# Emergent Universe from an Unstable de Sitter Phase

###### Abstract

In the Emergent scenario, the Universe should evolve from a non-singular state
replacing the typical singularity of General Relativity, for any initial
condition. For the scalar field model in [1] we show that only a subset of
initial conditions leads to emergence, either from a static state (an Einstein
model), or from a de Sitter state.

Assuming a scenario based on CDM interacting with a Dark Energy fluid, we show
that in general flat and open models expand from a non-singular unstable de
Sitter state at high energies; for closed models this state is a transition
phase with a bounce. A subset of these models are qualitatively in agreement
with the observable Universe, accelerating at high energies, going through a
matter-dominated decelerated era, then accelerating toward a de Sitter phase.

Molly Burkmara,1 (corresponding author) and Marco Brunia,b,2

aInstitute of Cosmology & Gravitation, University of Portsmouth, Dennis Sciama
Building, Burnaby Road, Portsmouth, PO1 3FX, United Kingdom

bINFN Sezione di Trieste, Via Valerio 2, 34127 Trieste, Italy

<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
Essay written for the Gravity Research Foundation 2024 Awards for Essays on
Gravitation.

##  1 Introduction

The Standard Model of Cosmology, a flat $\Lambda$CDM model with inflation, is
based on General Relativity (GR) and provides a successful framework to
describe the evolution of the Universe. However, the presence of singularities
pose a problem and are currently interpreted as points in space-time where GR
breaks down [2, 3, 4, 5]. Observations are consistent with a flat universe [6,
7], however they do not rule out open or closed spatial curvature [8]. In
particular, a closed universe is appealing as it avoids the problem of having
an infinite universe [9]. The Emergent Universe has been widely studied as an
alternative to an initial singularity, with models built with fine-tuned
initial conditions on the assumption that the past repellor must be a static
state represented by an Einstein model with positive curvature [1, 10, 11, 12,
13, 14, 15, 16, 17, 18, 19]. Thus the problem with this scenario is that very
special initial conditions are required, as the Einstein model is typically
represented by an unstable saddle or a centre in the phase space of the
relevant Ordinary Differential Equations (ODE), and so it is not a generic
past repellor, as in the classical model by Eddington [20].

In this essay we contend that in the Emergent scenario, the Universe should
evolve from a non-singular state replacing the typical singularity of GR for
any initial condition. In this light, we first consider whether models in [1],
based on a scalar field, are still non-singular for general initial
conditions. Similarly to the analysis in [21] and [22], we complete a
dynamical systems analysis of the Emergent scenario in [1]. We show that only
a set of measure zero of trajectories emerge from the static Einstein model.
We also find a class of models that do in fact emerge from a non-singular de
Sitter fixed point, representing a state where the scalar field is potential
dominated. The problem with this class of models is that the stability type of
this de Sitter fixed point is that of a 3-D saddle, so some fine-tuning is
still required for models to be non-singular. For trajectories not emerging
from this de Sitter state, we find that the past repellor is a fixed point
representing a stiff fluid singularity where the scalar field is kinetic
dominated111The analysis can be found at:
https://github.com/MollyBurkmar1/Emerging_Universe.git.

We then consider a fluid-based model to show an example where emergence from a
non-singular state is generic during expansion, regardless of initial
conditions. Assuming a scenario based on Cold Dark Matter (CDM, represented by
a dust fluid) interacting with a Dark Energy fluid with a nonlinear equation
of state, we show that all but a set of measure zero of trajectories in phase
space expand from an unstable de Sitter state, a classical vacuum at high
energies. This represents the asymptotic past of flat and open models, and a
transition phase with a bounce for closed models. Independently from the
initial conditions, all models are accelerating at high energies. A subset of
models are qualitatively in agreement with the observed universe, emerging
from a non-singular state, going through a matter-dominated decelerated era,
then accelerating toward the future de Sitter phase.

