#  Quantum deformed phantom dynamics in light of the generalized uncertainty
principle

Gaurav Bhandari<EMAIL_ADDRESS>Department of Physics, Lovely
Professional University, Phagwara, Punjab, 144411, India  S. D. Pathak
<EMAIL_ADDRESS>Department of Physics, Lovely Professional University,
Phagwara, Punjab, 144411, India  Manabendra Sharma<EMAIL_ADDRESS>Centre for Theoretical Physics and Natural Philosophy, Nakhonsawan Studiorum
for Advanced Studies, Mahidol University, Nakhonsawan, 60130, Thailand
Anzhong Wang<EMAIL_ADDRESS>GCAP-CASPER, Physics Department,
Baylor University, Waco, TX 76798-7316, USA

###### Abstract

Quantum gravity has been baffling the theoretical physicist for decades now:
both for its mathematical obscurity and phenomenological testing.
Nevertheless, the new era of precision cosmology presents a promising avenue
to test the effects of quantum gravity.

In this study, we consider a bottom-up approach. Without resorting to any
candidate quantum gravity, we invoke a generalized uncertainty principle (GUP)
directly into the cosmological Hamiltonian for a universe sourced by a phantom
scalar field with potential to study the early epoch of the evolution. This is
followed by a systematic analysis of the dynamics, both qualitatively and
quantitatively. Our qualitative analysis shows that the introduction of GUP
significantly alters the existence of fixed points for the potential
considered in this contribution. In addition, we confirm the existence of an
inflationary epoch and analyze the behavior of relevant cosmological
parameters with respect to the strength of GUP distortion.

###### Contents

  1. I  Introduction
  2. II  GUP-modified background dynamics
    1. II.1  Phantom scalar field with cosmological constant
      1. II.1.1 Classical Dynamics
      2. II.1.2 GUP deformed dynamics
      3. II.1.3 GUP corrected Friedmann equation with cosmological constant
    2. II.2  Phantom scalar field with an arbitrary potential
  3. III  Dynamical System Analysis
    1. III.1 For $V(\phi)=V_{0}\phi^{2}$
      1. III.1.1 Fixed point analysis
    2. III.2 Exponential Potential
      1. III.2.1 Fixed point analysis
      2. III.2.2 Phase portrait
  4. IV  GUP-modified inflationary scenario
    1. IV.0.1 Quadratic potential, $V(\phi)=\mu^{2}\phi^{2}$
    2. IV.0.2 Exponential potential, $V(\phi)=V_{0}e^{-k\phi}$
  5. V  Conclusion

##  I  Introduction

The advent of Einstein’s general theory of relativity, with its field
equation, gave birth to many fields of research. Since gravity is the only
dominant force at large distances, general relativity provides a viable
mathematical framework to construct models of cosmology. Over a period of 100
years, General Relativity (GR) has seen profound successes. A few examples
include the explanation of the perihelion precession of Mercury[1], the
deflection of light rays when passing close by massive bodies [2], and the
gravitational redshift of light [3].

In particular, to talk about cosmology, in the year 1929 the discovery of the
Hubble’s expansion law laid the foundation of modern cosmology. This
observational evidence of uniform and isotropic expansion of the universe as
incorporated by the Friedmann-Lemaitre-Robertson-Walker (FLRW) metric globally
gives rise to the standard model of cosmology (SM) when applied in GR. The
FLRW metric is a maximally symmetric geometry of spacetime that supports the
Copernican principle. One of the remarkable successes of the standard model of
cosmology is the prediction of cosmic microwave background radiation (CMB).

Although successful, the SM was confronted with some serious drawbacks. To
give an instance: the causal explanation for two otherwise spatially
disconnected regions of space was lacking within the scope of the SM.
Technically, these drawbacks of SM are summarized under the name of: horizon,
flatness and entropy problem[4, 5, 6, 7, 8].

The inflationary paradigm proposed by A. Guth (1981) rescues the situation by
providing a mechanism to solve the puzzles of the SM with the help of a nearly
exponential expansion of the universe early on. The scalar field serves as a
good candidate for an inflationary scenario.

The inflationary epoch not only rescues the standard model of cosmology but
also predicts the formation of the large-scale structure of the universe.
Although the universe looks almost homogeneous and isotropic at large
scale[9], the tiny fluctuation of the order of $10^{-5}$ is observed in CMB
radiation. This tininess of the scale allows us to employ perturbative theory,
wherein the zeroth order, background, and spacetime is still FLRW and any
inhomogeneity is given by the leading order correction. The physical reason
for the perturbation of spacetime is the quantum fluctuation of matter
content, which is inflaton in this situation. Of course, any perturbation of
the matter field would induce a perturbation in the metric field, resulting in
the clumping of energy and matter density, leading to the formation of the
structure we see today. In the process, inflation expands the tiny causally
connected quantum fluctuation into the super-Hubble mode, which re-enters the
Hubble radius at a later epoch, giving us a causal mechanism for the large-
scale structure we see today[10, 11, 12].

The background metric is still treated classically while quantizing the first-
order correction in the linearized theory of gravity. However, in a true
theory of quantum gravity, one would expect the quantum nature of the
background metric to play a significant role, at the scale approaching to
Planck region[13, 14].

This incomplete picture of the theory of physical cosmology is due to the
continued lack of a consistent candidate for quantum gravity. In fact, this is
one of the burning issues in modern theoretical physics. This is obscure
because the current understanding of our nature is based on two mathematically
incompatible frameworks; viz, general relativity (GR) and quantum mechanics
(QM) discussed in [15, 16, 17, 18].

In the literature, there exist different candidates based on different
philosophical approaches to quantize gravity, each with its own advantages and
issues. The two major streams of quantum gravity are–string theory and loop
quantum gravity. While string theory is based on the unification of gravity
with three other fundamental forces, loop quantum gravity (LQG) is the
quantization of the Riemannian geometry of general relativity on its own
right[19, 20, 21, 21, 22]. The LQG is background independent and non-
perturbative. The technique of LQG, when applied to cosmological spacetime,
gives rise to various models of quantum corrected cosmology, also called loop
quantum cosmology (LQC)[23]. One of the striking features of LQC is the
supplant of initial singularity by quantum bounce owing to the quantization of
geometry. In lieu of its endeavor to empirically grasp the semi-classical
physics near the Planck region, LQC faces serious criticism as it still
remains an open problem to recover it directly from the full theory (LQG), in
addition to its interpretational issue from a realist perspective.

Nevertheless, in view of the continued absence of a consistent theory of
quantum gravity, radically different paths have been adopted. The generalized
uncertainty principle is one such attempt that can generate quantum corrected
dynamics when applied to cosmology to study the early universe. In this
approach, we consider each classical point of space-time as a probability
density associated with basis vectors with additional fluctuations in the
geometry, giving rise to the extended generalized uncertainty
principle(EGUP)[24, 25].

The departure point from classical mechanics to standard quantum mechanics is
the Heisenberg’s uncertainty principle (HUP), which states the incompatibility
of position and momentum operators, reflecting the inherent imprecision of the
measurement of one when the other is known precisely.

However, at scales approaching the Planck length, theories of quantum gravity
suggest that the geometry of space-time cannot be measured below the Planck
scale or below the minimal length. The observation of minimally measured
length scale can be studied in different theories like string theory (minimal
length scale is the string length itself)[26, 27, 28, 29, 30, 31], minimally
measured length in loop quantum gravity[32, 33], and in black holes[34].

