrow\lambda_{1}=\frac{1-2\alpha}{\alpha},$
(40)

where, $\alpha\equiv 1/(\epsilon_{1}-1)$ corresponds to the scale factor
solution: $a(\eta)\propto(-\eta)^{\alpha}$ for the first fixed point.
Therefore, if the Universe is expanding, $\lambda<0$ leads to the stable
solution which in turn provide the bound on $\alpha$ as

$\displaystyle\alpha<0,\quad\text{or,}\quad\alpha>\frac{1}{2}.$ (41)

Adding the fact that, the solution needs to solve the horizon problem, i.e.,
$\eta$ is negative and $|\eta|\rightarrow 0$ (required solution for the early
Universe), it leads to one condition

$\displaystyle\alpha<0.$ (42)

As $\alpha$ is negative and approaches zero, it is evident that the solution
leads to accelerated expansion, and $\alpha=-1$ is the well-known de-Sitter
expansion, which, when expressed in cosmic time $t$, is purely exponential.
Therefore, in the minimal Einstein theory, accelerated expansion behaves as an
attractor. In contrast, when contraction occurs, the condition becomes

$\displaystyle 0<\alpha<\frac{1}{2},$ (43)

which is known as the ekpyrotic solution [62] — the explanation for the sole
stable bouncing (contracting) solution.

###  IV.2 Non-minimal frame

Now that the minimum condition has been examined, we may shift our attention
to conformal transformations of action, with the goal of modifying the scale
factor to behave as $\tilde{a}(\eta)\propto(-\eta)^{\beta}$ in the transformed
frame. It’s important to keep in mind that the conformal time coordinate,
$\eta$, remains invariant by conformal transformation, in contrast to the
cosmic time $t$. This new solution for the scale factor can be achieved in the
following way.

In the minimal frame, with the help of the fixed point solution in Eq. (38) as
well as Eq. (10), the conformal time can be written in terms $\phi$ as

$\displaystyle(-\eta)\propto V(\phi)^{-\frac{1}{2(1+\alpha)}}.$ (44)

Then according to Eq. (18), the coupling function takes the form:

$\displaystyle\Omega(\phi)=V(\phi)^{\frac{\alpha-\beta}{2(1+\alpha)}},$ (45)

and, using Eqs. (20), (21) and (22), we get the following functions in terms
of the scalar field $\phi$ as

$\displaystyle\tilde{f}(\phi)$ $\displaystyle=$ $\displaystyle
V(\phi)^{\frac{\beta-\alpha}{2(\alpha+1)}},$ (46)
$\displaystyle\tilde{\omega}(\phi)$ $\displaystyle=$
$\displaystyle\frac{\left(2(\alpha+1)^{2}\mu^{2}-3(\alpha-\beta)^{2}\right)V^{\prime
2}\,V^{\frac{-3\alpha+\beta-2}{\alpha+1}}}{2(\alpha+1)^{2}},\qquad$ (47)
$\displaystyle\tilde{V}(\phi)$ $\displaystyle=$ $\displaystyle
V(\phi)^{\frac{-\alpha+2\beta+1}{\alpha+1}}.$ (48)

These functions define the new action in (19) that leads to the new scale
factor solution. Using the above definitions, along with Eq. (23), we get the
transformed $\gamma$ and $\mu$, i.e., $\tilde{\gamma}$ and $\tilde{\mu}$ in
terms of $\alpha$ and $\beta$ as,

$\displaystyle\tilde{\gamma}$ $\displaystyle=$
$\displaystyle\frac{\beta-\alpha}{2(-\alpha+2\beta+1)},$ (49)
$\displaystyle\tilde{\mu}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{-2\alpha^{2}+6\alpha\beta+\alpha-3\beta^{2}}}{\sqrt{2}(-\alpha+2\beta+1)}.$
(50)

As was previously stated, these relations provide the relationship between the
new non-minimal frame and the previous minimal frame. $\alpha$ indicates the
scale factor solution in the minimal frame, whereas $\beta$ specifies the new
scale factor. Together, they constitute the non-minimal frame (19) expressed
within the functions in Eqs. (46), (47), and (48).