##  2 Scalar-Field-Based Emergent Universe

We consider a Friedmann-Lemaître-Robertson-Walker (FLRW) universe, containing
a scalar field $\phi$ with the potential

$V=V_{0}\left(e^{\phi}-1\right)^{2}\,,$ (1)

where $V_{0}$ is the asymptotic value of the potential at
$\phi\rightarrow-\infty$. Assuming $8\pi G=c=1$, the Klein-Gordon and
Raychaudhuri equations are

$\ddot{\phi}=-3H\dot{\phi}-2V_{0}e^{\phi}(e^{\phi}-1)\,,$ (2)

$\dot{H}=-H^{2}-\frac{1}{3}\left[\dot{\phi}^{2}-V_{0}(e^{\phi}-1)^{2}\right]\,,$
(3)

where $H=\dot{a}/a$ is the Hubble expansion scalar ($a$ is the scale factor)
and over-dots are derivatives with respect to time. The dynamics is
complemented by the usual Friedmann first integral (the Hamiltonian
constraint)

$H^{2}=-\frac{k}{a^{2}}+\frac{1}{3}\left[\frac{\dot{\phi}^{2}}{2}+V(\phi)\right].$
(4)

Setting $\psi=\dot{\phi}/\sqrt{V_{0}}$ defines a 3-dimensional set of ODE. To
analyse this dynamical system we define dimensionless variables ($\phi$ is
dimensionless by definition) and their compactified version,

$\eta=\sqrt{V_{0}}t,~{}~{}~{}\Phi=\frac{\phi}{\sqrt{1+\phi^{2}}},~{}~{}~{}\psi=\frac{\dot{\phi}}{\sqrt{V_{0}}},~{}~{}~{}\Psi=\frac{\psi}{\sqrt{1+\psi^{2}}},~{}~{}~{}y=\frac{H}{\sqrt{V_{0}}},~{}~{}~{}Y=\frac{y}{\sqrt{1+y^{2}}}\,,$
(5)

in order to study the behaviour of the system at infinity. Here,
$\phi\rightarrow\pm\infty$ corresponds to $\Phi=\pm 1$,
$\psi\rightarrow\pm\infty$ to $\Psi=\pm 1$ and $y\rightarrow\pm\infty$ to
$Y=\pm 1$.

First, we consider the phase space of spatially flat models. The Friedmann
equation (4) with $k=0$ in our compactified variables becomes

$\frac{Y}{\sqrt{1-Y^{2}}}=\pm\sqrt{\frac{1}{3}\left(e^{\frac{\Phi^{2}}{\sqrt{1-\Phi^{2}}}}-1\right)^{2}+\frac{\Psi^{2}}{6\left(1-\Psi^{2}\right)}}\,,$
(6)

which we substitute into the equation for $\Psi$, and numerically solve
alongside the equation for $\Phi$. Here, $Y>0$ corresponds to expansion and
$Y<0$ to contraction.

The fixed points of the flat sub-manifold with their stability are summarised
in Table 1. To understand the type of fixed points present at $\Phi=\pm 1$ and
$\Psi=\pm 1$, we need to understand what happens to the potential (1) and
kinetic energy $K=\psi^{2}/2$ at these points. When $\Psi=\pm 1$,
$K\rightarrow\infty$. For the fixed points where $\Phi=-1$, the potential is
finite: $V=V_{0}$. In this case the fixed points are kinetic dominated and so
are stiff fluid singularities. For $\Phi=+1$, $V\propto e^{\infty}$. Although
both the potential and kinetic energy become infinite, we find that the
dynamics is potential dominated in this limit, and therefore we define the
fixed points at $\Phi=+1$ as de Sitter.