This immediately implies that HUP is not applicable at the Planck scale as it
puts no limit to precisely measure length provided momentum is undetermined.
The inconsistency in Heisenberg’s uncertainty principle indicates the need to
modify the existing canonical Heisenberg uncertainty principle(HUP) by
incorporating gravitational correction.

The consideration of quantum fluctuation in the space-time geometry leads to
the Generalized uncertainty principle(GUP) which describes the limitation of
measurement of position and momentum. The uncertainties of position and
momentum depend on the fluctuation of spacetime. The greater the uncertainty
in the geometry of space, the greater the uncertainty in the position and
momentum of the particles[35, 36]. The notion that gravity might influence the
uncertainty principle was first proposed by Mead [37]. Later, candidate
theories of quantum gravity such as String Theory [38], Doubly Special
Relativity (DSR) Theory, and Black Hole Physics[39] introduced modifications
to the commutation relations between position and momentum, which are known as
the Generalized Uncertainty Principle (GUP)[40, 41].

Keeping in view the current status of quantum cosmology, the GUP-modified
cosmological dynamics require more attention than before to extract the low-
energy domain of quantum gravity. In particular, in this article, we focus on
the inflationary era due to GUP-corrected effective dynamics for phantom
scalar fields.

Phantom inflation leads to a cosmological scenario of the Big Rip, where the
universe undergoes a catastrophic expansion that leads to tearing apart all
the bound structures, including planets, galaxies, stars, and even fundamental
particles. However, investigating these consequences helps in understanding
the possible fate of our universe. Lately, there has been significant
attention to phantom cosmology, discussed in [42, 43, 44, 45]

In this paper, in Sec.II.1 we review the formulation of GUP-corrected
Hamiltonian by taking the Einstein-Hilbert action with a minimally coupled
phantom scalar field with a positive cosmological constant. We obtain the GUP-
corrected Friedmann equation, Raychaudhuri equation and the klien-Gordon
equation. Later, this is extended to include the arbitrary potential in
Sec.II.2. The techniques of dynamical system analysis have been employed to
extract qualitative information about the system in Sec.III. We limit
ourselves to quadratic and exponential potentials. In Sec.IV.0.1, we study
inflationary dynamics by calculating the Equation of State (EoS) parameter and
slow climb parameters with plots for quadratic potential and exponential
potential in Sec.IV.0.2.

##  II  GUP-modified background dynamics

The standard model of cosmology is based on the ”Copernican Principle” which
says universe is homogeneous and isotropic on a large scale. This is encoded
in the maximally symmetric FLRW (Friedmann-Lemaitre-Robertson-Walker) metric:

$ds^{2}=-N^{2}(t)dt^{2}+a^{2}(t)\left[dr^{2}+r^{2}d\Omega^{2}\right],$ (1)

Chronologically, GR was developed using the Lagrangian setup. Therefore, the
dynamics are obtained as the Euler–Lagrange equation by varying the metric and
the matter field of the Einstein-Hilbert action:

$S_{EH}=\frac{1}{2\kappa}\int d^{4}x\sqrt{-g}R+\mathcal{L}_{m},$ (2)

to give $G_{\mu\nu}=T_{\mu\nu}.$ Where $G_{\mu\nu}$ is the Einstein Tensor and
$T_{\mu\nu}$ is the energy momentum tensor. $\kappa\equiv\frac{1}{8\pi G}$ is
set equal to one for the rest of the paper owing to natural units.

GR says that the dynamics of the geometry of the universe are dictated by the
matter content of the universe, which in turn tells matter how to move. This
implies that the dominant matter content of the universe determines the rate
of expansion of the universe for the given epoch in question. An alternative
school of thought proposes to modify the geometric sector of the Einstein-
Hilbert action Eq.(2) giving rise to modified theories of gravity to produce
the desired rate of expansion in a given epoch. The modified theories of
gravity have been extensively employed to address both early- and late-time
scenarios.[46, 47, 48, 49].

In this paper, we focus on the former approach-modify the matter content to
address the early epoch of the universe. To this effect, we adopt a phantom
scalar field. This has been extensively studied in the context of the late-
time era of evolution. The fact that the phantom field produces a phase of
accelerated expansion of the universe makes it interesting to investigate its
implication in inflationary dynamics.

One of our prime focus areas is to explore the tail-end dynamics of the
universe where quantum gravity effects could still be important, though not
dominant. This, in the domain of loop quantum cosmology, has been reported as
transition phase from quantum to the classical universe in pre-inflationary
dynamics[50, 51].

Having said that, this paper takes a radically different approach by directly
invoking a generalized uncertainty principle into the Hamiltonian cosmological
dynamics. This is an effective way of modeling the quantum corrected
background evolution of the universe.

In this section and in what follows, we review the construction of the GUP-
deformed background equation of motion for a phantom scalar field.

Let us begin with the Lagrangian for the phantom field with an arbitrary
potential $V(\phi)$ [52],

$\mathcal{L}=-3a\dot{a}^{2}-a^{3}\left(\frac{\dot{\phi}^{2}}{2}+V(\phi)\right).$
(3)

The Hamiltonian is constructed from the Lagrangian with the help of the
Legendre transformation
$\mathcal{H}\equiv\dot{a}P_{a}+\dot{\phi}P_{\phi}-\mathcal{L},$ to give

$\mathcal{H}=-\frac{P_{a}^{2}}{12a}-\frac{P_{\phi}^{2}}{2a^{3}}+a^{3}V(\phi).$
(4)

Eq.(4) is the classical Hamiltonian that governs the background dynamics of a
universe dominated by a phantom scalar field with the following symplectic
structure.

$\displaystyle\\{a,P_{a}\\}=1,\quad\\{\phi,P_{\phi}\\}=1,$ (5)

where $P_{a}\equiv\frac{\delta\mathcal{L}}{\delta\dot{a}}$ and
$P_{\phi}=\frac{\mathcal{L}}{\delta\dot{\phi}}$ are the conjugate momentum to
$a$ and $\phi$ respectively. Thus, the complete phase space consists of
$(a,P_{a},\phi,P_{\phi})$. While the symplectic algebra 5 represents the
kinematical structure of a theory, the dynamical evolution is given by the
Poisson flow of the phase space variables w.r.t. the Hamiltonian
$\mathcal{H}.$

$\displaystyle\dot{a}$ $\displaystyle=$
$\displaystyle\\{a,\mathcal{H}\\},\quad\dot{P_{a}}=\\{P_{a},\mathcal{H}\\},$
$\displaystyle\dot{\phi}$ $\displaystyle=$
$\displaystyle\\{\phi,\mathcal{H}\\},\quad\dot{P_{\phi}}=\\{P_{\phi},\mathcal{H}\\}.$
(6)

Using the above Eq.(6) the the Klien-Gordon and Raychaudhuri equations for the
phantom scalar field are obtained as follows:

$\displaystyle\ddot{\phi}+3\dot{\phi}\frac{\dot{a}}{a}-\frac{dV(\phi)}{d\phi}=0,$
(7) $\displaystyle
2\frac{\ddot{a}}{a^{2}}+\left(\frac{\dot{a}}{a}\right)^{2}=\frac{\dot{\phi}^{2}}{2}+V(\phi).$
(8)

The treatment thus far has been purely classical. However, reformulating the
dynamics in terms of Poisson algebra is the point of departure from classical
to quantum mechanics. Next, we will see how to incorporate quantum correction
arising from GUP into the dynamics.