As the non-minimal theory is now established, using the expression (35), the
Lyapunov exponent of the first fixed point can now again be evaluated as

$\displaystyle\tilde{\lambda}_{1}=\frac{1-2\alpha}{\beta},$ (51)

which seems to be different from the minimal counterpart represented in Eq.
(40). The connection between these two distinct eigenvalues will be discussed
in the following section.

###  IV.3 Equivalence of stability

As shown above, the two theories lead to two different Lyapunov exponent
solutions, implying that the decay rate of the deviations is different in
these two theories, i.e.,

$\displaystyle\delta x\propto e^{\lambda_{1}N},\quad\delta\tilde{x}\propto
e^{\tilde{\lambda}_{1}\tilde{N}},$ (52)

where $\delta\tilde{x}$ signifies the deviation in the conformally modified
theory described in (19). However, it is clear that the e-folding number in
two separate frames differ and is related as

$\displaystyle\tilde{N}=\frac{\beta}{\alpha}N$ (53)

up to a constant. This quickly results in

$\displaystyle\lambda_{1}N=\tilde{\lambda}_{1}\tilde{N},\quad\Rightarrow\delta\tilde{x}\propto\delta
x.$ (54)

meaning that variations in both frames have the same effect. This is even
clearer if we express the time in conformal coordinates $\eta$. In this
instance

$\displaystyle\delta x\propto\delta\tilde{x}\propto(-\eta)^{(1-2\alpha)}.$
(55)

It shows us that, even if the deviations evolve differently in any other
coordinate time (cosmic time $t$ or the e-folding number $N$), they behave
similarly in conformal time $\eta$, and therefore the equivalence. It in no
way implies that the deviations in different frames behave identically, as
time runs independently in each frame. What this tells us is that if the
conformal evolution of the two theories is equal, then the deviations likewise
behave identically. The anticipated outcome is demonstrated in Ref. [67].
Indeed, it has also been shown that the equivalence can be extended to any
connected conformal theory as

$\displaystyle\lambda\,{\rm d}N=\tilde{\lambda}\,{\rm d}\tilde{N}.$ (56)

This is the first outcome of our efforts.

##  V With the addition of barotropic fluid

Now, as previously indicated, consider the model in the presence of a
barotropic fluid. Because fluid is present, unlike in the previous situation,
the degrees of freedom are now two, which are represented by $x$ and $y$.
Furthermore, only the scalar component is conformally connected, but the
barotropic fluid is unaffected by the conformal transformation, implying that
the two theories are not conformally associated, as mentioned in Sec. III.
Fixed points can then be obtained in this scenario by explicitly setting the
velocity of these two variables, i.e., ${\rm d}x/{\rm d}N={\rm d}y/{\rm
d}N=0$, resulting in seven fixed points [16, 72, 66, 67, 69, 70, 71]:

$\displaystyle 1.~{}$ $\displaystyle
x^{*}_{1}=\frac{(-1+4\gamma)\mu}{\sqrt{6}(\gamma+\,2\gamma^{2}+\,\mu^{2})},$
$\displaystyle y^{*}_{1}=\frac{\sqrt{\begin{aligned}
&48\gamma^{3}+120\gamma^{4}+8\gamma\mu^{2}+\mu^{2}(-1+6\mu^{2})+\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\gamma^{2}(-6+56\mu^{2})\end{aligned}}}{\sqrt{6}(\gamma+\,2\gamma^{2}+\,\mu^{2})},$
$\displaystyle 2.~{}$ $\displaystyle
x^{*}_{2}=\frac{(-1+4\gamma)\mu}{\sqrt{6}(\gamma+\,2\gamma^{2}+\,\mu^{2})},$
$\displaystyle y^{*}_{2}=-\frac{\sqrt{\begin{aligned}
&48\gamma^{3}+120\gamma^{4}+8\gamma\mu^{2}+\mu^{2}(-1+6\mu^{2})+\\\
&\qquad\qquad\qquad\qquad\qquad\quad\gamma^{2}(-6+56\mu^{2})\end{aligned}}}{\sqrt{6}(\gamma+\,2\gamma^{2}+\,\mu^{2})},$
$\displaystyle 3.~{}$ $\displaystyle
x^{*}_{3}=\frac{\sqrt{6}\gamma-\,\sqrt{6\gamma^{2}+\,\mu^{2}}}{\mu},\quad
y^{*}_{3}=0,$ (59) $\displaystyle 4.~{}$ $\displaystyle
x^{*}_{4}=\frac{\sqrt{6}\gamma+\,\sqrt{6\gamma^{2}+\,\mu^{2}}}{\mu},\quad
y^{*}_{4}=0,$ (60) $\displaystyle 5.~{}$ $\displaystyle
x^{*}_{5}=-\sqrt{\frac{3}{2}}(1+w_{m})\mu,$ $\displaystyle
y^{*}_{5}=\frac{\sqrt{\begin{aligned}
&\gamma(2-\,6w_{m})+\,12(1+\,w_{m})\gamma^{2}-\\\
&\qquad\qquad\qquad\qquad\,3(-1+w_{m}^{2})\mu^{2}\end{aligned}}}{\sqrt{2}},$
$\displaystyle 6.~{}$ $\displaystyle
x^{*}_{6}=-\sqrt{\frac{3}{2}}(1+w_{m})\mu,$ $\displaystyle
y^{*}_{6}=-\frac{\sqrt{\begin{aligned}
&\gamma(2-\,6w_{m})+\,12(1+\,w_{m})\gamma^{2}-\\\
&\qquad\qquad\qquad\qquad\,3(-1+w_{m}^{2})\mu^{2}\end{aligned}}}{\sqrt{2}},$
$\displaystyle 7.~{}$ $\displaystyle
x^{*}_{7}=\sqrt{\frac{2}{3}}\frac{(1-\,3\,w_{m})\gamma\mu}{(4\gamma^{2}+\,(1-\,w_{m})\mu^{2})},\quad
y^{*}_{7}=0.$ (63)

It is worth noting that $H$ appears in the denominator of the equation $y$ in
Eq. (11). As a result, while a positive sign of $y$ indicates that the
Universe is expanding, a negative sign indicates that the Universe is
contracting and thus the fixed points in the above expressions appear in pairs
(e.g., first and second fixed point). Using Eq. (13), one can quickly
calculate the slow-roll parameter for each of these fixed positions, which are
listed below:

$\displaystyle\epsilon_{1}$ $\displaystyle=$
$\displaystyle\frac{(1-6\gamma+8\gamma^{2})}{2(\gamma+2\gamma^{2}+\mu^{2})}$
(64) $\displaystyle\epsilon_{2}$ $\displaystyle=$
$\displaystyle\frac{(1-6\gamma+8\gamma^{2})}{2(\gamma+2\gamma^{2}+\mu^{2})}$
(65) $\displaystyle\epsilon_{3}$ $\displaystyle=$ $\displaystyle
3+\frac{2\gamma(6\gamma-\sqrt{6}\sqrt{6\gamma^{2}+\mu^{2}}}{\mu^{2}}$ (66)
$\displaystyle\epsilon_{4}$ $\displaystyle=$ $\displaystyle
3+\frac{2\gamma(6\gamma+\sqrt{6}\sqrt{6\gamma^{2}+\mu^{2}}}{\mu^{2}}$ (67)
$\displaystyle\epsilon_{5}$ $\displaystyle=$
$\displaystyle-\frac{3}{2}(1+w_{M})(-1+2\gamma)$ (68)
$\displaystyle\epsilon_{6}$ $\displaystyle=$
$\displaystyle-\frac{3}{2}(1+w_{M})(-1+2\gamma)$ (69)
$\displaystyle\epsilon_{7}$ $\displaystyle=$
$\displaystyle\frac{3(-1+w_{M})\mu^{2}-16\gamma^{2}}{2(-1+w_{M})\mu^{2}-8\gamma^{2}}.$
(70)

