# Little Rip, Pseudo Rip and bounce cosmology with generalized equation of
state in non-standard backgrounds

Oem Trivedia<EMAIL_ADDRESS>Alexander V. Timoshkin b,c
<EMAIL_ADDRESS>
(aSchool of Arts and Sciences, Ahmedabad University, Ahmedabad 380009,India
bInstitute of Scientific Research and Development,Tomsk State Pedagogical
University (TSPU), 634061 Tomsk, Russia
c Lab. for Theor. Cosmology, International Centre of Gravity and Cosmos, Tomsk
State University of Control Systems and Radio Electronics (TUSUR), 634050
Tomsk, Russia
)

###### Abstract

The discovery of universe’s late-time acceleration and dark energy has
overseen a great deal of research into cosmological singularities and
particularly future singularities. Perhaps the most extreme of such
singlarities is the big rip, which has propelled a lot of work into ways of
moderating it or seeking out alternatives to it and two such alternatives to
the big rip are the Little rip and Pseudo rip. Another possibility to consider
the far future of the universe is through bounce cosmologies, which presents
its own interesting ideas. So in this work we investigate the Little rip,
Pseudo rip and Bounce cosmology in non-standard cosmological backgrounds with
a generalized equation of state in the presence of a viscous fluid. In
particular we discuss about Chern-Simons cosmology and the RS-II Braneworld
and discuss how the exotic and non-conventional nature of gravity in such
cosmologies affect universal evolution in these scenarios. We find out that
there are very significant differences in the behaviour of such cosmic
scenarios in these universes in comparison to how they appear in the simple
general relativistic universe.

##  1 Introduction

The surprising observation of the universe’s accelerated expansion during its
later stages has prompted extensive research within the field of cosmology
[1].Numerous approaches have been explored to explain this expansion, ranging
from conventional methods like the Cosmological constant [1, 2, 3, 4, 5] to
more exotic scenarios such as Modified gravity theories [6, 7, 8] and recent
proposals for direct detection of dark energy [9]. It is widely recognized
that a dark energy universe exhibits peculiar characteristics regarding
singularities in the distant future, even in scenarios where it arises from
modified gravity [10, 11, 12, 13, 14, 15]. Among the various models
investigated in the literature, there are those that treat dark energy and
dark matter as ideal (nonviscous) fluids with an unconventional equation of
state (EoS). General dark fluid models with an inhomogeneous EoS were
introduced in [16, 17, 18] .

Numerous cosmological scenarios exist for the evolution of the universe,
including the Big Rip [19, 12], the Little Rip [20, 16, 21, 22, 23, 24, 25] ,
the Pseudo Rip [26], and the Quasi Rip [27]. The analytical representation of
the cosmological models Little Rip and Pseudo Rip , in terms of the parameters
of the generalized equation of state without considering the bulk viscosity of
the dark fluid in flat space, was derived in previous articles. The
investigation of bounce cosmological models in the flat FRW metric,
considering the viscosity properties of a dark fluid, was conducted in another
work and reconstructions within the framework of modified gravity theories
have also been done [28, 16, 29, 30, 31, 17, 32, 33, 34]. Therefore, the
equation of state plays a significant role not only in the Little Rip, Pseudo
Rip, and bounce phenomena but also in other cosmological scenarios. Analytical
expressions for the spatial curvature, thermodynamic parameter, and bulk
viscosity in these models are obtained. Specifically, the parameters for the
early and late stages of the universe’s history are derived.

While astronomical measurements of the luminosity of remote objects using the
Planck satellite indicate that our universe is nearly flat [35], other studies
such as observations of galaxy distribution in space challenge this
conclusion. It could be a viable possibility that the universe could
eventually possess finite spatial curvature[36, 37] and so investigations that
consider the combined influence of bulk viscosity and spatial curvature to
describe the evolution of the universe have gathered interest. A lot of work
has been dedicated to studying singularities and their ultimate formation
time. However, singularities are not the only possible endpoints of the
universe’s evolution. The cosmological models of the late-time universe,
including the Little Rip, Pseudo Rip, and the bounce cosmology describing a
cyclic universe, as discussed in this article, are nonsingular. Hence recent
work explored the Little Rip, Pseudo Rip, and the bounce cosmological
scenarios in cosmologies with viscous fluids and discussed their properties in
detail [38]. Here we would like to go a step further and discuss these
scenarios in non-conventional cosmological paradigms, namely the RS-II
Braneworld [39, 40] and a form of cosmology in Chern-Simons gravity [41]. We
choose these paradigms as they are both inspired heavily by quantum
gravitational ideas and also represent first-order corrections to cosmology
due to extra-dimensional considerations. So it makes for an interesting
endeavor to probe such scenarios in these cosmologies and note any differences
that they bring out in these paradigms from the usual general relativistic
cosmology. In Section 2, we would briefly review the viscous fluid regime we
would be probing, and in Section 3, we would discuss both forms of rip and the
bounce scenario in these cosmologies. We would then conclude the work in
Section 4.

##  2  Overview of the cosmologies considered

Here we shall briefly discuss the viscous fluid cosmological scenario that we
are interested in this work. We are examining a universe that is filled with a
one-component ideal fluid known as dark energy, in the framework of the FRW
metric. This universe possesses a nonzero spatial curvature and is described
by the scale factor $a$.

Let’s express the energy conservation law as follows:

$\dot{\rho}+3H(\rho+p)=0.$ (1)

Here, $H=\dot{a}/a$ represents the Hubble rate, and $k^{2}=8\pi G$ where $G$
denotes Newton’s gravitational constant. The quantities $\rho$ and $p$
correspond to the energy density and pressure of the dark energy,
respectively. The dot notation signifies differentiation with respect to
cosmic time $t$.

We will now investigate the homogeneous and isotropic FRW expanding universe,
given by the metric:

$ds^{2}=-dt^{2}+a^{2}(t)\left(\frac{dr^{2}}{1-\Pi_{0}r^{2}}+r^{2}d\Omega^{2}\right),$
(2)

In this equation, $d\Omega^{2}=d\theta^{2}+\sin^{2}\theta d\varphi^{2}$, $t$
represents cosmic time, $a(t)$ represents the scale factor (with length
units), $r$ is the special radius coordinate, and $\Pi_{0}$ signifies the
spatial curvature of the three-dimensional space.

As we know, Eq. (1) geometrically describes various types of universes.
Assuming $\Pi_{0}$ to be dimensionless for simplicity, when $\Pi_{0}=0$, the
universe is spatially flat; for $\Pi_{0}=1$, it is closed; and for
$\Pi_{0}=-1$, it is open. The nature of the universe’s expansion depends on
the spatial curvature: an open universe will expand indefinitely, a flat
universe will also expand indefinitely but with a constant expansion velocity
as $t\rightarrow+\infty$, and a closed universe will expand until a certain
moment, after which the expansion will transition to compression, eventually
leading to a collapse.

To describe an inhomogeneous viscous fluid, we adopt the following equation of
state (EoS) [15]:

$p=\omega(\rho,t)\rho-3H\zeta(H,t)$ (3)

Here, $\zeta(H,t)$ represents the bulk viscosity, which depends on $H$ and
$t$. From conventional thermodynamics, we know that $\zeta(H,t)>0$. We assume
the EoS parameter $\omega$ to have the form [15]:

$\omega(\rho,t)=\omega_{1}(t)(A_{0}\rho^{\alpha-1}-1),$ (4)

In this equation, $A_{0}\neq 0$ and $\alpha\geq 1$ is a constant. Regarding
dimensions, since $\omega$ and $\omega_{1}$ are dimensionless, the dimension
of $A_{0}$ becomes complicated when $\alpha>1$. However, in the simplest case
of $\alpha=1$, $A_{0}$ becomes dimensionless. Consequently, we set
$\omega(\rho,t)=\omega_{0}$ as a constant.Dissipative processes are described
by the bulk viscosity in the form [15]:

$\zeta(H,t)=\zeta_{1}(t)(3H)^{n},$ (5)

Here, $n>0$ and $\zeta_{1}(t)$ is an arbitrary function dependent on time.We
will now explore cosmological models of the viscous fluid.