###  II.1  Phantom scalar field with cosmological constant

####  II.1.1 Classical Dynamics

In this section, we consider the Einstein–Hilbert action with a minimally
coupled phantom scalar field and a positive cosmological constant.

$S_{EH}=\int\sqrt{-g}\left[\frac{1}{2\kappa}\left(R-2\Lambda\right)+\frac{1}{2}\partial_{\mu}\phi\partial_{\nu}\phi\right]d^{4}x,$
(9)

On the backdrop of a maximally symmetric spacetime
$ds^{2}=-N^{2}(t)dt^{2}+a^{2}(t)[dr^{2}+r^{2}d\Omega^{2}]$, where N(t) is the
lapse function. Given the FLRW background, our action takes the following
form:

$S=V_{0}\int
dt\left[-\frac{3a\dot{a}^{2}}{N}-a^{3}\left(\frac{\dot{\phi}^{2}}{2N}+N\Lambda\right)\right].$
(10)

where $V_{0}$ is the fictitious volume introduced to facilitate our
calculation in an otherwise non-compact FLRW spacetime. However, because it
does not affect the dynamics, it can be set equal to 1 without loss of
generality. Furthermore, for the rest of the paper we choose the natural unit
$\kappa=1.$

Thus, the expression of the Lagrangian from Eq.(10) is given by

$\mathcal{L}=-\frac{3a\dot{a}^{2}}{N}-a^{3}\left(\frac{\dot{\phi}^{2}}{2N}+N\Lambda\right).$
(11)

It is obvious that the Lagrangian, Eq.(11), is devoid of $\dot{N}(t).$ and
hence there is no dynamics in the lapse function $N(t)$,
$P_{N}\equiv\frac{\partial\mathcal{L}}{\partial\dot{N}}=0$ being a constant of
motion. Therefore, the dynamics of the system are completely contained in the
equation of motion for $(a,P_{a},\phi,P_{\phi})$ governed by the Hamiltonian:

$\mathcal{H}=N\left[\frac{P_{a}^{2}}{12a}+\frac{P_{\phi}^{2}}{2a^{3}}-a^{3}\Lambda\right],$
(12)

which is obtained from Eq.(11) through the Legendre transformation.
The Equation of motion can be obtained using Eq.(6) and by substituting the
form of $P_{a}$ and $P_{\phi}$ so obtained from the Lagrangian Eq.(11). The
Raychaudhuri equation is

$2\frac{\ddot{a}}{a}+\left(\frac{\dot{a}}{a}\right)^{2}=\frac{\dot{\phi}^{2}}{2}+\Lambda.$
(13)

The Friedmann equation is

$3H^{2}=-\frac{\dot{\phi}^{2}}{2}+\Lambda.$ (14)

Furthermore, the Klein-Gordon (KG) equation is

$\ddot{\phi}+3H\dot{\phi}=0.$ (15)

The absence of the $\frac{dV}{d\phi}$ term in the KG equation above is because
$\Lambda$ acts as a constant potential.

####  II.1.2 GUP deformed dynamics

In this subsection, we review the inclusion of higher-order correction of the
uncertainty principle in the cosmological Hamiltonian. To achieve this, we
perform a canonical transformation of the phase space as follows:

$x=\frac{a^{3/2}}{\mu}\sin(\mu\phi),\hskip
14.22636pty=\frac{a^{3/2}}{\mu}\cos(\mu\phi),$ (16)

While preserving the dynamics. To see it explicitly, we note that
differentiation of Eq.(16) gives

$\displaystyle\dot{x}=\frac{3}{2}\frac{a^{1/2}\dot{a}}{\mu}\sin(\mu\phi)+(a^{3/2}\dot{\phi})\cos(\mu\phi),$
(17)
$\displaystyle\dot{y}=\frac{3}{2}\frac{a^{1/2}\dot{a}}{\mu}\cos(\mu\phi)-(a^{3/2}\dot{\phi})\sin(\mu\phi).$
(18)

Now, multiplying both sides of Eq.(17) by $\sin(\mu\phi)$ and Eq.(18) by
$\cos(\mu\phi)$ and adding them together gives

$\dot{x}\sin(\mu\phi)+\dot{y}\cos(\mu\phi)=\frac{3}{2}\frac{a^{1/2}}{\mu}\dot{a}.$
(19)

Next by squaring both sides of Eq.(19), followed by dividing the expression by
$2N$ and considering $\mu=\sqrt{3/8}$ one obtains,

$\frac{\dot{x}^{2}\sin^{2}(\mu\phi)+\dot{y}^{2}\cos^{2}(\mu\phi)+2\dot{x}\dot{y}\sin(\mu\phi)\cos(\mu\phi)}{2N}=\frac{3a\dot{a}^{2}}{N}.$
(20)

Instead, multiply both sides of Eq.(17) by $\cos(\mu\phi)$ and Eq.(18) by
$\sin(\mu\phi)$ and subtracting

$\dot{x}\cos(\mu\phi)-\dot{y}\sin(\mu\phi)=a^{3/2}\dot{\phi}.$ (21)

This is followed by squaring Eq.(21) and dividing it by $2N$ to obtain

$\frac{\dot{x}^{2}\cos^{2}(\mu\phi)+\dot{y}^{2}\sin^{2}(\mu\phi)-2\dot{x}\dot{y}\cos(\mu\phi)\sin(\mu\phi)}{2N}=\frac{a^{3}\dot{\phi}^{2}}{2N}.$
(22)

In addition, by considering the sum of the squares of the components in
Eq.(16), we have:

$\displaystyle
x^{2}+y^{2}=\frac{a^{3}}{\mu^{2}}\sin^{2}(\mu\phi)+\frac{a^{3}}{\mu^{2}}\cos^{2}(\mu\phi)$
$\implies x^{2}+y^{2}=\frac{a^{3}}{\mu^{2}}=\frac{8a^{3}}{3}.$ (23)

Now, from Eq.(23) we observe that the physical volume of the universe under
study can be elegantly expressed as the radius of the circle, with $(0,0)$ as
the center in the plane containing the configuration variable $(x,y).$ Since
the Friedmann equation is nothing but the fractional rate of change of volume,
intuitively, one can speculate that the knowledge of the dynamics of the pair
$(x,y)$ suffices to predict the evolution of the universe.

Having set the stage in terms of Cartesian pair (x,y), we now return to the
question of dynamics. Using Eq.(20) Eq.(21) and Eq.(23) we obtain the final
form of the Lagrangian in terms of configuration coordinates
$(x,y,\dot{x},\dot{y})$ to be

$\mathcal{L}=-\left[\frac{\dot{x}^{2}+\dot{y}^{2}}{2N}+\frac{3}{8}(x^{2}+y^{2})\Lambda
N\right].$ (24)

A quick look at the form of the Lagrangian suggests that dynamics is symmetric
w.r.t the origin of the circle in the plane of $(x,y)$.