Only the stability of the de Sitter fixed points $dS_{1\pm}$ can be found from
linearisation of the system because there are infinities in the Jacobian. We
can find their stability through plots of the phase space, however for
repellor and attractor fixed points it is helpful to define Liapunov functions
[23, 24] in order to obtain more rigorous results. For the Minkowski fixed
point at $\phi=0$, we define the Liapunov function $L$ as the energy density
of the scalar field, where we approximate $e^{\phi}\simeq 1+\phi$ such that
$V\simeq\phi^{2}$ around the origin:

$L=\frac{\Phi^{2}}{1-\Phi^{2}}+\frac{\Psi^{2}}{2\left(1-\Psi^{2}\right)}.$ (7)

We then take the first derivative of $L$ to be able to determine the stability
of the fixed point. If $L^{\prime}<0$ at least in a disc around the fixed
point, then it is is an attractor, and if $L^{\prime}>0$ then the fixed point
is a repellor. Taking the first derivative of $L$, we find

$L^{\prime}=-\frac{3\Psi^{2}Y}{(1-\Psi^{2})\sqrt{1-Y^{2}}}.$ (8)

Taking $Y>0$ ($Y<0)$ for the expanding (contracting) case, we find
$L^{\prime}<0$ ($L^{\prime}>0)$ for the whole phase space, therefore the
Minkowski fixed point is a global attractor (repellor) in the expanding
(contracting) phase space. For the past (future) singularity, we define the
Liapunov function as a circle centered around the fixed point:

$L=\left(1+\Phi\right)^{2}+\left(\Psi\mp 1\right)^{2}\,.$ (9)

We fix a radius of $L_{*}=r^{2}=0.5^{2}$ so that we only study the behaviour
of trajectories around the singularities. The first derivative of $L$ gives

$\L^{\prime}=\frac{2\Psi\left(1+\Phi\right)\left(1-\Phi^{2}\right)^{3/2}}{\sqrt{1-\Psi^{2}}}+2\left(\Psi\mp
1\right)\left[-2\left(1-\Psi^{2}\right)^{3/2}e^{\frac{\Phi}{\sqrt{1-\Phi^{2}}}}\left(e^{\frac{\Phi}{\sqrt{1-\Phi^{2}}}}-1\right)-\frac{3Y\Psi\left(1-\Psi^{2}\right)}{\sqrt{1-Y^{2}}}\right]\,.$
(10)

For the expanding (contracting) case, we find $L^{\prime}>0$ ($L^{\prime}<0$)
for $L<L_{*}$, therefore the past (future) singularity is a repellor (an
attractor).

The phase space for the flat sub-manifold is shown in Fig. 1, where Fig. 1(a)
shows the expanding case, and Fig. 1(b) the contracting case. In the expanding
case, trajectories between the two separatrices are potential dominated in the
past and so expand from $dS_{3+}$, whereas trajectories outside the two
separatrices are kinetic dominated in the past and so expand from $S_{1+}$.
The past repellor $S_{1+}$ represents the scalar field starting on the flat
part of the potential and rolling towards the minimum, and the fixed point
$dS_{3+}$ represents the field starting from the exponential part of the
potential and rolling towards the minimum. There is also a special trajectory
in the phase space, which emerges from the de Sitter fixed point $dS_{1+}$ and
expands towards the Minkowski fixed point $M$. In this case, the field is
initially on the asymptotic part of the potential, and rolls toward the
minimum where it oscillates.

Fig. 1(b) shows the contracting case. Here, trajectories contract from the
Minkowski fixed point $M$ in all cases, and in general contract towards a
singularity at $S_{2-}$ or the de Sitter fixed point $dS_{2-}$. Similarly to
the expanding case, trajectories between the two separatrices become potential
dominated and contract towards $dS_{2-}$, and trajectories outside the
separatrices become kinetic dominated and contract towards the singularity
$S_{2-}$. There is also the possibility that a trajectory contracts towards
the de Sitter fixed point $dS_{1-}$, although this requires a set of measure
zero initial conditions.

It is clear there are flat models for which trajectories are non-singular and
can be past or future asymptotic to a de Sitter state. However, the fixed
points $dS_{3+}$ and $dS_{2-}$ both have saddle stability so non-singular
models do require some level of fine-tuning. For all other trajectories the
past repellor is the singularity $S_{1+}$.