Furthermore the Little rip scenario we will consider is specified by the
following form for the Hubble parameter [21, 20]:

$H=H_{0}\exp(\lambda t),\quad H_{0}>0,\lambda>0.$ (6)

We will follow the development of the universe beginning at some time $t=0$
which will at first be left unspecified, except that it refers to an initial
instant in the very early universe. Its precise meaning will be dependent on
which model we consider. The symbol $H_{0}$ means the Hubble parameter at this
particular instant. The Pseudo Rip, proposed in [26], is a second variant of
the Little Rip. This interesting possible scenario is related to the LR
cosmology when the Hubble parameter tends to infinity in the remote future

$H(t)\rightarrow H_{\infty},\quad t\rightarrow+\infty.$ (7)

The pseudo rip scenario will also be specified by an ansatz for the Hubble
parameter, which will be given in this case [26]

$H=H_{0}-H_{1}\exp(-\lambda t),$ (8)

where $H_{0},H_{1}$ and $\lambda$ are positive constants, $H_{0}>H_{1}$, and
$t>0$. For more information on the Little and Pseudo rips, the reader can
refer to [16, 42, 43, 44, 45, 29, 46, 47, 48, 22, 49, 50]. In the case of
bounce cosmology, the universe goes from an era of accelerated collapse to the
expanding era without displaying a singularity. This is essentially a cyclic
universe model. After the bounce the universe soon enters a matter dominated
expansion phase [51, 52, 53]. For this we can consider two ansatz, the first
one where the scale factor $a(t)$ has an exponential form [52]:

$a(t)=a_{0}\exp[\alpha(t-t_{0})^{2n}],\quad n\in N,$ (9)

where $\alpha$ is a positive (dimensional) constant and $t_{0}$ is the instant
when bouncing occurs.In this case the Hubble parameter becomes

$H(t)=2n\alpha(t-t_{0})^{2n-1}.$ (10)

Another viable ansatz for a bounce scenario is where the scale factor has the
following form [52]:

$a(t)=a_{0}+\alpha(t-t_{0})^{2n},$ (11)

where $\alpha$ is dimensional constant, $n\in N$ and $t=t_{0}$ is a fixed
bounce time. The Hubble parameter in this case becomes

$H(t)=\frac{2n\alpha(t-t_{0})^{2n-1}}{a_{0}+\alpha(t-t_{0})^{2n}}.$ (12)

We will now investigate all of these scenarios in a universe with a viscous
fluid form of dark energy, in both the Chern-Simons cosmology and the RS-II
Braneworld cosmology.

##  3 Chern Simons cosmology

The Friedmann equation which we will be considering here is given by that we
will be covering is given by

$\frac{6}{k}\left(\frac{\Pi_{0}}{a^{2}}+H^{2}\right)+\alpha\left(\frac{\Pi_{0}}{a^{4}}+\frac{2H^{2}\Pi_{0}}{a^{2}}+H^{4}\right)=\rho$
(13)

where in the context of the Chern-Simons scenario, k is a constant which is
positive while $\alpha$ is also a parameter which is considered to be a
constant but is not necessarily positive but we would consider it to be
positive for our analysis for simplicity. The above equation is very special
in the sense that it can be derived from various distinct approaches. This
equation can be achieved by considering a quantum corrected entropy-area
relation of $S=\frac{A}{4}-\alpha\ln\left(\frac{A}{4}\right)$ where A is the
area of the apparent horizon and $\alpha$ is a dimensionless positive constant
determined by the conformal anomaly of the fields, where the conformal anomaly
is interpreted as a correction to the entropy of the apparent horizon [54] .
Also, this could be derived in terms of spacetime thermodynamics together with
a generalized uncertainly principle of quantum gravity[55]. This Friedmann
equation can also be derived by considering an anti-de Sitter-Schwarzschild
black hole via holographic renormalization with appropriate boundary
conditions [56]. Finally, a Chern-Simons type of theory can also yield this
Friedmann equation [41]. Hence the equation (13) can derived by a wide range
of approaches to gravitational physics and can be representative of the
effects of these different theories on the cosmological dynamics.

Our analysis will be based in the simplest case, when
$\omega(\rho,t)=\omega_{0}$ is a constant, and $\zeta(H,t)=\zeta_{0}$, with a
constant $\zeta_{0}>0$, for which the generalized equation of state takes the
form:

$p=\omega_{0}\rho-3\zeta_{0}H.$ (14)

Considering the Friedmann equation (13), we can write the derivative of the
energy density as

$\dot{\rho}=\frac{12H^{2}\lambda\left(\alpha\left(\frac{\Pi_{0}}{a^{2}}+2H^{2}\right)+1\right)}{k}$
(15)

As we would firstly like to discuss the Little rip scenario, using the ansatz
(6), we have the continuity equation as

$3H\left(\frac{4H\lambda\left(\frac{\alpha\Pi_{0}}{a^{2}}+2\alpha
h^{2}+1\right)}{k}+\frac{(w+1)\left(a^{2}H^{2}+\Pi_{0}\right)\left(a^{2}\left(\alpha
H^{2}k+6\right)+\alpha k\Pi_{0}\right)}{a^{4}k}-3\zeta_{0}H\right)=0$ (16)

In the limit $t\to 0$, we have

$w=-\frac{8\alpha gH_{0}^{2}+\alpha
H_{0}^{3}k-3\zeta_{0}k+4\lambda+6H_{0}}{\alpha H_{0}^{3}k+6H_{0}}$ (17)

In the limit $t\to\infty$, we have $w\to-1$ for all positive $\lambda$ . So in
the very late universe in this case, we see pure de Sitter behaviour
irrespective of anything. In the case $\Pi_{0}\neq 0$, we have the w parameter

$w=\frac{-8a^{4}\alpha\lambda H^{3}-a^{4}\alpha
H^{4}k+3a^{4}\zeta_{0}Hk-4a^{4}\lambda H-6a^{4}H^{2}-4a^{2}\alpha\lambda
H\Pi_{0}-2a^{2}\alpha H^{2}k\Pi_{0}-6a^{2}\Pi_{0}-\alpha
k\Pi_{0}^{2}}{\left(a^{2}H^{2}+\Pi_{0}\right)\left(a^{2}\alpha
H^{2}k+6a^{2}+\alpha k\Pi_{0}\right)}$ (18)

Using the ansatz, we have

$w=-1-\frac{a^{2}H_{0}e^{\lambda t}\left(a^{2}\left(\lambda\left(8\alpha
H_{0}^{2}e^{2\lambda
t}+4\right)-3\zeta_{0}k\right)+4\alpha\lambda\Pi_{0}\right)}{\left(a^{2}H_{0}^{2}e^{2\lambda
t}+\Pi_{0}\right)\left(a^{2}\left(\alpha H_{0}^{2}ke^{2\lambda
t}+6\right)+\alpha k\Pi_{0}\right)}$ (19)

In the far future limit $t\to\infty$ we have $w\to-1$, pure de Sitter
behaviour. In the limit $t\to 0$, we have

$w=-1-\frac{a^{2}H_{0}\left(a^{2}\left(\lambda\left(8\alpha
H_{0}^{2}+4\right)-3\zeta_{0}k\right)+4\alpha\lambda\Pi_{0}\right)}{\left(a^{2}H_{0}^{2}+\Pi_{0}\right)\left(a^{2}\left(\alpha
H_{0}^{2}k+6\right)+\alpha k\Pi_{0}\right)}$ (20)

In either case of non flat or flat universe, we have the universe tending
towards neither a Quintessence nor a phantom behaviour in the far future
limit, something which is not observed in the case of the conventional
cosmology [38].