Given this, it is straightforward to obtain the canonically transformed
Hamiltonian using the Legendre transformation. Thus, the final form of the
Hamiltonian looks like

$\mathcal{H}_{0}=N\left[\frac{P_{x}^{2}}{2}+\frac{P_{y}^{2}}{2}+\frac{\omega^{2}}{2}(x^{2}+y^{2})\right],$
(25)

where $\omega^{2}=-\frac{3}{4}\Lambda$. The utility of the canonical
transformation of Eq.(16) is clear from the elegant expression of 25. It is
straightforward to see that the dynamics of a universe with a phantom scalar
and a positive cosmological constant can be expressed as a system of two
decoupled simple harmonic oscillators in the new phase space.

However, the Eq.(25) is still classical, though expressed in a different
guise. To write down the dynamics due to momentum deformation owing to the
Generalized Uncertainty Principle (GUP), we introduce semi-classical canonical
variables $q_{i}$ and $P_{i}$ and introduce GUP in the WDW equation in our
cosmological model as followed by [53].

$q_{i}=q_{0i},\quad P_{i}=P_{0i}\left(1-\beta\gamma
P_{0}+2\gamma^{2}\frac{\beta^{2}+2\epsilon}{3}P_{0}^{2}\right),$ (26)

where $P_{0}^{2}=P_{0j}P_{0j}$ , $\beta,\epsilon$ and $\gamma$ are constants.
we calculate the GUP distorted Hamiltonian up to the order of $\gamma^{2}$
[54] as:

$\mathcal{H}=\mathcal{H}_{0}-\beta\gamma(P_{0x}^{2}+p_{0y}^{2})^{3/2}+\gamma^{2}(P_{0x}^{2}+p_{0y}^{2})^{2}\left(\frac{\beta^{2}}{6}+\frac{2\epsilon}{3}\right)+\mathcal{O}(\gamma^{3}),$
(27)

where
$\mathcal{H}_{0}=\frac{P_{0x}^{2}}{2}+\frac{P_{0y}^{2}}{2}+\frac{\omega^{2}}{2}(x^{2}+y^{2})$
is the unperturbed Hamiltonian before introducing GUP. From now on, the
subscript $0$ will be used to denote the unperturbed version to represent
Eq.(25). For example, the unperturbed $x,y$ are represented as
$(q_{0x},q_{0y})$ while the unperturbed pair $(P_{x},P_{y})$ are to be
recognized as $(P_{0x},P_{0y}).$

In the final step, we re-express the GUP deformed Hamiltonian Eq.(27) in the
cosmological phase-space variable. This is achieved by applying the inverse
transformation to express Eq(27) in the cosmological variables, namely, the
scale factor, the scalar field, and their corresponding conjugate momenta.
Using Eq.(17) and Eq.(17), we obtain

$P_{0x}=\dot{x}=\frac{3}{2}\frac{a^{1/2}\dot{a}}{\mu}\sin(\mu\phi)+(a^{3/2}\dot{\phi})\cos(\mu\phi),\\\
$ (28)
$P_{0y}=\dot{y}=\frac{3}{2}\frac{a^{1/2}\dot{a}}{\mu}\cos(\mu\phi)-(a^{3/2}\dot{\phi})\sin(\mu\phi).$
(29)

Because there are no dynamics in the lapse function, without loss of
generality $N(t)=1$,

$\frac{\partial\mathcal{L}}{\partial\dot{a}}=P_{a}=-6\dot{a}a,$

and

$\frac{\partial\mathcal{L}}{\partial\dot{\phi}}=P_{\phi}=-a^{3}\dot{\phi},$

by substituting $P_{a}$ and $P_{\phi}$ in Eq.(28) the equations becomes

$P_{0x}=-\frac{P_{a}}{4a^{1/2}\mu}\sin(\mu\phi)-\frac{P_{\phi}}{a^{3/2}}\cos(\mu\phi),$
(30)
$P_{0y}=-\frac{P_{a}}{4a^{1/2}\mu}\cos(\mu\phi)+\frac{P_{\phi}}{a^{3/2}}\sin(\mu\phi).$
(31)

Applying $P_{0x}$ and $P_{0y}$ in the momentum-deformed Hamiltonian due to GUP
correction in Eq.(27) by taking $\epsilon=1$,

$\scalebox{1.1}{{\hbox{\mathcal{H}_{GUP}=\left[\frac{\omega^{2}}{2}a^3+\frac{P_{a}^{2}}{12a}+\frac{P_{\phi}^{2}}{2a^{3}}+
2\gamma^2\left(\frac{P_{a}^{4}}{108a^{2}}+\frac{P_{\phi}^{4}}{3a^{6}}+\frac{P_{a}^{2}P_{\phi}^{2}}{9a^{4}}\right)\right]}}}.$
(32)

This is the required GUP distorted Hamiltonian in the cosmological phase space
dominated by a phantom scalar field with a positive cosmological constant up
to second-order perturbation.
Using Eq.(6)we can obtain the final Raychaudhuri equation

$2\frac{\ddot{a}}{a}+\left(\frac{\dot{a}}{a}\right)^{2}=\frac{\dot{\phi}^{2}}{2}-\frac{\omega^{2}}{2}+\gamma^{2}\left(16a^{3}H^{4}+\frac{4a^{3}\dot{\phi}^{4}}{3}+\frac{32a^{3}H^{2}\dot{\phi}^{2}}{3}\right).$
(33)

The final Friedmann equation is

$3H^{2}=-\left[\frac{\dot{\phi}^{2}}{2}+\frac{\omega^{2}}{2}\right]-2\gamma^{2}\left[12H^{4}a^{3}+\frac{a^{3}\dot{\phi}^{4}}{3}+4a^{3}H^{2}\dot{\phi}^{2}\right].$
(34)

and the KG equation is

$\ddot{\phi}+3\dot{\phi}H=0.$ (35)

We notice that the Raychaudhuri Eq.(33) and the Friedmann Eq.(34) directly
incorporate quantum corrections, whereas the Klein-Gordon Eq.(35), there is no
explicit dependence on quantum correction. Any quantum deformation entering
the KG Eq.(35) arises only implicitly through the Hubble parameter. This
situation is similar to the models of loop quantum cosmology [57, 58, 59, 60,
61, 62].

####  II.1.3 GUP corrected Friedmann equation with cosmological constant

In this subsection we simplify the Friedmann Eq.(34) Defining a new parameter
for the perturbation term $2\gamma^{2}\equiv\alpha$, the Eq.(34) can be
written as a quadratic equation in terms of $\tilde{H}=H^{2}$ as

$3\tilde{H}^{2}+\left(\dot{\phi}^{2}+\frac{3}{4\alpha
a^{3}}\right)\tilde{H}-C_{0}=0,$ (36)

where
$C\equiv-\left(\frac{\dot{\phi}^{2}}{2}+\frac{\omega^{2}}{2}+\frac{\alpha
a^{3}\dot{\phi}^{4}}{3}\right)$ and $C_{0}\equiv\frac{C}{4\alpha a^{3}}$.
Solving the quadratic equation gives us:

$\tilde{H}=\frac{-\left(\dot{\phi^{2}}+\frac{3}{4\alpha
a^{3}}\right)\pm\sqrt{\left(\dot{\phi}^{2}+\frac{3}{4\alpha
a^{3}}\right)^{2}+12C_{0}}}{6}.$ (37)

Since $\tilde{H}=H^{2}$, the right-hand side of the Eq.(37) must be greater
than zero. This implies