Fixed point | $\Phi$ | $\Psi$ | $Y$ | Stability in 2-D | Stability in 3-D
---|---|---|---|---|---
$M_{\pm}$ | 0 | 0 | 0 | Spiral attractor (+) / Spiral repellor (-) | Spiral saddle
$dS_{1\pm}$ | $-1$ | 0 | $\pm 1/2$ | Saddle ($\pm$) | Saddle ($\pm$)
$dS_{2\pm}$ | $1$ | $1$ | $\pm 1$ | Saddle ($\pm$) | Saddle ($\pm$)
$dS_{3\pm}$ | $1$ | $-1$ | $\pm 1$ | Saddle ($\pm$) | Saddle ($\pm$)
$S_{1\pm}$ | $-1$ | $1$ | $\pm 1$ | Repellor (+)/ Saddle (-) | Repellor (+)/ Saddle (-)
$S_{2\pm}$ | $-1$ | $-1$ | $\pm 1$ | Saddle (+)/ Attractor (-) | Saddle (+)/ Attractor (-)
Table 1: Fixed points for the expanding (+) and contracting (-) flat 2-D sub-
manifolds in Fig. 1, and the full 3-D phase space in Fig. 3. $M$ denotes the
Minkowski fixed point, $dS$ the de Sitter fixed points, and $S$ the
singularities.

(a) Expansion ($Y>0$).

(b) Contraction ($Y<0$).

Figure 1: The flat sub-manifolds for the system, where we substitute $Y>0$ for
the expanding case in (a) and $Y<0$ for the contracting case in (b). The
Minkowski fixed point is shown in black, the de Sitter fixed points in magenta
and the singularities in orange. The two black curves are separatrices that
separate where the trajectories expand from (contract to). Trajectories
between the two separatrices expand from (contract to) the de Sitter fixed
point $dS_{3+}$ ($dS_{2-}$). Otherwise, trajectories outside these
separatrices expand from (contract to) $S_{1+}$ ($S_{2-}$).

Going back to the 3-D dynamics, we first consider the $\Phi=-1$ sub-manifold,
where the potential (1) reduces to $V=V_{0}$. The fixed points of the
$\Phi=-1$ sub-manifold are shown in Table 2, and the phase space is shown in
Fig. 2. The Einstein points $E_{\pm}$ are both unstable saddles, with only two
separatrix trajectories emerging from each. Therefore, these are not good
candidates for emerging models as these are a set of measure zero for the
whole phase space. In this sub-manifold, the past repellors for initially
expanding models are the singularities $S_{1+}$ and $S_{2+}$, and the de
Sitter fixed point $dS_{1-}$ is the past repellor for models which initially
contract. Trajectories emerging from the non-singular de Sitter state
$dS_{1-}$ either collapse, bounce and expand towards another de Sitter state
$dS_{1+}$, or collapse to a singularity ($S_{1-}$ or $S_{2-}$).

Fixed point | $\Psi$ | $Y$ | Stability in 2-D | Stability in 3-D
---|---|---|---|---
$E_{\pm}$ | $\pm\frac{1}{\sqrt{2}}$ | 0 | Saddle ($\pm$) | Saddle ($\pm$)
$dS_{1\pm}$ | 0 | $\pm\frac{1}{2}$ | Attractor (+)/ Repellor (-) | Saddle ($\pm$)
$dS_{4\pm}$ | 0 | $\pm 1$ | Saddle ($\pm$) | Saddle ($\pm$)
$S_{1\pm}$ | 1 | $\pm 1$ | Repellor (+)/ Attractor (-) | Repellor (+)/ Saddle (-)
$S_{2\pm}$ | -1 | $\pm 1$ | Repellor (+)/ Attractor (-) | Saddle (+)/ Attractor (-)
Table 2: Fixed points with their stability for the 2-D $\Phi=-1$ sub-manifold
in Fig. 2 and the full 3-D phase space in Fig. 3. $E$ denotes the Einstein
fixed points, $dS$ the de Sitter fixed points, and $S$ the singularities.
Figure 2: The $\Phi=-1$ submanifold. The thick green curve shows the Flat
Friedmann Separatrix (FFS) separating the open models outside the FFS from the
closed models between the two green curves. The thick black curve is the
Closed Friedmann Separatrix (CFS) which seperates different types of closed
models in the phase space. The Einstein fixed points are shown in cyan, the de
Sitter fixed points in magenta and the singularities in orange.