The continuity equation in the case for the pseudo rip with
$H(t)=H_{0}-H_{1}\exp(-\lambda t)$ will be given by

$\frac{4\lambda\mathrm{h}_{1}e^{-4\lambda t}\left(H_{0}e^{\lambda
t}-H_{1}\right)\left(a^{2}\left(e^{2\lambda t}\left(\alpha
H_{0}^{2}k+3\right)-2\alpha H_{0}H_{1}ke^{\lambda t}+\alpha
H_{1}^{2}k\right)+\alpha k\Pi_{0}e^{2\lambda t}\right)}{a^{2}k}+\\\
3\left(H_{0}-H_{1}e^{-\lambda
t}\right)\left((w+1)\left(\frac{6\left(\frac{\Pi_{0}}{a^{2}}+\left(H_{0}-H_{1}e^{-\lambda
t}\right)^{2}\right)}{k}+\frac{\alpha e^{-4\lambda
t}\left(a^{2}\left(H_{1}-H_{0}e^{\lambda t}\right)^{2}+\Pi_{0}e^{2\lambda
t}\right)^{2}}{a^{4}}\right)\right)-\\\ 3\left(H_{0}-H_{1}e^{-\lambda
t}\right)(3\epsilon_{0}\left(H_{0}-H_{1}e^{-\lambda t}\right))=0$ (21)

This gives us the equation of state parameter as,

$\begin{split}w=-\frac{1}{3\left(a^{2}\left(H_{1}-H_{0}e^{\lambda
t}\right)^{2}+\Pi_{0}e^{2\lambda t}\right)\left(a^{2}\left(e^{2\lambda
t}\left(\alpha H_{0}^{2}k+6\right)-2\alpha H_{0}H_{1}ke^{\lambda t}+\alpha
H_{1}^{2}k\right)+\alpha k\Pi_{0}e^{2\lambda t}\right)}\\\
\times\Bigg{[}4a^{4}\alpha H_{1}^{3}k(\lambda-3H_{0})e^{\lambda
t}+3a^{4}\alpha H_{1}^{4}k+a^{2}H_{1}e^{3\lambda t}\left(a^{2}\left(4\alpha
H_{0}^{2}k(\lambda-3H_{0})+9(\zeta_{0}k-4H_{0})+12\lambda\right)+\right.\\\
\left.4\alpha k\Pi_{0}(\lambda-3H_{0})\right)-2a^{2}H_{1}^{2}e^{2\lambda
t}\left(a^{2}(\alpha H_{0}k(4\lambda-9H_{0})-9)-3\alpha
k\Pi_{0}\right)+3e^{4\lambda t}\left(a^{4}H_{0}\left(\alpha
H_{0}^{3}k-3\zeta_{0}k+6H_{0}\right)+\right.\\\
\left.2a^{2}\Pi_{0}\left(\alpha H_{0}^{2}k+3\right)+\alpha
k\Pi_{0}^{2}\right)\Bigg{]}\end{split}$ (22)

For $\Pi_{0}=0$, we have in the $t\to 0$ limit

$w=\frac{(H_{0}-H_{1})\left(-\alpha k(H_{0}-H_{1})\left(4\lambda
H_{1}+3(H_{0}-H_{1})^{2}\right)+9\zeta_{0}k+18(H_{1}-H_{0})\right)-12\lambda
H_{1}}{3(H_{0}-H_{1})^{2}\left(\alpha k(H_{0}-H_{1})^{2}+6\right)}$ (23)

While in the far-future limit, we have

$w=-1-\frac{3\zeta_{0}k}{\alpha H_{0}^{3}k+6H_{0}}$ (24)

Here we see that similar to the case of the conventional cosmology, in the far
future the phantom regime will take over in this case even for a Chern Simons
cosmology. Finally, the bulk viscosivity parameter in this case is given as

$\zeta_{0}=\frac{\frac{4H_{1}\lambda e^{-4\lambda t}\left(H_{0}e^{\lambda
t}-H_{1}\right)\left(a^{2}\left(e^{2\lambda t}\left(\alpha
H_{0}^{2}k+3\right)-2\alpha H_{0}H_{1}ke^{\lambda t}+\alpha
H_{1}^{2}k\right)+\alpha k\text{$\pi$0}e^{2\lambda
t}\right)}{a^{2}k}}{9\left(H_{0}-H_{1}e^{\lambda(-t)}\right)^{2}}+\\\
\frac{3(w+1)\left(H_{0}-H_{1}e^{\lambda(-t)}\right)\left(\frac{6\left(\frac{\text{$\pi$0}}{a^{2}}+\left(H_{0}-H_{1}e^{\lambda(-t)}\right)^{2}\right)}{k}+\frac{\alpha
e^{-4\lambda t}\left(a^{2}\left(H_{1}-H_{0}e^{\lambda
t}\right)^{2}+\text{$\pi$0}e^{2\lambda
t}\right)^{2}}{a^{4}}\right)}{9\left(H_{0}-H_{1}e^{\lambda(-t)}\right)^{2}}$
(25)

We see that the viscosity parameter is influenced by the value of the
curvature $\Pi_{0}$ very much, just like it is in the case of the conventional
cosmology as well.

Now we discuss the case of bounce cosmology, where we have for the exponential
ansatz $a(t)=a_{0}\exp\left(\alpha(t-t_{0})^{2n}\right)$ the continuity
equation as

$\displaystyle-4\left(2\alpha
n(t-t_{0})^{2n}\left(-2a_{0}n+a_{0}+a_{1}(t-t_{0})^{2n}\right)+p(t-t_{0})^{2}\right)$
$\displaystyle\quad\times\left(3a_{0}^{2}(t-t_{0})^{2}+6a_{0}\alpha(t-t_{0})^{2n+2}\right.$
$\displaystyle\quad\quad+\alpha\left(k\left(4\alpha^{2}n^{2}(t-t_{0})^{4n}+p(t-t_{0})^{2}\right)+3\alpha(t-t_{0})^{4n+2}\right)\bigg{)}$
$\displaystyle\quad+3(w+1)$
$\displaystyle\quad\times\left(6(t-t_{0})^{4}\left(a_{0}+a_{1}(t-t_{0})^{2n}\right)^{2}\right.$
$\displaystyle\quad\quad\left.\times\left(4a_{1}^{2}n^{2}(t-t_{0})^{4n-2}+\Pi_{0}\right)+a_{1}k\left(4a_{1}^{2}n^{2}(t-t_{0})^{4n}+\Pi_{0}(t-t_{0})^{2}\right)^{2}\right)$
$\displaystyle\quad-18\alpha
fkn(t-t_{0})^{2n+3}\left(a_{0}+\alpha(t-t_{0})^{2n}\right)^{3}=0$ (26)

The expressions for the equation of state parameter and the viscosity
parameter are awfully long in this case but we can still discuss some
important issues here. Taking $n=1$, we have the EOS parameter at the time of
bounce $t->t_{0}$ as

$w=\frac{-24a_{0}^{3}a_{1}-6a_{0}^{2}\Pi_{0}-8\alpha a_{0}a_{1}k\Pi_{0}+\alpha
k\Pi_{0}^{2}}{18a_{0}^{2}\Pi_{0}+3\alpha k\Pi_{0}^{2}}$ (27)

We see that for the case of the flat universe (27) blows up, which is similar
to what happens in the same case for the conventional cosmology [38]. While in
the case of the open universe, $\Pi=-1$, we have

$w=\frac{4a_{0}^{2}(2a_{0}a_{1}+1)}{\alpha
k-6a_{0}^{2}}+\frac{8\text{a0}a_{1}}{3}+\frac{1}{3}$ (28)

which shows that in this case $w>0$ while for the case of the the closed
universe

$w=\frac{1}{3}-\frac{4a_{0}^{2}}{6a_{0}^{2}+\alpha
k}-\frac{24a_{0}^{3}a_{1}+8\alpha a_{0}a_{1}k}{18a_{0}^{2}+3\alpha k}$ (29)

In the usual form of cosmology, when one considers the case of non-flat
universes one gets $w\to-1$ but we see that in the case of the Chern Simons
scenario that is not the case.