$\sqrt{\left(\dot{\phi}^{2}+\frac{3}{4\alpha
a^{3}}\right)^{2}+12C_{0}}>\left(\dot{\phi}^{2}+\frac{3}{4\alpha
a^{3}}\right).$ (38)

this is equivalent to

$C_{0}>0,$ (39)

or

$\frac{\dot{\phi}^{2}}{2}+\frac{\alpha
a^{3}\dot{\phi}^{4}}{3}<\frac{3\Lambda}{8}.$ (40)

Rewriting Eq.(37)

$H^{2}=-\frac{\dot{\phi}^{2}}{6}-\frac{1}{8\alpha
a^{3}}+\left(\frac{\dot{\phi}^{2}}{6}+\frac{1}{8\alpha
a^{3}}\right)\sqrt{1+\frac{12C_{0}}{\left(\dot{\phi}^{2}+\frac{3}{4\alpha
a^{3}}\right)^{2}}},$ (41)

and applying the conditions Eq.(38)

$\sqrt{1+\frac{12C_{0}}{M^{2}}}>1,$ (42)

where $M\equiv\dot{\phi}^{2}+\frac{3}{4\alpha a^{3}}$. Applying binomial
expansion of the Eq.(41) we get,

$H^{2}=-\frac{\dot{\phi}^{2}}{6}-\frac{1}{8\alpha
a^{3}}+\left(\frac{\dot{\phi}^{2}}{6}+\frac{1}{8\alpha
a^{3}}\right)\left[1+\frac{1}{2}\left(\frac{12C_{0}}{M^{2}}\right)-\frac{1}{2}\left(\frac{1}{2}-1\right)\left(\frac{1}{2!}\right)\left(\frac{12C_{0}}{M^{2}}\right)^{2}+\mathcal{O}(3)\right].$
(43)

Considering term up to the first order in $12C_{0}/M^{2}$,

$H^{2}=\frac{\left(\frac{-\dot{\phi}^{2}}{2}+\frac{3\Lambda}{8}-\frac{\alpha
a^{3}\dot{\phi}^{4}}{3}\right)}{4\alpha a^{3}\dot{\phi}^{2}+3}.$ (44)

Eq.(44) is the required Friedmann equation with GUP modification for a phantom
scalar field with a positive cosmological constant.

###  II.2  Phantom scalar field with an arbitrary potential

Now, we construct the GUP-modified Friedmann equation for arbitrary potential
by applying the change in variables shown in the previous section II.1.2

$x=\frac{a^{\frac{3}{2}}}{\mu}\sin(\mu\phi),\hskip
28.45274pty=\frac{a^{\frac{3}{2}}}{\mu}\cos(\mu\phi).$ (45)

The Lagrangian Eq.(3) can be written in terms of ($x$,$y$,$\dot{x}$,$\dot{y}$)
by applying the procedure prescribed in the section II.1.2 as

$\mathcal{L}=-\left[\frac{\dot{x}^{2}+\dot{y}^{2}}{2}+\frac{3}{8}(x^{2}+y^{2})V(\phi)\right].$
(46)

In addition, the unperturbed Hamiltonian can be obtained by the Legendre
transformation of the Eq.(46)

$H_{0}=\frac{P_{x}^{2}}{2}+\frac{P_{y}^{2}}{2}-\frac{3}{8}(x^{2}+y^{2})V(\phi).$
(47)

Introducing GUP, as given in section II.1.2 to the unperturbed Hamiltonian
Eq.(47) generates

$\mathcal{H}_{GUP}=\frac{P_{a}^{2}}{12a}+\frac{P_{\phi}^{2}}{2a^{3}}-a^{3}V(\phi)+2\gamma^{2}\left(\frac{P_{a}^{4}}{108a^{2}}+\frac{P_{\phi}^{4}}{3a^{6}}+\frac{P_{a}^{2}P_{\phi}^{2}}{9a^{4}}\right).$
(48)

From the above GUP corrected Hamiltonian Eq.(48) one can easily obtain the
Raychaudhuri equation as:

$2\frac{\ddot{a}}{a}+\left(\frac{\dot{a}}{a}\right)^{2}=\frac{\dot{\phi}^{2}}{2}+V(\phi)+\gamma^{2}\left(16a^{3}H^{4}+\frac{4a^{3}\dot{\phi}^{4}}{3}+\frac{32a^{3}H^{2}\dot{\phi}^{2}}{3}\right).$
(49)

Following the same procedure given in section II.1.3, after some algebraic
manipulation, the Friedmann equation for arbitrary potential can be derived as

$H^{2}=\frac{\left(-\frac{\dot{\phi}^{2}}{2}+V(\phi)-\frac{\alpha
a^{3}\dot{\phi}^{4}}{3}\right)}{4\alpha a^{3}\dot{\phi}^{2}+3},$ (50)

and the KG equations remain the same for any arbitrary potential as

$\ddot{\phi}+3\dot{\phi}H-\frac{dV(\phi)}{d\phi}=0.$ (51)

We note that this feature is consistent with the loop quantum cosmology
scenario in which the KG equation does not receive any explicit quantum
correction. Any quantum correction occurs only through the Hubble parameter;
thus, the classical form of the KG equation holds for any given potential in
the GUP corrected regime.

In standard cosmology, the factor $\frac{\alpha a^{3}\dot{\phi}^{4}}{3}$ is
absent in the numerator of the Eq.(50), retaining only the energy density
term. The additional term arises solely from the quantum corrections due to
GUP. In addition to this, the Freidemann Eq.(50) reveals the possibility of
singularity resolution. The occurrence of non-singular bounce in phantom
models has already been studied in [63, 64, 65, 66, 67]. The possibility of a
non-singular bounce is further enhanced by the presence of a negative quantum
corrected term, $-\frac{\alpha a^{3}\dot{\phi}^{4}}{3}$, in the Friedmann
Eq.(50). Though $\dot{a}=0$ is a necessary condition for the occurrence of a
bounce, the solution must meet $\ddot{a}>0$ at the turning point of the bounce
for the contracting universe to reverse its trajectory and begin expanding
again. This condition is satisfied by the Raychaudhuri Eq.(49) as the right-
hand side of the equation all the terms are positive due to even powers for
the non-zero arbitrary values of $V(\phi)$ and $\dot{\phi}$.

##  III  Dynamical System Analysis

A nonlinear system is generally difficult to predict. However, the method of
dynamical system analysis serves as a powerful tool for extracting qualitative
information from nonlinear systems. The Einstein’s equation when applied to
FLRW spacetime, becomes a set of coupled second-order differential equations.
However, with a suitable choice of variables, they can be transformed to
first-order autonomous differential equations. The powerful technique of
dynamical system analysis (DSA) can be widely employed to analyze the
background dynamics of cosmological models.

The approach of DSA is qualitative in nature. It predicts the overall dynamics
of the of the system in question without actually solving it. The most widely
adopted method is linear stability analysis. In this method the system’s
dynamics is cast as a set of first-order autonomous differential equations,
and the fixed points are defined as the points where the vector flow of the
dynamical variables vanishes. The precise nature of the fixed points is
obtained by examining the behavior of the leading order perturbation around
the fixed points. Mathematically, the signs of the eigenvalues of the Jacobian
matrix evaluated at the fixed points indicate the nature of the fixed points.
For an extensive review of DSA, we refer the readers to [68, 69, 70, 71].

Phase portraits, on the other hand, are visual representations of the
trajectories of a dynamic system. It provides insights into the qualitative
behavior of the system, pictorially. This is achieved by drawing a tangent at
each point given by the flow vectors of the autonomous differential equations.
When applied to cosmology, it offers an intuitive understanding of the fate of
the Universe even without solving the equations.