The full phase space for the 3-D system is shown in Fig. 3, where Fig. 3(a)
shows examples of open (blue) and closed (purple) trajectories coming from and
going to a singularity, and Fig. 3(b) shows examples of non-singular
trajectories. The fixed points that have not already been considered in the
sub-manifolds are shown in Table 3. In Fig. 3(a), there are two examples of
closed models: one simply expands from a singularity, reaches a turn-around on
the surface $Y=0$ and then collapses to a singularity. The other expands and
oscillates around the Minkowski fixed point (at the minimum of the potential),
where it contracts, bounces and expands, before turning around on the $Y=0$
surface and collapsing to the singularity. Whether there is a sufficient
inflationary phase after the bounce remains to be seen. The expanding open
model evolves from a singularity, and oscillates around the minimum of the
potential, and the contracting open model oscillates around the Minkowski
fixed point before collapsing to a singularity.

Fig. 3(b) shows examples of non-singular models in the phase space, which
emerge from Einstein fixed points (cyan) and de Sitter fixed points (magenta).
The trajectories that emerge from the Einstein fixed points require extremely
fine-tuned initial conditions: they form a set of measure zero in phase space.
Trajectories can emerge from the de Sitter states $dS_{4+}$ and $dS_{5+}$, but
these are a set of measure zero in the phase space. However, there is a 3-D
subset of trajectories emerging from $dS_{3+}$, like the blue trajectory in
the figure, but this fixed point is a 3-D saddle, repelling the aforementioned
subset and attracting in another sub-manifold. Thus, fine-tuning is still
needed to ensure that trajectories emerge from this fixed point. For all other
trajectories the past repellor is the $S_{1+}$ point, representing a stiff
fluid singularity where the scalar field is kinetic dominated.

Fixed point | $\Phi$ | $\Psi$ | $Y$ | Stability
---|---|---|---|---
$dS_{5\pm}$ | 0 | 0 | $\pm 1$ | Saddle ($\pm$)
Table 3: The fixed points of the full system not already shown in the flat and
$\Phi=-1$ sub-manifolds.

(a) Singular trajectories.

(b) Non-singular trajectories.

Figure 3: The full phase space for the system, where we have plotted examples
of open (blue) and closed (purple) trajectories, as well as separatrices
(black). The green surface shows the Flat Friedmann Separatrix. The Minkowski
fixed point is shown in black, the Einstein fixed points in cyan, the de
Sitter fixed points in magenta, and the singularities in orange.

##  3 Emergence from Interacting Dark Energy and Dark Matter

In this essay, we contend that it is desirable for all trajectories to emerge
from a non-singular state in the Emerging Universe scenario, regardless of
initial conditions. In the following, we provide an example of a model with
dark energy with a non-linear equation of state [25] interacting with a dust
fluid [26], where the generic trajectory either represents cyclic models, or
models that emerge from an unstable de Sitter phase during expansion. For flat
and open models this is the asymptotic past repellor, whilst for a subset of
closed models the de Sitter phase represents a transition through a bounce
from contraction to expansion.

In this scenario, the conservation equations for the dark matter and dark
energy are

$\dot{\rho}_{m}=-3H\rho_{m}+\frac{H\rho_{x}\rho_{m}}{\rho_{i}}\,,$ (11)

$\dot{\rho_{x}}=-3H(\rho_{x}-\rho_{\Lambda})\left(1+w_{x}+\epsilon\frac{\rho_{x}}{\rho_{*}}\right)-\frac{H\rho_{x}\rho_{m}}{\rho_{i}}\,.$
(12)

where $\rho_{m}$ is the dark matter energy density and $\rho_{x}$ is the dark
energy density. The dark energy has a non-linear equation of state, with
$w_{x}$ defining its linear part, $\rho_{*}$ the characteristic scale of the
non-linear part, and $\rho_{\Lambda}$ representing its low energy attractor.
$\epsilon$ is a free dimensionless parameter that fixes the sign and the
strength of the quadratic term and the energy scale $\rho_{i}$ characterises
the non-linear interaction. To close the system, we also require the
Raychaudhuri equation to describe the evolution of the expansion scalar $H$,