Considering now the power law bounce model with the ansatz (11), we have the
continuity equation as

$-4\left(2na_{1}(t-t_{0})^{2n}\left(-2a_{0}n+a_{0}+a_{1}(t-t_{0})^{2n}\right)+\Pi_{0}(t-t_{0})^{2}\right)\left(3a_{0}^{2}(t-t_{0})^{2}+6a_{0}a_{1}(t-t_{0})^{2n+2}\right.\\\
+\alpha
k\left(4n^{2}a_{1}^{2}(t-t_{0})^{4n}+\Pi_{0}(t-t_{0})^{2}\right)+3a_{1}^{2}(t-t_{0})^{4n+2}\bigg{)}+3(w+1)\\\
\bigg{(}6(t-t_{0})^{4}\left(a_{0}+a_{1}(t-t_{0})^{2n}\right)^{2}\left(4n^{2}a_{1}^{2}(t-t_{0})^{4n-2}\right)+\left.\Pi_{0}\right)+\alpha
k\left(4n^{2}a_{1}^{2}(t-t_{0})^{4n}+\Pi_{0}(t-t_{0})^{2}\right)^{2}\bigg{)}-\\\
18\zeta_{0}kna_{1}(t-t_{0})^{2n+3}\left(a_{0}+a_{1}(t-t_{0})^{2n}\right)^{3}=0$
(30)

From which we can get the equation of state parameter and the viscosity
parameter as

$\displaystyle w$
$\displaystyle=-1+\frac{6\zeta_{0}kna_{1}(t-t_{0})^{2n+3}\left(a_{0}+a_{1}(t-t_{0})^{2n}\right)^{3}}{6(t-t_{0})^{4}\left(a_{0}+a_{1}(t-t_{0})^{2n}\right)^{2}\left(4n^{2}a_{1}^{2}(t-t_{0})^{4n-2}+\Pi_{0}\right)+\alpha
k\left(4n^{2}a_{1}^{2}(t-t_{0})^{4n}+\Pi_{0}(t-t_{0})^{2}\right)^{2}}$
$\displaystyle\quad+\frac{1}{3\left(6(t-t_{0})^{4}\left(a_{0}+a_{1}(t-t_{0})^{2n}\right)^{2}\left(4n^{2}a_{1}^{2}(t-t_{0})^{4n-2}+\Pi_{0}\right)+\alpha
k\left(4n^{2}a_{1}^{2}(t-t_{0})^{4n}+\Pi_{0}(t-t_{0})^{2}\right)^{2}\right)}$
$\displaystyle\times\Bigg{[}4\left(2na_{1}(t-t_{0})^{2n}\left(-2a_{0}n+\left(a_{0}+a_{1}(t-t_{0})^{2n}\right)\right)+\Pi_{0}(t-t_{0})^{2}\right)$
$\displaystyle\quad\times\left((t-t_{0})^{2}\left(3a_{0}^{2}+\alpha
k\Pi_{0}\right)+6a_{0}a_{1}(t-t_{0})^{2n+2}+a_{1}^{2}(t-t_{0})^{4n}\left(4\alpha
kn^{2}+3(t-t_{0})^{2}\right)\right)\Bigg{]}$ (31)

For the case of flat space $\Pi_{0}=0$, we have

$w=-1+\frac{3a_{1}^{3}(t-t_{0})^{2}(3\zeta_{0}k(t-t_{0})-8n+4)+3a_{0}^{2}a_{1}(t-t_{0})^{2n+2}(9\zeta_{0}k(t-t_{0})-16n+12)}{36a_{0}^{2}a_{1}n(t-t_{0})^{2n+2}+72a_{0}a_{1}^{2}n(t-t_{0})^{4n+2}+12a_{1}^{3}n(t-t_{0})^{6n}\left(2\alpha
kn^{2}+3(t-t_{0})^{2}\right)}+\\\
\frac{a_{0}a_{1}^{2}(t-t_{0})^{4n}\left(16\alpha
kn^{2}(1-2n)+27\zeta_{0}k(t-t_{0})^{3}-12(2n-3)(t-t_{0})^{2}\right)}{36a_{0}^{2}a_{1}n(t-t_{0})^{2n+2}+72a_{0}a_{1}^{2}n(t-t_{0})^{4n+2}+12a_{1}^{3}n(t-t_{0})^{6n}\left(2\alpha
kn^{2}+3(t-t_{0})^{2}\right)}\\\ \frac{a_{1}^{3}(t-t_{0})^{6n}\left(16\alpha
kn^{2}+3(t-t_{0})^{2}(3\zeta_{0}k(t-t_{0})+4)\right)}{36a_{0}^{2}a_{1}n(t-t_{0})^{2n+2}+72a_{0}a_{1}^{2}n(t-t_{0})^{4n+2}+12a_{1}^{3}n(t-t_{0})^{6n}\left(2\alpha
kn^{2}+3(t-t_{0})^{2}\right)}$ (32)

The important thing to note is that the equation of state parameter for both
(3) and (32) diverges at the time of the bounce $t\to t_{0}$. Such a blow up
of the w parameter is also seen in the conventional cosmology, but only for
the flat space case but interestingly for the Chern-Simons scenario this is
also happening for the non-flat case too.

##  4 RS-II Braneworld

The RS-II model is founded on a modification of the RS-I Braneworld cosmology
model [39]. In the RS-I model, the hierarchy problem is addressed by embedding
two 3-branes within a five-dimensional bulk, with one of these branes housing
the Standard Model particles. In contrast, the RS-II braneworld cosmology
simplifies the model by eliminating one of the 3-branes and successfully
recovering both Newtonian gravity and General Relativity as its limiting cases
[40].

Given the potential impact of other braneworld cosmology scenarios, such as
the Dvali-Gabadadze-Porrati (DGP) model [57, 58], on the late-time evolution
of the universe, it becomes intriguing to explore whether the RS-II model can
also address issues related to late-time acceleration. In recent times,
several studies have delved into the topic of Quintessence within the RS-II
Braneworld framework [59, 60, 61].

We can express the complete action of the RS-II Model, incorporating the
background fluid terms too, as follows:

$S=S_{RS}+S_{B}=\int
d^{5}x\sqrt{-g^{(5)}}\left(\Lambda^{(5)}+2R^{(5)}\right)+\int
d^{4}x\sqrt{-g}\left(\omega+\mathcal{L}_{B}\right)$ (33)

Here, $R^{(5)}$, $g^{(5)}_{\mu\nu}$, and $\Lambda^{(5)}$ denote the bulk Ricci
Scalar, the metric of the five-dimensional bulk, and the cosmological constant
in that bulk, respectively. Additionally, $\omega$ represents the brane
tension on the 3-brane, while $g_{\mu\nu}$ stands for the metric on the
3-brane.

If one wants to consider, for example, an RS-II scenario with scalar field
dark energy then the action would be

$S=S_{RS}+S_{B}+S_{\phi}=\int
d^{5}x\sqrt{-g^{(5)}}\left(\Lambda^{(5)}+2R^{(5)}\right)+\int
d^{4}x\sqrt{-g}\left(\lambda-\frac{1}{2}\mu(\phi)(\nabla\phi)^{2}-V(\phi)+\mathcal{L}_{B}\right)$
(34)

with $\mu(\phi)$ being a scalar coupling function, but we are not restricting
the cosmology to a scalar field dark energy scenario in any case. The
Friedmann equation in this case for an FLRW background inclusive of the
curvature term would be

$H^{2}=\frac{k^{2}}{3}\left(\rho+\frac{\rho}{3}\right)-\frac{\Pi_{0}}{a^{2}}$
(35)

where k is again a positive term. dependent on the five dimensional Planck
mass in this case $m_{p}^{(5)}$. The energy density in this case can then be
written as

$\rho=\sqrt{\frac{6\sigma\left(\frac{\Pi_{0}}{a^{2}}+H^{2}\right)}{k^{2}}+\sigma^{2}}-\sigma$
(36)

And the continuity equation (3) can be written as

$3H\left(\sqrt{\sigma\left(\frac{6\left(a^{2}H^{2}+\Pi_{0}\right)}{a^{2}k^{2}}+\sigma\right)}\left(\frac{2a^{2}H\lambda}{a^{2}\left(\sigma
k^{2}+6H^{2}\right)+6\Pi_{0}}+w+1\right)-b(w+1)-3\zeta_{0}H\right)=0$ (37)

In the case of flat space we can then write

$w=-1+\frac{\frac{3\zeta_{0}k\left(\sqrt{\sigma\left(\sigma
k^{2}+6H^{2}\right)}+\sigma k\right)}{\sigma}+2\lambda\left(-\frac{\sigma
k}{\sqrt{\sigma\left(\sigma k^{2}+6H^{2}\right)}}-1\right)}{6H}$ (38)

This in the $t\to 0$ limit gives us

$w=-1+\frac{\frac{3\zeta_{0}k\left(\sqrt{\sigma\left(\sigma
k^{2}+6H_{0}^{2}\right)}+\sigma k\right)}{\sigma}+2\lambda\left(-\frac{\sigma
k}{\sqrt{\sigma\left(\sigma k^{2}+6H_{0}^{2}\right)}}-1\right)}{6H_{0}}$ (39)

While in the far future limit $t\to\infty$ we have

$w=-1+\frac{\sqrt{\frac{3}{2}}\zeta_{0}k}{\sqrt{\sigma}}$ (40)