Among them, only the first, second, third, and fourth solutions, i.e., Eqs.
(V), (V), (59), and (60) are scalar field dominated solutions as the
fractional energy density in Eq. (12) vanishes for each of these solutions,
i.e.,

$\Omega_{m1}=0,\quad\Omega_{m2}=0,\quad\Omega_{m3}=0,\quad\Omega_{m4}=0.$

The remaining solutions, Eqs. (V), (V) and (63) — refer to the mixed state
solutions, in which both the scalar field and the barotropic fluid’s energy
densities are still non-zero:

$\displaystyle\Omega_{m5}=-6\gamma^{2}(w_{m}+1)-\gamma(3w_{m}+7)-$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad 3\mu^{2}(w_{m}+1)+1,$ (71)
$\displaystyle\Omega_{m6}=-6\gamma^{2}(w_{m}+1)-\gamma(3w_{m}+7)-$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad 3\mu^{2}(w_{m}+1)+1,$ (72)
$\displaystyle\Omega_{m7}=\frac{\left(6\gamma^{2}+\mu^{2}\right)\left(8\gamma^{2}(2-3w_{m})+3\mu^{2}(w_{m}-1)^{2}\right)}{3\left(\mu^{2}(w_{m}-1)-4\gamma^{2}\right)^{2}}.$

Comparing the above fixed points with the results from the previous section
reveals that $\\{x_{1}^{*},y_{1}^{*}\\}$ (Eq. (V)) and
$\\{x_{2}^{*},y_{2}^{*}\\}$ (Eq.(V)) are the desired fixed points.
Nonetheless, it can be seen that both fixed points are similar (for instance,
compare the slow-roll parameters $\epsilon_{1}$ in Eq. (64) and $\epsilon_{2}$
Eq. (65)), with the exception that the signs of $y$ are opposite, signifying
that they represent identical solutions, one with forward-time and the other
with backward-time. Now, in order to analyze the stability of these fixed
points as we did in Eq. (33), we must linearize Eqs. (16) and (17) as follows:

$\displaystyle\left(\begin{aligned} &\frac{{\rm d}\delta x}{{\rm d}N}\\\
&\frac{{\rm d}\delta y}{{\rm d}N}\end{aligned}\right)=\left(\begin{aligned}
\frac{\partial A(x,y)}{\partial x}\Big{|}_{*}&&\frac{\partial A(x,y)}{\partial
y}\Big{|}_{*}\\\ \frac{\partial B(x,y)}{\partial x}\Big{|}_{*}&&\frac{\partial
B(x,y)}{\partial y}\Big{|}_{*}\end{aligned}\right)\left(\begin{aligned} \delta
x&\\\ \delta y&\end{aligned}\right),$ (74)

where, $A(x,y)$ and $B(x,y)$ represent the right-hand sides of Eqs. (16) and
(17), respectively, and $|_{*}$ represents the value at the fixed point.
$\delta x$ and $\delta y$ represent the deviations of $x$ and $y$ from their
fixed points $(x_{*},y_{*})$, respectively. The above-mentioned square matrix
must be diagonalized to determine the eigenvalues and eigenvectors, which will
help us find the solutions of $\delta x$ and $\delta y$, as the solutions can
then be expressed as

$\displaystyle\delta x=C_{11}\,e^{\lambda_{1}\,N}+C_{12}\,e^{\lambda_{2}\,N},$
$\displaystyle\delta y=C_{21}\,e^{\lambda_{1}\,N}+C_{22}\,e^{\lambda_{2}\,N}.$
(75)

The eigenvalues of the matrix are $\lambda_{1}$ and $\lambda_{2}$, known as
the Lyapunov exponents, as before, and $C$’s are related to the eigenvectors
as well as the initial conditions. In this scenario, if both the eigenvalues
are negative (positive) during expansion (contraction), the deviations decay,
implying that the fixed point is stable, or an attractor. If, on the other
hand, one or both exponents become positive (negative), i.e., the solution is
a non-attractor, then $\delta x$ and $\delta y$ grow with time, and the entire
solution rapidly moves away from the desired fixed point, and the system may
become highly unstable.