$\dot{H}=-H^{2}-\frac{1}{6}\left[\rho_{m}+\rho_{x}(1+3w_{x}-3\epsilon\frac{\rho_{\Lambda}}{\rho_{*}})\\\
-3\rho_{\Lambda}(1+w_{x})+3\epsilon\frac{\rho_{x}^{2}}{\rho_{*}}\right]\,.$
(13)

To analyse the dynamical system consisting of (11), (12) and (13), we first
define dimensionless variables,

$x=\frac{\rho_{x}}{\rho_{*}},~{}~{}y=\frac{H}{\sqrt{\rho_{*}}},~{}~{}z=\frac{\rho_{m}}{\rho_{*}},~{}~{}\mathcal{R}=\frac{\rho_{\Lambda}}{\rho_{*}},~{}~{}q=\frac{\rho_{*}}{\rho_{i}},~{}~{}\eta=\sqrt{\rho_{*}}t\,,$
(14)

as well as compactified variables,

$X=\frac{x}{1+x},~{}~{}Y=\frac{y}{\sqrt{1+y^{2}}},~{}~{}Z=\frac{z}{1+z}\,.$
(15)

in order to see the behaviour of the system at infinity. For this system,
$Y=\pm 1$ corresponds to $H\rightarrow\pm\infty$, and $X=Z=1$ corresponds to
$\rho_{x},\rho_{m}\rightarrow+\infty$.

An example of the full phase space for a subset of models that expand towards
a future de Sitter state is shown in Fig. 4 with the fixed points shown in
Table 4. We set the parameter values to $\epsilon=-0.25$, $q=1$,
$\mathcal{R}=0.05$222Note that we set $\mathcal{R}$ such that the phase space
is readable. In reality we would expect $10^{-120}<\mathcal{R}<10^{-60}$. and
$w_{x}=-0.5$, which defines a surface in the phase space. The reddish surface
shows where the acceleration is zero, with models accelerating when
trajectories are above this surface, and decelerating when they are below it.
The Flat Friedmann Separatrix would be a surface in the phase space, however
for the sake of clarity we just show the flat trajectories in green for the
parameter values we set. There is a trajectory, not shown in the figure, which
connects the flat fixed points $dS_{3+}$ and $dS_{3-}$. This is a high energy
de Sitter trajectory, representing a closed de Sitter model.

In this case, all trajectories are non-singular and emerge from either a de
Sitter or Einstein fixed point. Open (blue) and flat models emerge from a de
Sitter fixed point (magenta) and expand (contract) to another de Sitter state.
The Closed Friedmann Separatrix (CFS) passing through the saddle Einstein
point is shown by the thick black curve. Closed models (purple) outside the
CFS emerge from a contracting phase asymptotic in the past to the de Sitter
fixed point $dS_{1-}$, then bounce at high energy before expanding to
$dS_{1+}$. Trajectories inside the CFS either bounce once, contracting from
$dS_{1-}$ and expanding to $dS_{1+}$, or are cyclic around the Einstein fixed
point $E_{2}$, and repeatedly contract, bounce and expand. An emergent case
which expands from the Einstein fixed point $E_{1}$ also exists along the CFS.

In this example, two Einstein fixed points (cyan) exist in the phase space,
which coincide with where the acceleration is zero. This means in this
example, open, flat and closed trajectories outside of the CFS all evolve with
an early- and late-time acceleration, connected by a decelerated period. These
are the cases of interest as they qualitatively match the observed Universe.