Here we start to immediately see the stark difference provided by the RS II
cosmology from what the Chern Simons and conventional GR cosmology provided us
in the similar case. In the similar case for the conventional cosmology, one
saw a phantom evolution $w<-1$ while for Chern Simons one saw a pure dS
evolution but for the RS II braneworld, we see that here we will have a
quintessence type evolution which is different from both the other scenarios.
Moving further and talking about the non flat universe, we have for
$\Pi_{0}\neq 0$

$w=\frac{-\frac{3H\sqrt{\sigma\left(\frac{6\left(a^{2}H^{2}+\Pi_{0}\right)}{a^{2}k^{2}}+\sigma\right)}\left(a^{2}\left(\sigma
k^{2}+2H(3H+\lambda)\right)+6\Pi_{0}\right)}{a^{2}\left(\sigma
k^{2}+6H^{2}\right)+6\Pi_{0}}+3\sigma
H+9\zeta_{0}H^{2}}{3H\sqrt{\sigma\left(\frac{6\left(a^{2}H^{2}+\Pi_{0}\right)}{a^{2}k^{2}}+\sigma\right)}-3\sigma
H}$ (41)

In the limit $t\to 0$, we have

$w=\frac{-\frac{\sqrt{\sigma\left(\frac{6\left(a^{2}H_{0}^{2}+\Pi_{0}\right)}{a^{2}k^{2}}+\sigma\right)}\left(a^{2}\left(\sigma
k^{2}+2H_{0}(3H_{0}+\lambda)\right)+6\Pi_{0}\right)}{a^{2}\left(\sigma
k^{2}+6H_{0}^{2}\right)+6\Pi_{0}}+\sigma+3\zeta_{0}H_{0}}{\sqrt{\sigma\left(\frac{6\left(a^{2}H_{0}^{2}+\Pi_{0}\right)}{a^{2}k^{2}}+\sigma\right)}-\sigma}$
(42)

While in the far future limit $t\to\infty$ we have

$w=-1+\zeta_{0}k\sqrt{\frac{3}{2\sigma}}$ (43)

In the far future limit, even for the non flat space we have the universe
tending towards a quintessence evolution, again something which we didnt
observe in the Chern Simons theory.

For the pseudo rip with $H(t)=H_{0}-H_{1}\exp(-\lambda t)$, we have the
conservation law as

$3\left(H_{0}-H_{1}e^{-\lambda
t}\right)\left((w+1)\left(\sqrt{\sigma\left(\frac{6\left(\frac{\Pi_{0}}{a^{2}}+\left(H_{0}-H_{1}e^{-\lambda
t}\right)^{2}\right)}{k^{2}}+\sigma\right)}-\sigma\right)\right.\\\
\left.\frac{2\sigma\lambda H_{1}e^{-\lambda
t}}{k^{2}\sqrt{\sigma\left(\frac{6\left(\frac{\Pi_{0}}{a^{2}}+\left(H_{0}-H_{1}e^{-\lambda
t}\right)^{2}\right)}{k^{2}}+\sigma\right)}}-3\zeta_{0}\left(H_{0}-H_{1}e^{-\lambda
t}\right)\right)=0$ (44)

In the case of $\Pi_{0}=0$, we have

$w=-1+\frac{3\zeta_{0}\left(H_{0}-H_{1}e^{-\lambda
t}\right)}{\sqrt{\sigma\left(\sigma+\frac{6\left(H_{0}-H_{1}e^{-\lambda
t}\right)^{2}}{k^{2}}\right)}-\sigma}-\frac{2\sigma\lambda H_{1}e^{-\lambda
t}}{k^{2}\left(\sqrt{\sigma\left(\sigma+\frac{6\left(H_{0}-H_{1}e^{-\lambda
t}\right)^{2}}{k^{2}}\right)}-\sigma\right)\sqrt{\sigma\left(\sigma+\frac{6\left(H_{0}-H_{1}e^{-\lambda
t}\right)^{2}}{k^{2}}\right)}}$ (45)

In the limit of the far future, we have

$w=\frac{3\zeta_{0}H_{0}}{\sqrt{\sigma\left(\sigma+\frac{6H_{0}^{2}}{k^{2}}\right)}-\sigma}-1$
(46)

In this case, we see that as opposed to what happens in the conventional
cosmology, one sees here that a quintessence phase of evolution is to happen
for sure in the late universe. Furthermore, in the limit of $t\to 0$, we have

$w=-1\frac{3\zeta_{0}(H_{0}-H_{1})}{\sqrt{\sigma\left(\sigma+\frac{6(H_{0}-H_{1})^{2}}{k^{2}}\right)}-\sigma}-\frac{2\sigma\lambda
H_{1}}{k^{2}\left(\sqrt{\sigma\left(\sigma+\frac{6(H_{0}-H_{1})^{2}}{k^{2}}\right)}-\sigma\right)\sqrt{\sigma\left(\sigma+\frac{6(H_{0}-H_{1})^{2}}{k^{2}}\right)}}$
(47)

For the non-flat universe, we similarly have

$w=-1+\frac{3\zeta_{0}\left(H_{0}-H_{1}e^{-\lambda
t}\right)}{\sqrt{\sigma\left(\frac{6\left(\frac{\Pi_{0}}{a^{2}}+\left(H_{0}-H_{1}e^{-\lambda
t}\right)^{2}\right)}{k^{2}}+\sigma\right)}-\sigma}-\\\ \frac{2\sigma\lambda
H_{1}e^{-\lambda
t}}{k^{2}\sqrt{\sigma\left(\frac{6\left(\frac{\Pi_{0}}{a^{2}}+\left(H_{0}-H_{1}e^{-\lambda
t}\right)^{2}\right)}{k^{2}}+\sigma\right)}\left(\sqrt{\sigma\left(\frac{6\left(\frac{\Pi_{0}}{a^{2}}+\left(H_{0}-H_{1}e^{-\lambda
t}\right)^{2}\right)}{k^{2}}+\sigma\right)}-\sigma\right)}$ (48)

In the $t\to 0$ limit, we have

$w=-1+\frac{3\zeta_{0}(H_{0}-H_{1})}{\sqrt{\sigma\left(\frac{6\left(\frac{\Pi_{0}}{a^{2}}+(H_{0}-H_{1})^{2}\right)}{k^{2}}+\sigma\right)}-\sigma}-\\\
\frac{2\sigma\lambda
H_{1}}{k^{2}\sqrt{\sigma\left(\frac{6\left(\frac{\Pi_{0}}{a^{2}}+(H_{0}-H_{1})^{2}\right)}{k^{2}}+\sigma\right)}\left(\sqrt{\sigma\left(\frac{6\left(\frac{\Pi_{0}}{a^{2}}+(H_{0}-H_{1})^{2}\right)}{k^{2}}+\sigma\right)}-\sigma\right)}$
(49)

While in the far future,

$w=\frac{3\zeta_{0}H_{0}}{\sqrt{\sigma\left(\frac{6\left(\frac{\Pi_{0}}{a^{2}}+H_{0}^{2}\right)}{k^{2}}+\sigma\right)}-\sigma}-1$
(50)

Again, we see that a quintessence phase will occur in the pseudo rip case even
for the non-flat universe. Finally, we have the $\zeta_{0}$ parameter as

$\zeta_{0}=\frac{(w+1)\left(\sqrt{\sigma\left(\frac{6\left(\frac{\Pi_{0}}{a^{2}}+\left(H_{0}-H_{1}e^{-\lambda
t}\right)^{2}\right)}{k^{2}}+\sigma\right)}-\sigma\right)+\frac{2\sigma\lambda
H_{1}e^{-\lambda
t}}{k^{2}\sqrt{\sigma\left(\frac{6\left(\frac{\Pi_{0}}{a^{2}}+\left(H_{0}-H_{1}e^{-\lambda
t}\right)^{2}\right)}{k^{2}}+\sigma\right)}}}{3\left(H_{0}-H_{1}e^{-\lambda
t}\right)}$ (51)