Using the above method, one can evaluate the eigenvalues for all fixed points;
however, because we are only interested in the first (and the second) fixed
point $\\{x_{1}^{*},y_{1}^{*}\\}$ given in Eqs. (V), these exponents have the
following form:

$\displaystyle\lambda_{1}=-3-3w_{m}+\frac{1-4\gamma}{(\gamma+\,2\gamma^{2}+\,\mu^{2})},$
$\displaystyle\lambda_{2}=-3+\frac{(1-4\gamma)(1+2\gamma)}{2(\gamma+\,2\gamma^{2}+\,\mu^{2})}.$
(76)

Now that we have calculated the eigenvalues for the desired fixed point in the
general non-minimal frame, we will assess and compare in two separate
scenarios, namely the minimal and non-minimal frames, in the following
section.

###  V.1 Minimal Einstein frame

In this instance, $\gamma=0$; hence, first fixed point in (V) (or the second
fixed point in (V)) and the slow-roll parameter in Eq. (64) are expressed as
follows:

$\displaystyle x_{1}^{*}=-\frac{1}{\sqrt{6}\mu},\quad
y_{1}*=\pm\frac{\sqrt{6\mu^{2}-1}}{\sqrt{6}\mu},\quad\epsilon_{1}=\frac{1}{2\mu^{2}}.$
(77)

The associated Lyapunov exponents in Eq. (V) can therefore be derived as

$\displaystyle\lambda_{1}=\frac{2-\alpha(1+3w_{m})}{\alpha},\quad\lambda_{2}=\frac{1-2\alpha}{\alpha},$
(78)

where, $\alpha\equiv 1/(\epsilon_{1}-1),$ yields the scale factor solution,
$a(\eta)\propto(-\eta)^{\alpha}$, as described in the preceding section.
Notice that $\lambda_{2}$ is similar to that in the minimal scenario, i.e.,
Eq. (35), without the additional fluid. In contrast, due to the barotropic
fluid, we now have a new eigenvalue that likewise depends on its equation of
state parameter $w_{m}$. Immediately, we now have the following stability
conditions:

$\displaystyle\text{For
expansion:}\begin{cases}&w_{m}<-\frac{1}{3},~{}\text{and}~{}\alpha>\frac{2}{1+3w_{m}},\\\
&\text{or,}\\\ &w_{m}\geq-\frac{1}{3},~{}\text{and}~{}\alpha<0.\end{cases}$
(79) $\displaystyle\text{For contraction:}\begin{cases}&w_{m}\leq
1,~{}\text{and}\quad\alpha<\frac{1}{2},\\\ &\text{or,}\\\
&w_{m}>1,~{}\text{and}\quad\alpha<\frac{2}{1+3w_{m}}.\end{cases}$

Please keep in mind that while evaluating scalar perturbations, we have
additional constraint on $\alpha$: $-1<\alpha<0$ is forbidden because it leads
to ghost instability. However, because $\alpha\leq-1$ indicates that the
Universe is in an accelerated (inflationary) expansion, we find that all
inflationary solutions are guaranteed to be stable only if $w_{m}>-1/3$,
whereas stability is conditional for $w_{m}\leq-1/3$, as the lower bound of
$\alpha$ is: $\alpha_{\rm min}\equiv 2/(3w_{m}+1).$ It follows instantly that
if the barotropic fluid is de-Sitter, as with $w_{m}=-1$, $\alpha_{\rm min}$
becomes $-1.$ In the most slow-roll inflationary scenario, at or near the
pivot scale, the effective value of $\alpha$ is extremely close to $-1$, i.e.,
$\alpha\lesssim-1.$ As a classic example, consider chaotic inflation, where
$\epsilon_{1}\simeq 0.01$ and, consequently, $\alpha\simeq-1.01$ at the pivot
scale. In that scenario, as $\alpha<\alpha_{\rm min},$ the slow-roll solution
is not a stable solution. When we consider $w_{m}\leq-1$, the severity of the
situation grows.