In the subsequent subsections, we construct the autonomous equations and
henceforth perform dynamic system analysis for background cosmology with GUP
modification.

We know that the Einstein’s equation are second order in nature. However, to
perform a dynamical system analysis, they must be transformed into a set of
first-order differential equations. A widely practiced method is to begin by
normalizing the Friedmann equation by the square of the Hubble parameter to
make each term dimensionless. In the process, it brings all the components
contributing to the Hubble rate on equal footing. The next step is to write
the equation of motion for each independent dimensionless dynamical variable
obtained with the help of the Raychaudhuri and Klien-Gordon equation. The
final set of equations for each variable, when expressed entirely in terms of
newly defined dimensionless variables, constitutes the required autonomous
system.

###  III.1 For $V(\phi)=V_{0}\phi^{2}$

In this subsection, we study qualitative dynamics (DSA) for a universe
dominated by a phantom scalar field with quadratic potential considering GUP
correction.

Following the above-mentioned recipe, we perform dynamic system analysis by
constructing autonomous equations for the chosen potential. This will allow us
to perform fixed point analysis and study the behavior of phase portraits.
From the eq.(50), the expansion normalized Friedmann equation can be written
as

$4\alpha
a^{3}\dot{\phi}^{2}+3=-\frac{\dot{\phi}^{2}}{2H^{2}}+\frac{V(\phi)}{H^{2}}-\frac{2\alpha
a^{3}\dot{\phi}^{2}}{3}\frac{\dot{\phi}^{2}}{2H^{2}}.$ (52)

A suitable choice of dimensionless dynamical also called, Einstein-Nordstrom,
variable is

$\displaystyle x$ $\displaystyle\equiv$
$\displaystyle\sqrt{\frac{\dot{\phi}^{2}}{6H^{2}}},\quad
y\equiv\sqrt{\frac{V_{0}\phi^{2}}{3H^{2}}},\quad z\equiv\sqrt{\alpha
a^{3}\dot{\phi}^{2}}.$ (53)

Expressing in terms of dynamic variables, the expansion-normalized Friedmann
equation is

$\displaystyle x^{2}+y^{2}-\frac{2}{3}z^{2}x^{2}-\frac{4}{3}z^{2}=1.$ (54)

In addition to this from eq.(8) the KG equation is

$\ddot{\phi}+3\dot{\phi}H-2V_{0}\phi=0.$ (55)

From eq.(49) we can obtain,

$\displaystyle\frac{\dot{H}}{H^{2}}$ $\displaystyle=$
$\displaystyle\frac{3}{2}(x^{2}+y^{2}-1)$ (56)
$\displaystyle+\frac{(2+6x^{4}+8x^{2})(-1-x^{2}+y^{2})}{(2x^{4}+4x^{2})}.$

With this, the ground has been prepared to construct the autonomous
differential equation for EN variables. However, it should be noted that the
EN variables alone fail to close the autonomous system for power law
potential. For example, a new dynamic variable depending on $\phi$ appears in
the form of $\lambda\equiv-\frac{V,_{\phi}}{V}$ with its independent equation
of motion.

The physical phase space for a power law system is always represented by a
positive y half-cylinder stretching from $\lambda=0$ to $+\infty$ due to
symmetry[72, 73, 74, 75], which means that the phase space is not compact. To
make phase space compact, we choose a new dynamic variable $u$, which is

$u=\frac{\lambda}{\lambda+1}.$

This transformation makes our phase space compact with the range $0\leq u\leq
1$. finally we can write our system of equations in terms of $u$. The
autonomous set of dynamical equations for quadratic potential with GUP
correction read as

$\displaystyle f(x,y)$ $\displaystyle\equiv$
$\displaystyle\frac{dx}{dN}=-3(1-u)x-u\left(\sqrt{6}y^{2}\right)-(1-u)x\left\\{\frac{3}{2}(x^{2}+y^{2}-1)+\frac{(2+6x^{4}+8x^{2})(-1-x^{2}+y^{2})}{(2x^{4}+4x^{2})}\right\\},$
(57) $\displaystyle g(x,y)$ $\displaystyle\equiv$
$\displaystyle\frac{dy}{dN}=\sqrt{6}yxu-y(1-u)\left\\{\frac{3}{2}(x^{2}+y^{2}-1)+\frac{(2+6x^{4}+8x^{2})(-1-x^{2}+y^{2})}{(2x^{4}+4x^{2})}\right\\},$
(58) $\displaystyle h(x,y)$ $\displaystyle\equiv$
$\displaystyle\frac{du}{dN}=-\sqrt{6}(\Gamma-1)(1-u)xz^{2},$ (59)

where $\Gamma=\frac{VV_{,\phi\phi}}{V_{,\phi}^{2}}$. The complete cosmological
dynamics for the universe with quadratic potential and GUP correction are
contained in the three equations 58, 59 and 59.

####  III.1.1 Fixed point analysis

In this subsection, we perform a thorough analysis of the fixed points of the
cosmological system dictated by a phantom scalar field with quadratic
potential in a GUP-modified scenario. Later, we compare the results with those
without GUP.

Fixed points are obtained by simultaneously solving the autonomous system to
zero. This physically means that the system becomes stationary at that point.
In our present case, this is given by the set of all points for which
autonomous equations(58-59) simultaneously equal to zero . We apply linear
stability analysis to comment on the nature of the fixed points. The fixed
points along with their behavior, as obtained, are tabulated in 2. To compare
our results with the original dynamics, we turn off the quantum perturbation
by setting $\alpha=0$ in Eqs.(58-59). The results are summarized in Table (1).

Comparison of the tables 1 and 2 reveals the dynamics without GUP are richer
than the dynamics after GUP modification. This is clear as 1 depicting the
dynamics without GUP contains a greater number of fixed points than in 2 with
GUP. However, introducing distortion due to GUP does not change the fixed
points as the eigenvalues in both tables contain zero. The theory of linear
stability analysis has no say on the properties of such fixed points. One
requires higher order analysis. However, in this paper, we confine ourselves
to linear stability analysis.

Table 1: Fixed points and the stability analysis for the Potential
$V(\phi)=V_{0}\phi^{2}$ without GUP

| x | y | $u$ | $E_{1}$ | $E_{2}$ | $E_{3}$ | Stability
---|---|---|---|---|---|---|---
A | 0 | 0 | c | $-3(1-c)/2$ | $3(1-c)/2$ | 0 | saddle point
B | 0 | 0 | 0 | $-3/2$ | $3/2$ | 0 | saddle point
C | 0 | 0 | 1 | 0 | 0 | 0 | neutral point
D | $c$ | 0 | 1 | $2\sqrt{6}c$ | $-\sqrt{6}c$ | 0 | saddle point
E | 0 | -1 | 0 | $-3$ | $-3$ | 0 | stable point
F | 0 | 1 | 0 | $-3$ | $-3$ | 0 | stable point