Fixed point | $X$ | $Y$ | $Z$ | Stability
---|---|---|---|---
$dS_{1\pm}$ | 0.048 |  $\pm$0.13 | 0 | Attractor (+)/ Repellor (-)
$dS_{2\pm}$ | 0.67 |  $\pm$ 0.63 | 0 | Saddle ($\pm$)
$dS_{3\pm}$ | 0.75 |  $\pm$ 0.74 | 0.42 | Spiral saddle ($\pm$)
$dS_{4\pm}$ | 0.75 |  $\pm$ 1 | 0.42 | Spiral repellor (+)/ Spiral attractor (-)
$E_{1}$ | 0.052 | 0 | 0.094 | Saddle
$E_{2}$ | 0.73 | 0 | 0.87 | Centre
Table 4: Fixed points for the fluid model, where dust is interacting with a
dark energy with a non-linear equation of state. The parameters are set to
$\epsilon=-0.25$, $q=1$, $\mathcal{R}=0.05$ and $w_{x}=-0.5$.

(a)

(b)

Figure 4: The full phase space for the interacting cold dark matter and dark
energy model, where the parameters are set to $\epsilon=-0.25$, $q=1$,
$\mathcal{R}=0.05$ and $w_{x}=-0.5$. The reddish surface shows where the
acceleration is zero, with trajectories accelerating above this surface and
decelerating below it. The green trajectories show the flat models and the
black curves show the Closed Friedmann Separatrix (CFS). Open models are shown
in blue, and closed models are shown in purple. All trajectories are non-
singular, emerging from either an Einstein fixed point (cyan) or a de Sitter
fixed point (magenta). We only show the fixed points that are relevant to the
dynamics; the full list of fixed points can be seen in Table 4.

##  4 Conclusions

In this essay, we have carried out a dynamic systems analysis of the Emergent
Universe scenario in [1], which consists of a scalar field with an asymmetric
potential with a plateau for negative $\phi$. Originally, this scenario was
considered with specific initial conditions such that the universe emerged
from a static Einstein state. The aim of this essay was to understand whether
the past repellor was generically non-singular for any initial conditions. The
de Sitter model is normally associated to a cosmological constant associated
with an asymptotically stable future attractor [27, 28], but it can also
represent a past repelling fixed point [29, 25]. We found a class of models
which emerged from a non-singular de Sitter fixed point, however the stability
of this point is that of a 3-D saddle, so some level of fine-tuning is
required. All other trajectories expand from a singularity. We remark that
simply with the exchange $\phi\rightarrow-\phi$ the potential (1) for the
model in [1] becomes the potential for the Einstein-frame version [30] of
Starobinsky inflation [31], hence our analysis can easily be mapped into one
for that model. Models of this type are currently the most favored by
experimental results [32, 33].

With the aim of showing a scenario where emergence from a non-singular state
is generic during expansion, we then considered a fluid model with interacting
dark energy and dark matter. In this case, it is not necessary to have a
quantum gravity era at high energy to avoid a singularity. Instead we have a
classical de Sitter vacuum-dominated era at high energy from which our models
emerge from, which is a past attractor for flat and open models, or a
transition phase through a bounce for a subset of closed models. We also
highlighted the models of interest that evolve with an early- and late-time
acceleration connected by a decelerated period, which include open, flat and
closed models. Therefore, at least qualitatively, these models are consistent
with the observable Universe and emerge from a non-singular de Sitter state.

For FLRW models that have a singularity, it is well known that this is very
special in that it is matter dominated, while in more general GR models the
shear anisotropy becomes dominant in approaching the singularity, which is
said to be velocity dominated, e.g. see [34] and refs. therein. More in
general, especially for bouncing models with a contracting phase, the question
is if the singularity avoidance found in the FLRW case is stable against shear
anisotropy [35]. In a long-wavelength approximation to inhomogeneities [36],
this question can be investigated by generalising non-singular FLRW models
with a bounce to anisotropic Bianchi IX models, see [37] and refs. therein.
This will be the goal of our future investigation, generalising the non-
singular scenario presented here based on dark matter interacting with dark
energy, to study the stability of the results [26].

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