The following result is not surprising given that, during conventional slow-
roll inflation (at or near the pivot scale), the energy density
$\rho_{\phi}\sim 10^{-9}M_{{}_{\rm Pl}}^{4}$ steadily decays to one or two
orders below that level, at the minimum of the potential, resulting in a
(p)reheating scenario (which in the current context, we are not studying). In
the presence of a de-Sitter type fluid, however, the energy density of the
fluid $\rho_{m}$ remains constant, and so $\rho_{m}/\rho_{\phi}$ increases by
an order or two during inflation, explaining the instability of the fixed
point. As a result, it instantly suggests that, unless
$\rho_{m}\ll\rho_{\phi},$ a de-Sitter like fluid with
$\rho_{m}\sim\rho_{\phi}$ can induce system instability. However, it should be
emphasized that any natural energy scale below the Planck level, such as
electroweak energy, is considerably below the Hubble scale, and so the
$\rho_{m}\ll\rho_{\phi}$ condition is satisfied, which does not affect the
system.

In contrast, in the case of contraction, $\alpha<1/2$ (the ekpyrotic solution)
always results in stability, regardless of the equation of state of the
additional fluid. In reality, as Eq. (78) shows, $\alpha\rightarrow 0^{+}$
renders the system unaffected by $w_{m}$, regardless of the value as $w_{m}$
is multiplied by $\alpha$. This conclusion is crucial because it suggests that
ekpyrosis can dilute any energy density associated with increased barotropic
fluid, whereas it is conditional in the case of inflation.

###  V.2 Non-minimal frame

Similar to previous calculations, utilizing the relations $\tilde{\gamma}$ and
$\tilde{\mu}$ in terms of $\gamma$ and $\mu$ in Eqs. (49) and (50), the
Lyapunov exponents in the newly conformally modified theory can be found
directly as

$\displaystyle\tilde{\lambda}_{1}=\frac{2(1-\alpha)+\beta\left(1-3w_{m}\right)}{\beta},\quad\tilde{\lambda}_{2}=\frac{1-2\alpha}{\beta}$
(81)

It is clear that when we compare the above relation to the minimally coupled
case (cf. Eq. (78)), we find that

$\displaystyle\lambda_{1}N\neq\tilde{\lambda}_{1}\tilde{N},\quad\lambda_{2}N=\tilde{\lambda}_{2}\tilde{N},$
(82)

leading one to conclude that stability equivalence has been violated. Again,
this is due to the fact that we left the additional matter unaffected during
the conformal transformation; hence, the two theories are not conformally
related. Let us now investigate the stability criteria in the new theory.
Using the above relation stated in Eq. (81), we get the criteria as

$\displaystyle\text{Expansion:}\begin{cases}&w_{m}<\frac{1}{3},~{}\alpha<\frac{1}{2},~{}\text{and}~{}\beta>\frac{2(1-\alpha)}{(3w_{m}-1)},\\\
&\text{or,}\\\
&w_{m}\geq\frac{1}{3},~{}\text{and}~{}\alpha<\frac{1}{2}.\end{cases}$ (83)
$\displaystyle\text{Contraction:}\begin{cases}&w_{m}\leq\frac{1}{3},~{}\text{and}~{}\alpha<\frac{1}{2},\\\
&\text{or,}\\\
&w_{m}>\frac{1}{3},~{}\alpha<\frac{1}{2},~{}\text{and}~{}\beta<\frac{2(1-\alpha)}{(3w_{m}-1)}.\end{cases}$

Let’s rewrite the above expression in a straightforward yet different way as

$\displaystyle\alpha<\frac{1}{2},\quad\text{and}$ (85)
$\displaystyle\beta\begin{cases}>-\frac{2(1-\alpha)}{(1-3w_{m})},\quad\text{for}\quad
w_{m}<\frac{1}{3},\\\
<\frac{2(1-\alpha)}{(3w_{m}-1)},\quad~{}~{}\text{for}\quad
w_{m}>\frac{1}{3}.\end{cases}$ (86)