For the power law model $a(t)=a_{0}+a_{1}(t-t_{0})^{2n}$ we have

$(w+1)(t-t_{0})^{2}\left(a_{0}+a_{1}(t-t_{0})^{2n}\right)^{2}\left(\sqrt{\sigma\left(\frac{6\left(4a_{1}^{2}n^{2}(t-t_{0})^{4n}+\Pi_{0}(t-t_{0})^{2}\right)}{k^{2}(t-t_{0})^{2}\left(a_{0}+a_{1}(t-t_{0})^{2n}\right)^{2}}+\sigma\right)}-\sigma\right)-\\\
\frac{2\sigma\left(2a_{1}n(t-t_{0})^{2n}\left(-2a_{0}n+a_{0}+a_{1}(t-t_{0})^{2n}\right)+\Pi_{0}(t-t_{0})^{2}\right)}{k^{2}\sqrt{\sigma\left(\frac{6\left(4a_{1}^{2}n^{2}(t-t_{0})^{4n}+\Pi_{0}(t-t_{0})^{2}\right)}{k^{2}(t-t_{0})^{2}\left(a_{0}+a_{1}(t-t_{0})^{2n}\right)^{2}}+\sigma\right)}}\\\
-6a_{1}\zeta_{0}n(t-t_{0})^{2n+1}\left(a_{0}+a_{1}(t-t_{0})^{2n}\right)=0$
(52)

The equation of state parameter in this case comes out to be for $\Pi_{0}\neq
0$

$w=-1+\frac{6a_{1}\zeta_{0}n(t-t_{0})^{2n-1}}{\left(a_{0}+a_{1}(t-t_{0})^{2n}\right)\left(\sqrt{\sigma\left(\frac{6\left(4a_{1}^{2}n^{2}(t-t_{0})^{4n}+\Pi_{0}(t-t_{0})^{2}\right)}{k^{2}(t-t_{0})^{2}\left(a_{0}+a_{1}(t-t_{0})^{2n}\right)^{2}}+\sigma\right)}-\sigma\right)}+\\\
2\sigma\left(2a_{1}n(t-t_{0})^{2n}\left(-2a_{0}n+a_{0}+a_{1}(t-t_{0})^{2n}\right)+\Pi_{0}(t-t_{0})^{2}\right)\\\
\frac{1}{k^{2}(t-t_{0})^{2}\left(a_{0}+a_{1}(t-t_{0})^{2n}\right)^{2}\left(\sqrt{\sigma\left(\frac{6\left(4a_{1}^{2}n^{2}(t-t_{0})^{4n}+\Pi_{0}(t-t_{0})^{2}\right)}{k^{2}(t-t_{0})^{2}\left(a_{0}+a_{1}(t-t_{0})^{2n}\right)^{2}}+\sigma\right)}-\sigma\right)}\\\
\frac{1}{\sqrt{\sigma\left(\frac{6\left(4a_{1}^{2}n^{2}(t-t_{0})^{4n}+\Pi_{0}(t-t_{0})^{2}\right)}{k^{2}(t-t_{0})^{2}\left(a_{0}+a_{1}(t-t_{0})^{2n}\right)^{2}}+\sigma\right)}}$
(53)

In the limit of $t\to t_{0}$ we have

$w=-1+\frac{2a_{0}^{2}\Pi_{0}}{a_{0}^{2}k^{2}\left(\sigma-\sqrt{\sigma\left(\frac{6\Pi_{0}}{a_{0}^{2}k^{2}}+\sigma\right)}\right)+6\Pi_{0}}$
(54)

Again we see that instead of the phantom regime as was seen in the case of the
conventional cosmology, we have a quintessence regime happening instead. Also,
we notice that for the flat universe $\Pi_{0}\to 0$, one finds that at the
time of the bounce $w\to-1$ while in the simple cosmology w diverges in this
scenario. Finally for the sake of completeness we also write the viscosity
parameter here to be

$\zeta_{0}=\frac{(t-t_{0})^{-2n-1}\left((w+1)(t-t_{0})^{2}\left(a_{0}+a_{1}(t-t_{0})^{2n}\right)^{2}\left(\sqrt{\sigma\left(\frac{6\left(4a_{1}^{2}n^{2}(t-t_{0})^{4n}+\Pi_{0}(t-t_{0})^{2}\right)}{k^{2}(t-t_{0})^{2}\left(a_{0}+a_{1}(t-t_{0})^{2n}\right)^{2}}+\sigma\right)}-\sigma\right)\right)}{6a_{1}n\left(a_{0}+a_{1}(t-t_{0})^{2n}\right)}-\\\
\frac{-\frac{2\sigma\left(2a_{1}n(t-t_{0})^{2n}\left(-2a_{0}n+a_{0}+a_{1}(t-t_{0})^{2n}\right)+\Pi_{0}(t-t_{0})^{2}\right)}{k^{2}\sqrt{\sigma\left(\frac{6\left(4a_{1}^{2}n^{2}(t-t_{0})^{4n}+\Pi_{0}(t-t_{0})^{2}\right)}{k^{2}(t-t_{0})^{2}\left(a_{0}+a_{1}(t-t_{0})^{2n}\right)^{2}}+\sigma\right)}}}{6a_{1}n\left(a_{0}+a_{1}(t-t_{0})^{2n}\right)}$
(55)

For the exponential model we have
$a(t)=a_{0}\exp\left(a_{1}(t-t_{0})^{2n}\right)$, and for this we have the
continuity equation as

$6a_{2}k^{2}n(t-t_{0})^{2n-1}\sqrt{\sigma\left(\frac{6\left(\frac{\Pi_{0}e^{-2a_{2}(t-t_{0})^{2n}}}{a_{0}^{2}}+4a_{2}^{2}n^{2}(t-t_{0})^{4n-2}\right)}{k^{2}}+\sigma\right)}\\\
\left((w+1)\left(\sqrt{\sigma\left(\frac{6\left(\frac{\Pi_{0}e^{-2a_{2}(t-t_{0})^{2n}}}{a_{0}^{2}}+4a_{2}^{2}n^{2}(t-t_{0})^{4n-2}\right)}{k^{2}}+\sigma\right)}-\sigma\right)-6a_{2}\zeta_{0}n(t-t_{0})^{2n-1}\right)+\\\
3\sigma\left(8a_{2}^{2}n^{2}(2n-1)(t-t_{0})^{4n-3}-\frac{4a_{2}n\Pi_{0}(t-t_{0})^{2n-1}e^{-2a_{2}(t-t_{0})^{2n}}}{a_{0}^{2}}\right)=0$
(56)

We have the equation of state parameter as

$w=-1+\frac{6a_{2}\zeta_{0}n(t-t_{0})^{2n-1}}{\sqrt{\sigma\left(\frac{6\left(\frac{\Pi_{0}e^{-2a_{2}(t-t_{0})^{2n}}}{a_{0}^{2}}+4a_{2}^{2}n^{2}(t-t_{0})^{4n-2}\right)}{k^{2}}+\sigma\right)}-\sigma}+\\\
2\sigma
e^{-2a_{2}(t-t_{0})^{2n}}\left(\Pi_{0}(t-t_{0})^{2}-2a_{0}^{2}a_{2}n(2n-1)(t-t_{0})^{2n}e^{2a_{2}(t-t_{0})^{2n}}\right)\\\
\Bigg{[}\frac{1}{a_{0}^{2}k^{2}(t-t_{0})^{2}\left(\sqrt{\sigma\left(\frac{6\left(\frac{\Pi_{0}e^{-2a_{2}(t-t_{0})^{2n}}}{a_{0}^{2}}+4a_{2}^{2}n^{2}(t-t_{0})^{4n-2}\right)}{k^{2}}+\sigma\right)}-\sigma\right)}\\\
\frac{1}{\sqrt{\sigma\left(\frac{6\left(\frac{\Pi_{0}e^{-2a_{2}(t-t_{0})^{2n}}}{a_{0}^{2}}+4a_{2}^{2}n^{2}(t-t_{0})^{4n-2}\right)}{k^{2}}+\sigma\right)}}\Bigg{]}$
(57)

For the case of a flat universe

$w=-1+\frac{6a_{2}\zeta_{0}n(t-t_{0})^{2n-1}}{\sqrt{\sigma\left(\frac{24a_{2}^{2}n^{2}(t-t_{0})^{4n-2}}{k^{2}}+\sigma\right)}-\sigma}-\frac{4a_{2}\sigma
n(2n-1)(t-t_{0})^{2n-2}}{k^{2}\left(\sqrt{\sigma\left(\frac{24a_{2}^{2}n^{2}(t-t_{0})^{4n-2}}{k^{2}}+\sigma\right)}-\sigma\right)\sqrt{\sigma\left(\frac{24a_{2}^{2}n^{2}(t-t_{0})^{4n-2}}{k^{2}}+\sigma\right)}}$
(58)

For the flat universe scenario, when one goes to the time of the bounce $t\to
t_{0}$, $w\to-1$ which is again in contrast to the conventional scenario but
in the case of the non-flat universe (57), we have $w\to\infty$ at the time of
the bounce which is similar to the simple cosmology.