Table 2: Fixed points and the stability analysis for the Potential
$V(\phi)=V_{0}\phi^{2}$ with GUP

| x | y | $u$ | $E_{1}$ | $E_{2}$ | $E_{3}$ | Stability
---|---|---|---|---|---|---|---
A |  $+c$(other than 0) | 0 | 1 | 0 | $2\sqrt{6}c$ | $-\left(\frac{2\sqrt{6}c^{3}+\sqrt{6}c^{5}}{c^{2}(2+c^{2})}\right)$ | saddle point
B |  $-c$(other than 0) | 0 | 1 | 0 | $-2\sqrt{6}c$ | $\left(\frac{2\sqrt{6}c^{3}+\sqrt{6}c^{5}}{c^{2}(2+c^{2})}\right)$ | saddle point

###  III.2 Exponential Potential

In this subsection, we consider a potential of the form

$V(\phi)=V_{0}e^{-k\phi},$ (60)

to study the GUP modified background dynamics. EN variables are defined as

$\displaystyle x\equiv\sqrt{\frac{\dot{\phi}^{2}}{6H^{2}}},\quad
y\equiv\sqrt{\frac{V_{0}e^{-k\phi}}{3H^{2}}},\quad z\equiv\sqrt{\alpha
a^{3}\dot{\phi}^{2}}.~{}~{}~{}~{}$ (61)

The expansion-normalized Friedmann equation in the form of EN variables is

$\displaystyle x^{2}+y^{2}-\frac{2}{3}z^{2}x^{2}-\frac{4}{3}z^{2}=1,$ (62)

and from eq.(8) the Klein Gordon equation reads as

$\ddot{\phi}+3\dot{\phi}H+V_{0}ke^{-k\phi}=0.$ (63)

Furthermore, the Raychaudhuri equation (49) in terms of EN variables is

$\displaystyle\frac{\dot{H}}{H^{2}}$ $\displaystyle=$
$\displaystyle\frac{3}{2}(x^{2}+y^{2}-1)$ (64)
$\displaystyle+\frac{(2+6x^{4}+8x^{2})(-1-x^{2}+y^{2})}{(2x^{4}+4x^{2})}.$

Because our potential is exponential, we can get
$\Gamma=\frac{VV_{,\phi\phi}}{V_{,\phi}^{2}}$ equal to 1. While constructing
the autonomous equation, we obtain a factor of
$Q\equiv-\frac{V_{,\phi}}{V}=k$, which is equal to a constant $k$. With these,
we can write the autonomous differential equations as:

$\displaystyle\tilde{f}(x,y)$ $\displaystyle=$
$\displaystyle\frac{dx}{dN}=-3x-\sqrt{\frac{3}{2}}Qy^{2}-x\left\\{\frac{3}{2}(x^{2}+y^{2}-1)+\frac{(2+6x^{4}+8x^{2})(-1-x^{2}+y^{2})}{(2x^{4}+4x^{2})}\right\\},$
(65) $\displaystyle\tilde{g}(x,y)$ $\displaystyle=$
$\displaystyle\frac{dy}{dN}=-\sqrt{\frac{3}{2}}Qyx-y\left\\{\frac{3}{2}(x^{2}+y^{2}-1)+\frac{(2+6x^{4}+8x^{2})(-1-x^{2}+y^{2})}{(2x^{4}+4x^{2})}\right\\}.$
(66)

The complete dynamics of the system with exponential potential and GUP
corrections are contained in the two equations 66 and 66. The original
dynamics without quantum fluctuations can be easily obtained as a limiting
case by setting $\alpha=0$.

####  III.2.1 Fixed point analysis

In this subsection, we perform dynamical system analysis for the exponential
potential both with and without GUP. Because of the exponential potential, the
system can be completely explained by two dynamical variables $x$ and $y$. The
fixed points of the system with the eigenvalues of the Jacobian are summarized
in the table 3 and 4, respectively, for GUP and without GUP.

Table 3: Fixed points and corresponding eigenvalues of the Potential
$V(\phi)=V_{0}e^{-k\phi}$ without GUP

| x | y | $E_{1}$ | $E_{2}$ | Stability
---|---|---|---|---|---
A | $-Q/\sqrt{6}$ | $\sqrt{(Q^{2}+6)/6}$ | $-(6+Q^{2})/2$ | $-(3+Q^{2})$ | stable point
B | $-Q/\sqrt{6}$ | $-\sqrt{(Q^{2}+6)/6}$ | $-(6+Q^{2})/2$ | $-(3+Q^{2})$ | stable point
C | 0 | 0 | $-3/2$ | $3/2$ | saddle point

Table 4: Fixed points and the corresponding eigenvalues of the Potential
$V(\phi)=V_{0}e^{-k\phi}$

| x | y | $E_{1}$ | $E_{2}$ | Stability
---|---|---|---|---|---
A | $-\frac{Q}{\sqrt{6}}$ | $\sqrt{Q^{2}+6/6}$ | $\frac{-72Q^{2}-18Q^{4}-Q^{6}}{2Q^{2}(12+Q^{2}))}$ | $-\frac{72+84Q^{2}+21Q^{4}+Q^{6}}{Q^{2}(12+Q^{2}))}$ | stable point
B | $-\frac{Q}{\sqrt{6}}$ | $-\sqrt{Q^{2}+6/6}$ | $\frac{-72Q^{2}-18Q^{4}-Q^{6}}{2Q^{2}(12+Q^{2}))}$ | $-\frac{72+84Q^{2}+21Q^{4}+Q^{6}}{Q^{2}(12+Q^{2}))}$ | stable point

The rigorous fixed point analysis shows that the introduction of GUP
correction completely alters the fixed points and hence their stability in the
case of exponential potential. It is found that the dynamics without GUP have
two stable fixed points and one saddle point, whereas the GUP-modified
dynamics for exponential potential leave us with two stable fixed points.

####  III.2.2 Phase portrait

However, from the physical perspective, only the upper half of the plane,
$y\geq 0$, is of meaning. This is because $y\leq 0$ is not physically
achievable for a positive potential. Thus, we discard the set of the points,
including the fixed points, with $y\leq 0$ as not being physical. The dynamic
variable $x$ is proportional to the velocity. In both cases, with and without
GUP,

The phase portraits of the system are presented in Fig.(1) for $Q=1$. the
saddle point $C=(0,0)$ disappears upon the introduction of the GUP distortion.
However, the physically relevant point $A=(-0.40,1.08)$ remains there even
after quantum correction. Physically, the stable fixed point $A=(-0.40,1.08)$
implies that at a later time, the scalar field settles down to a negative
velocity value with a positive field value.

Figure 1: Phase portrait for potential $V(\phi)=V_{0}e^{-k\phi}$ for value of
$Q=1$. Left without GUP fluctuation where the stable fixed points are at
A=(-0.40,1.08), B=(-0.40,-1.08) and a saddle fixed point at C=(0,0) and Right
with GUP where the stable fixed point are A=(-0.40,1.08), B=(-0.40,-1.08)

##  IV  GUP-modified inflationary scenario

In this section, we present inflationary dynamics under the influence of the
Generalized Uncertainty Principle (GUP) for quadratic and exponential
potentials. To understand the change in the behavior of the background
dynamics of our cosmology in the presence of GUP distortion, we need to study
the behavior of cosmologically relevant parameters like; of scale factor a(t),
scalar field $\phi(t)$ , Equation of State (EoS) ($w_{eff}$) and slow climb
parameters $\epsilon$ and $\eta$ with and without GUP deformation [76]. In
what follows, we do this case by case for each potential considered in this
article.