As a result, comparing this expression to the preceding one becomes easier.
Again, in the minimal frame, $\alpha$ reflects the scale factor exponent,
whereas in the non-minimal frame, $\beta$ does. Furthermore, as always,
$\beta<0$ denotes accelerated expansion, whereas positive $\beta$ indicates a
contracting (bouncing) solution. As a result, although in the simplest
situation, not all values of $\alpha$ ensure the stability requirement (cf.
Eq. (V.1)), in this case, $\alpha$, along with $\beta$, do. Consider the state
parameter equation to be in between the desired range, i.e., $-1<w_{m}<1$. The
stability condition in that situation is

$\displaystyle-\frac{(1-\alpha)}{2}<\beta<(1-\alpha).$ (87)

Since $\alpha$ must meet the stringent condition of $\alpha<1/2$ (cf. Eq.
(81)), the strong condition is now

$\displaystyle-\frac{1}{4}<\beta<\frac{1}{2}.$ (88)

It is worth noting that this condition, coupled with $\alpha<1/2$ and
$-1<w_{m}<1$, always assures stability. In fact, the biggest achievement of
non-minimal coupling is that it is always feasible to achieve stability even
when $|w_{m}|>1$ is chosen, i.e.,

$\displaystyle-\frac{1}{(1-3w_{m})}<\beta<\frac{1}{(3w_{m}-1)}.$ (89)

As the magnitude of $w_{m}$ increases, the range of $\beta$ necessary for
stability decreases. However, it still leaves an appropriate range of $\beta$
such that the above criterion is satisfied, and the solution is always an
attractor.

We have accomplished a novel and intriguing result, thus there are few points
worth mentioning. $\beta$ is determined by how we apply the conformal
transformation, whereas $\alpha$ determines the model’s observational
signature, which is previously known [69, 70, 71]. As a result, while there is
a bound on $\alpha$ appearing immediately from the perturbation, i.e.,
$-1<\alpha<0$ is forbidden, there is no similar constraint on $\beta.$ As a
result, the choice of choosing $\beta$ is absolutely random. Second, while the
stability of slow-roll expansion was a condition in the minimal case, we now
have tangible stability if $\beta$ is chosen correctly for the same choice of
$\alpha$. That is, regardless of the choice of $w_{m},$ if $\beta$ is chosen
to be very close to zero (i.e., ekpyrotic expansion or contraction), the
system will always find stability. For this reason, using classical
perturbation analysis, we can state unequivocally that ekpyrosis (whether
expansion or contraction) always dilutes perturbations, whereas inflation does
not guarantee to, even in general non-minimal frames. This is the second and
most significant outcome from this study.

##  VI Summary and conclusions

Let us first summarize what this article has accomplished. First, we analyzed
the minimal Einstein’s theory in the absence of an additional fluid and
performed a dynamical analysis of the system. Only $\alpha<1/2$ provides the
stability requirement, where $\alpha$ denotes the exponent of the scale factor
solution in the conformal time. Then, we conformally modified the theory so
that the scale factor solution in the new theory can be written as
$a(\eta)\propto(-\eta)^{\beta}$. Again in the non-minimal frame, we examined
the stability condition and discovered that stability is equivalent, i.e., the
condition is independent of $\beta$ and entirely dependent on $\alpha$ with
$\alpha<1/2.$ This is the equivalence of stability in a conformally connected
frame, as previously investigated in [67].

Next, an additional barotropic fluid was added to the minimal theory. We
analyzed the stability condition once again. In addition to the one condition
previously acquired, we now have a second condition that is dependent on the
equation of state of the barotropic fluid, as predicted. Instantaneously, we
discovered that $\alpha<1/2$ does not always guarantee stability. Stability
exists only when $\alpha$ approaches zero, i.e., ekpyrosis, whereas
$\alpha<-1$ (i.e., inflation) is conditional. However, $-1<\alpha<0$ is ruled
out because it leads to ghosts, while $\alpha>0$ is ruled out by the
observations. As a result, in minimal Einstein’s theory, the inclusion of
additional fluid perturbations does not ensure attractor solution.