##  5 Conclusions and final remarks

We conclude our work with final comments and an overview on the implications
of the analysis here. We can make the following observations :

  * •

Let’s consider firstly the case of the little rip and focus on (19) and (38).
These are the equation of state parameters in the case of little rip for Chern
Simons and RS-II cosmologies respectively and we notice that while in the case
of chern simons a phantom evolution is supported for LR (as in the case of the
GR cosmology as you discussed in your paper), in the case of RS-II a
Quintessence scenario is supported. Going further, in the $t\to 0$ limit as
well, Chern Simons supports phantom but in the $t\to\infty$ limit( far
future), Chern Simons shows a pure dS evolution with w tending to -1 which is
different than what happens in the conventional cosmology as you showed it
goes to phantom even in the $t\to\infty$ limit. RS-II cosmology (40) ,
meanwhile, supports only quintessence in both $t\to 0$ and $t\to\infty$ limit,
no dS and no phantom, which is again a very fascinating fact. Also, this
evolution scheme remains the same for both Chern Simons and RS-II in both flat
and non flat universes, which is again surprising.

  * •

In the case of the pseudo rip, a similar situation is observed. In both the
cases of the flat and non flat universes, w tends to phantom regime for Chern
Simons (24), as it does for the usual cosmology while in both flat and non
flat universes w tends to quintessence regime for RS-II braneworld (note how
(50) reduces to the same as (46) in the flat universe limit ). An extra
interesting bit is that in this case RS-II could allow for the Phantom regime
too for the case in which t is not tending to infinity, as seen by (48) and
(49). So in both LR and PR, it’s only in the pseudo rip scenario where for a
non extreme limit the RS-II universe could go phantom but eventually in the
far future it would again have to be in a quintessence evolution phase in the
case of PR.

  * •

In the case of the bounce models, for the power law model RS-II again has
quintessence evolution both in the limit of the bounce time as $t\to t_{0}$
(54) while in the conventional cosmology one observed phantom regime for the
same case. The incredibly interesting thing that now occurs is that in this
case even the Chern Simons cosmology has a quintessence evolution rather than
a phantom evolution (3), which is surprising seeing how Chern Simons was
preferring phantom regime in the cases of rips. Both of these results hold
true in both flat and non flat universes.

  * •

In the case of the exponential model, for the flat universe Chern Simons goes
phantom while for the case of the non flat universe it can go into
Quintessence type evolution which is exactly the behaviour shown by the GR
cosmology for the exponential model as you showed. For the flat universe, RS-
II has a possibility of allowing for any of Quintessence or phantom types of
evolution but for a non flat universe the RS-II cosmology goes completely
quintessence which again distinguishes its behaviour from the usual cosmology
for the same exponential model.

All in all, the RS-II cosmology shows a stronger departure in the way it
behaves under all of these scenarios than what happens in these scenarios in
the usual GR cosmology. Chern Simons for a lot of cases shows similar
behaviour to the conventional cosmology but also shows differences at various
places. What is interesting is that both of these cosmologies are primarily
higher dimensional theories ( in fact both being 5 dimensional in their usual
forms ) and both have one quantity which distinguishes their features from the
GR cosmology ( brane tension for the RS-II case and the $\alpha$ parameter for
Chern Simons) but while being similar models in their makings, they respond to
the cases of Little Rip, Pseudo Rip and Bounce cosmologies very differently.

## Acknowledgements

The authors would like to thank Sergei Odintsov for various helpful
discussions on singularities. This work was supported by Russian Foundation
for Basic Research; Project No. 20-52-05009