We then moved to non-minimal theory by conformally transforming the theory, as
we had done previously, with the difference that additional fluid remained
unaffected by the change. This means that conformal invariance has been
broken, as well as the stability equivalence. Consequently, the stability
criteria have also changed, and the result is intriguing. These criteria now
depend on $\alpha$, $\beta$, and $w_{m}$. Additionally, with the tuning of
$\beta$, which is completely arbitrary and within our control to fix by
conformal transformation, any value of $\alpha$ and $w_{m}$ can now lead to a
stable solution, which is not conceivable in the minimal theory. In addition,
only $\alpha$, not $\beta$, determines the observable signature of the theory,
presuming that only a scalar field-dominated solution is viable for the early
Universe. This is because, curvature and tensor perturbations remain invariant
under conformal transformation. This invariance even holds for higher-order
perturbations, as the interaction Hamiltonian at any order of perturbations
(needed to evaluate for the higher-order correlation functions. See, for
instance, Refs. [73, 74, 75]) does not change under conformal transformation.
As a result, even if the perturbations and thus the observables remain
unchanged, the non-minimal theory becomes far more desirable than the original
minimal counterpart.

In other words, using only the scalar field, one can construct a theory, such
as slow-roll inflation, in Einstein’s minimal frame and then transform it into
a non-minimal model in which the solution is ekpyrotic. As a consequence, it
is possible to assert that the new theory is more beneficial and classically
stable than the original inflationary theory. These examples have already been
demonstrated in Refs. [69, 70, 71], and the broader ramifications of such
transformation are now demonstrated in this paper.

Any such result would have far-reaching effects. At first glance, such a
result seems straightforward and unimportant, given that we can simply apply
the conformal transformation to a single theory without influencing the
barotropic fluid. However, as the early Universe era comes to an end, the
impact of the barotropic fluid automatically kicks in, leaving a testable
theory to be examined in the setting of the late-time Universe. Solutions
other than the scalar field dominated solution, which were left out on
purpose, govern these solutions and, if studied thoroughly, can have exciting
and observable effects. Simple examples of these are research into the
reheating era and the $H_{0}$ tension. In addition, as was previously noted,
non-minimal stability with a de-Sitter like barotropic fluid might be seen as
an answer to the naturalness of the cosmological constant problem. The reason
for this is that, as the scalar field solution is very stable, the effective
energy density, $\rho_{m}/f^{2}(\phi)\equiv\Lambda/f^{2}(\phi)$ (i.e.,
$\Omega_{m}H^{2}$), decreases as compared to the Hubble energy (cf. Eq. (12)).
It is only after a long period of time that the effective energy density
becomes identical to $\Lambda$ as a result of the decay of $f(\phi)$ to unity.
One can think of this like the “flatness problem” of the early Universe.
Third, we didn’t cover many interesting but more complicated models like the
non-canonical model, Galileon theory, generic Horndeski theory, etc., adhering
instead to canonical scalar field theory and its conformally modified
counterpart. As a fourth point, disformal transformation, like conformal
transformation, renders scalar and tensor perturbations invariant, opening the
door to the possibility of experimenting with disformal transformation to
transform the original theory. These questions have yet to be resolved, but
they hold the promise of illuminating insights and fresh takes on the
universe. At the moment, we are doing a significantly more thorough
investigation of these prospects.

## Acknowledgements

DN is supported by the DST, Government of India through the DST-INSPIRE
Faculty fellowship (04/2020/002142). MK is supported by a DST-INSPIRE
Fellowship under the reference number: IF170808, DST, Government of India. DN
and MK are also very thankful to the Department of Physics and Astrophysics,
University of Delhi. MK and DN also acknowledge facilities provided by the
IUCAA Centre for Astronomy Research and Development (ICARD), University of
Delhi.

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