## References

  * [1] Adam G. Riess et al.  Observational evidence from supernovae for an accelerating universe and a cosmological constant.  Astron. J., 116:1009–1038, 1998.
  * [2] Steven Weinberg.  The Cosmological Constant Problem.  Rev. Mod. Phys., 61:1–23, 1989.
  * [3] Lucas Lombriser.  On the cosmological constant problem.  Phys. Lett. B, 797:134804, 2019.
  * [4] E. J. Copeland, M. Sami, and S. Tsujikawa.  Dynamics of dark energy.  Int. J. Mod. Phys. D, 15:1753–1936, 2006.
  * [5] T. Padmanabhan.  Cosmological constant: The Weight of the vacuum.  Phys. Rept., 380:235–320, 2003.
  * [6] Salvatore Capozziello and Mariafelicia De Laurentis.  Extended Theories of Gravity.  Phys. Rept., 509:167–321, 2011.
  * [7] Shin’ichi Nojiri and Sergei D. Odintsov.  Unified cosmic history in modified gravity: from F(R) theory to Lorentz non-invariant models.  Phys. Rept., 505:59–144, 2011.
  * [8] S. Nojiri, S. D. Odintsov, and V. K. Oikonomou.  Modified Gravity Theories on a Nutshell: Inflation, Bounce and Late-time Evolution.  Phys. Rept., 692:1–104, 2017.
  * [9] Zhen Zhang.  Geometrization of light bending and its application to SdSw spacetime.  Class. Quant. Grav., 39(1):015003, 2022.
  * [10] Shin’ichi Nojiri and Sergei D. Odintsov.  Introduction to modified gravity and gravitational alternative for dark energy.  eConf, C0602061:06, 2006.
  * [11] Shin’ichi Nojiri and Sergei D. Odintsov.  Modified gravity with negative and positive powers of the curvature: Unification of the inflation and of the cosmic acceleration.  Phys. Rev. D, 68:123512, 2003.
  * [12] Shin’ichi Nojiri and Sergei D. Odintsov.  Quantum escape of sudden future singularity.  Phys. Lett. B, 595:1–8, 2004.
  * [13] Shin’ichi Nojiri and Sergei D. Odintsov.  The Final state and thermodynamics of dark energy universe.  Phys. Rev. D, 70:103522, 2004.
  * [14] Shin’ichi Nojiri and Sergei D. Odintsov.  Inhomogeneous equation of state of the universe: Phantom era, future singularity and crossing the phantom barrier.  Phys. Rev. D, 72:023003, 2005.
  * [15] Shin’ichi Nojiri and Sergei D. Odintsov.  The New form of the equation of state for dark energy fluid and accelerating universe.  Phys. Lett. B, 639:144–150, 2006.
  * [16] I. Brevik, E. Elizalde, S. Nojiri, and S. D. Odintsov.  Viscous Little Rip Cosmology.  Phys. Rev. D, 84:103508, 2011.
  * [17] I. Brevik and A. V. Timoshkin.  Viscous Fluid Holographic Bounce.  Int. J. Geom. Meth. Mod. Phys., 17(02):2050023, 2019.
  * [18] I. Brevik and A. V. Timoshkin.  Holographic representation of the unified early and late universe via a viscous dark fluid.  Int. J. Geom. Meth. Mod. Phys., 19(08):2250113, 2022.
  * [19] Robert R. Caldwell, Marc Kamionkowski, and Nevin N. Weinberg.  Phantom energy and cosmic doomsday.  Phys. Rev. Lett., 91:071301, 2003.
  * [20] Paul H. Frampton, Kevin J. Ludwick, and Robert J. Scherrer.  The Little Rip.  Phys. Rev. D, 84:063003, 2011.
  * [21] Paul H. Frampton, Kevin J. Ludwick, Shin’ichi Nojiri, Sergei D. Odintsov, and Robert J. Scherrer.  Models for Little Rip Dark Energy.  Phys. Lett. B, 708:204–211, 2012.
  * [22] Artyom V. Astashenok, Shin’ichi Nojiri, Sergei D. Odintsov, and Robert J. Scherrer.  Scalar dark energy models mimicking $\Lambda$CDM with arbitrary future evolution.  Phys. Lett. B, 713:145–153, 2012.
  * [23] Artyom V. Astashenok, Emilio Elizalde, Sergei D. Odintsov, and Artyom V. Yurov.  Equation-of-state formalism for dark energy models on the brane and the future of brane universes.  Eur. Phys. J. C, 72:2260, 2012.
  * [24] Shinichi Nojiri, S. D. Odintsov, and D. Saez-Gomez.  Cyclic, ekpyrotic and little rip universe in modified gravity.  AIP Conf. Proc., 1458(1):207–221, 2012.
  * [25] Andrey N. Makarenko, Valery V. Obukhov, and Ilya V. Kirnos.  From Big to Little Rip in modified F(R,G) gravity.  Astrophys. Space Sci., 343:481–488, 2013.
  * [26] Paul H. Frampton, Kevin J. Ludwick, and Robert J. Scherrer.  Pseudo-rip: Cosmological models intermediate between the cosmological constant and the little rip.  Phys. Rev. D, 85:083001, 2012.
  * [27] Hao Wei, Long-Fei Wang, and Xiao-Jiao Guo.  Quasi-Rip: A New Type of Rip Model without Cosmic Doomsday.  Phys. Rev. D, 86:083003, 2012.
  * [28] I. Brevik, V. V. Obukhov, and A. V. Timoshkin.  Quasi-Rip and Pseudo-Rip Universes Induced by the Fluid Inhomogeneous Equation of State.  Astrophys. Space Sci., 344:275–279, 2013.
  * [29] I. Brevik, V. V. Obukhov, K. E. Osetrin, and A. V. Timoshkin.  Little Rip cosmological models with time-dependent equation of state.  Mod. Phys. Lett. A, 27:1250210, 2012.
  * [30] I. Brevik, V. V. Obukhov, A. V. Timoshkin, and Y. Rabochaya.  Rip brane cosmology from 4d inhomogeneous dark fluid universe.  Astrophys. Space Sci., 346:267–271, 2013.
  * [31] I. Brevik, A. V. Timoshkin, Ye. Rabochaya, and S. Zerbini.  Turbulence Accelerating Cosmology from an Inhomogeneous Dark Fluid.  Astrophys. Space Sci., 347:203–208, 2013.
  * [32] Kazuharu Bamba, Shin’ichi Nojiri, and Sergei D. Odintsov.  The Universe future in modified gravity theories: Approaching the finite-time future singularity.  JCAP, 10:045, 2008.
  * [33] Kazuharu Bamba, Sergei D. Odintsov, Lorenzo Sebastiani, and Sergio Zerbini.  Finite-time future singularities in modified Gauss-Bonnet and F(R,G) gravity and singularity avoidance.  Eur. Phys. J. C, 67:295–310, 2010.
  * [34] Kazuharu Bamba, Ratbay Myrzakulov, Shin’ichi Nojiri, and Sergei D. Odintsov.  Reconstruction of $f(T)$ gravity: Rip cosmology, finite-time future singularities and thermodynamics.  Phys. Rev. D, 85:104036, 2012.
  * [35] N. Aghanim et al.  Planck 2018 results. I. Overview and the cosmological legacy of Planck.  Astron. Astrophys., 641:A1, 2020.
  * [36] Eleonora Di Valentino, Alessandro Melchiorri, and Joseph Silk.  Planck evidence for a closed Universe and a possible crisis for cosmology.  Nature Astron., 4(2):196–203, 2019.
  * [37] Will Handley.  Curvature tension: evidence for a closed universe.  Phys. Rev. D, 103(4):L041301, 2021.
  * [38] A. V. Timoshkin and A. V. Yurov.  Little Rip, Pseudo Rip and bounce cosmology from generalized equation of state in the Universe with spatial curvature.  Int. J. Geom. Meth. Mod. Phys., 20:2350204, 5 2023.
  * [39] Lisa Randall and Raman Sundrum.  A Large mass hierarchy from a small extra dimension.  Phys. Rev. Lett., 83:3370–3373, 1999.
  * [40] Lisa Randall and Raman Sundrum.  An Alternative to compactification.  Phys. Rev. Lett., 83:4690–4693, 1999.
  * [41] F Gomez, P Minning, and P Salgado.  Standard cosmology in chern-simons gravity.  Physical Review D, 84(6):063506, 2011.
  * [42] I. Brevik and A. V. Timoshkin.  Holographic cosmology with two coupled fluids in the presence of viscosity.  Int. J. Geom. Meth. Mod. Phys., 18(09):2150149, 2021.
  * [43] I. Brevik and A. V. Timoshkin.  Rip brane cosmology from a viscous holographic dark fluid.  Int. J. Geom. Meth. Mod. Phys., 17(06):2050087, 2020.
  * [44] I. Brevik, V. V. Obukhov, and A. V. Timoshkin.  Cosmological Models Coupled with Dark Matter in a Dissipative Universe.  Astrophys. Space Sci., 359(1):11, 2015.
  * [45] I. Brevik, V. V. Obukhov, and A. V. Timoshkin.  Dark Energy Coupled with Dark Matter in Viscous Fluid Cosmology.  Astrophys. Space Sci., 355:399–403, 2015.
  * [46] Santosh V Lohakare, Francisco Tello-Ortiz, B Mishra, and SK Tripathy.  Rip models in ricci-gauss-bonnet cosmology.  arXiv preprint arXiv:2203.11676, 2022.
  * [47] Teodor Borislavov Vasilev, Mariam Bouhmadi-López, and Prado Martín-Moruno.  Classical and Quantum f(R) Cosmology: The Big Rip, the Little Rip and the Little Sibling of the Big Rip.  Universe, 7(8):288, 2021.
  * [48] Teodor Borislavov Vasilev, Mariam Bouhmadi-López, and Prado Martín-Moruno.  Little rip in classical and quantum f(R) cosmology.  Phys. Rev. D, 103(12):124049, 2021.
  * [49] L. N. Granda and E. Loaiza.  Big Rip and Little Rip solutions in scalar model with kinetic and Gauss Bonnet couplings.  Int. J. Mod. Phys. D, 2:1250002, 2012.
  * [50] I. Brevik, R. Myrzakulov, S. Nojiri, and S. D. Odintsov.  Turbulence and Little Rip Cosmology.  Phys. Rev. D, 86:063007, 2012.
  * [51] M. Novello and S. E. Perez Bergliaffa.  Bouncing Cosmologies.  Phys. Rept., 463:127–213, 2008.
  * [52] Kazuharu Bamba, Andrey N. Makarenko, Alexandr N. Myagky, Shin’ichi Nojiri, and Sergei D. Odintsov.  Bounce cosmology from $F(R)$ gravity and $F(R)$ bigravity.  JCAP, 01:008, 2014.
  * [53] Yi-Fu Cai, Shih-Hung Chen, James B. Dent, Sourish Dutta, and Emmanuel N. Saridakis.  Matter Bounce Cosmology with the f(T) Gravity.  Class. Quant. Grav., 28:215011, 2011.
  * [54] Rong-Gen Cai, Li-Ming Cao, and Ya-Peng Hu.  Corrected entropy-area relation and modified friedmann equations.  Journal of High Energy Physics, 2008(08):090, 2008.
  * [55] James E Lidsey.  Holographic cosmology from the first law of thermodynamics and the generalized uncertainty principle.  Physical Review D, 88(10):103519, 2013.
  * [56] Pantelis S Apostolopoulos, George Siopsis, and Nikolaos Tetradis.  Cosmology from an anti-de sitter–schwarzschild black hole via holography.  Physical review letters, 102(15):151301, 2009.
  * [57] Cedric Deffayet.  Cosmology on a brane in Minkowski bulk.  Phys. Lett. B, 502:199–208, 2001.
  * [58] G. R. Dvali, Gregory Gabadadze, and Massimo Porrati.  4-D gravity on a brane in 5-D Minkowski space.  Phys. Lett. B, 485:208–214, 2000.
  * [59] Varun Sahni and Yuri Shtanov.  Brane world models of dark energy.  JCAP, 11:014, 2003.
  * [60] M. Sami and V. Sahni.  Quintessential inflation on the brane and the relic gravity wave background.  Phys. Rev. D, 70:083513, 2004.
  * [61] M. C. Bento, R. Gonzalez Felipe, and N. M. C. Santos.  Brane assisted quintessential inflation with transient acceleration.  Phys. Rev. D, 77:123512, 2008.