####  IV.0.1 Quadratic potential, $V(\phi)=\mu^{2}\phi^{2}$

During cosmic inflation, the behavior of a phantom field differs from that of
a normal scalar field. While a normal scalar field undergoes a slow roll along
its potential, a phantom field exhibits a slow climb along its potential. This
distinction is evident both mathematically and graphically, as demonstrated in
[77] for power-law potentials.
After introducing GUP corrections to the Friedmann equations, the slow climb
parameters defined as $\epsilon\equiv-\frac{\dot{H}}{H^{2}}$,
$\eta\equiv\frac{V"}{3H^{2}}$ and $\delta\equiv\eta-\epsilon$ are expressed as
follows:

$\epsilon=\frac{3}{2}\left(\frac{\dot{\phi}^{2}}{6H^{2}}+\frac{V_{0}\phi^{2}}{3H^{2}}-1\right)+4\alpha
a^{3}H^{2}+\frac{\alpha a^{3}\dot{\phi}^{4}}{3H^{2}}+\frac{8}{3}\alpha
a^{3}\dot{\phi}^{2}.\\\ $ (67)

For $V(\phi)=V_{0}\phi^{2}$, we have

$\eta=\frac{2V_{0}}{3H^{2}}.$ (68)

Figure 2: Left panel: Plot of $\epsilon(t)$ vs time(t) for the initial
condition as $a(0)=1$, $\dot{a}(0)=1$ and $\dot{\phi}(0)=0.1$ for three
different values of $\alpha$ where, $\alpha=0$ shows no GUP fluctuation. Right
panel: For initial condition as $a(0)=4$, $\dot{a}(0)=4$ and
$\dot{\phi}(0)=0.1$.

Figure 3: $\eta(t)$ vs $t$ for the square potential.

In the cosmological context, EoS is a useful parameter to understand and
classify the acceleration and reacceleration phases of our universe. EoS is
the relationship between energy density and pressure. For $w=0$ corresponds to
non-relativistic matter such as cold dark matter (CDM) or non-relativistic
baryonic matter and $0<w<1/3$ refers to radiation dominated. For
$w=-1$,$-1<w<-1/3$ and $w<-1$ refer to the cosmological constant,
Quintessence, and Phantom eras, respectively.
The Raychaudhuri equation, in the standard case, can be written in terms of
EoS as follows:

$\frac{\ddot{a}}{a}=-(1+3w)\frac{\rho}{6},$ (69)

the GUP modified Raychaudhuri Eq.(49) looks like

$\frac{\ddot{a}}{a}=-\left[1+3\left(\frac{\frac{-\dot{\phi}^{2}}{2}-V(\phi)-\frac{2\alpha
a^{3}\dot{\phi}^{4}}{3}-\frac{4\alpha
a^{3}\dot{\phi}^{2}V(\phi)}{3}}{-\frac{\dot{\phi}^{2}}{2}+V(\phi)-\frac{\alpha
a^{3}\dot{\phi}^{4}}{3}}\right)\right]\frac{\rho}{6},$ (70)

comparing Eq.(70) with Eq.(49) we get

$w=\frac{\frac{-\dot{\phi}^{2}}{2}-V(\phi)-\frac{2\alpha
a^{3}\dot{\phi}^{4}}{3}-\frac{4\alpha
a^{3}\dot{\phi}^{2}V(\phi)}{3}}{-\frac{\dot{\phi}^{2}}{2}+V(\phi)-\frac{\alpha
a^{3}\dot{\phi}^{4}}{3}}.$ (71)

For the quadratic potential, the EoS parameter is

$w=\frac{\frac{-\dot{\phi}^{2}}{2}-V_{0}\phi^{2}-\frac{2\alpha
a^{3}\dot{\phi}^{4}}{3}-\frac{4\alpha
a^{3}\dot{\phi}^{2}V_{0}\phi^{2}}{3}}{-\frac{\dot{\phi}^{2}}{2}+V_{0}\phi^{2}-\frac{\alpha
a^{3}\dot{\phi}^{4}}{3}}.$ (72)

Figure 4: Comparison of EoS vs time for potential $V=\mu^{2}\phi^{2}$ for
different values of $\alpha$.

Figure 5: Left panel : Plot of $a(t)$ along the Y-axis vs time($t$) along the
X-axis for three values of $\alpha=0,0.01,0.9$ respectively in
$V=\mu^{2}\phi^{2}$ potential with initial condition as $a(0)=1$ and
$\phi(0)=1$ and $\dot{\phi}(0)=0.1$. Right panel : Magnified view.

Figure 6: Left panel : Plot of $a(t)$ along the Y-axis vs time($t$) along the
X-axis for three values of $\alpha=0,0.01,0.9$ respectively in
$V=\mu^{2}\phi^{2}$ potential with initial condition as $a(0)=0.1$ and
$\phi(0)=0.1$. Right panel : Magnified view.

Figure 7: Left panel : Plot of logarithm of $a(t)$ along the Y-axis vs
time($t$) along the X-axis for three values of $\alpha=0,0.01,0.9$
respectively in $V=\mu^{2}\phi^{2}$ potential with initial condition as
$a(0)=1$ and $\phi(0)=1$. Right panel : Magnified view.

Figure 8: Left panel : Plot of logarithm of $a(t)$ along the Y-axis vs
time($t$) along the X-axis for three values of $\alpha=0,0.01,0.9$
respectively in $V=\mu^{2}\phi^{2}$ potential with initial condition as
$a(0)=0.1$ and $\phi(0)=0.1$. Right panel : Magnified view.

Figure 9: Left panel : Plot of $\phi(t)$ along the Y-axis vs time($t$) along
the X-axis for three values of $\alpha=0,0.01,0.9$ respectively in
$V=\mu^{2}\phi^{2}$ potential with initial condition as $a(0)=1$ and
$\phi(0)=1$. Right panel : Magnified view.

Figure 10: Left panel : Plot of $\phi(t)$ along the Y-axis vs time($t$) along
the X-axis for three values of $\alpha=0,0.01,0.9$ respectively in
$V=\mu^{2}\phi^{2}$ potential with initial condition as $a(0)=0.1$ and
$\phi(0)=0.1$. Right panel : Magnified view.

Fig.(5) represents dynamics of scale factor with time, illustrating a nearly
exponential rise of the scale factor, indicating inflation. However, the
effect of GUP is not readily discernible from the left side of Fig.(5) To
observe this effect, it is necessary to magnify the range presented on the
right side of Fig.(5). The blue line represents the original scale factor
without GUP modification. As we increase the strength of ($\alpha$), the value
of the original scale factor becomes nonlinearly distorted. In addition, we
present the behavior of the scale factor for different initial conditions in
Fig.(6).
The behavior of $\phi$ w.r.t cosmic time t for different initial conditions is
shown in Fig.(9) and Fig.(10). A magnified view of the effect of GUP
deformation for different values of $\alpha$ is provided on the right-hand
side of the figures. On the left side of the figures, the scalar field starts
from a very low value and then increases linearly upward for different initial
conditions. As we can observe from the graph of $\phi$ vs t, the inclusion of
a higher value of GUP-strength, $\alpha$, in the dynamics causes the evolution
to distort upwards from the original dynamics without GUP.
The Fig.(4) depicts the behavior of the EoS parameter with the introduction of
a small strength of GUP distortion. We observe that the EoS for the phantom
remains always less than $-1$ in the absence of GUP fluctuations due to the
negative pressure term. Even in this case of GUP-modified dynamics, the
behavior of the EoS remains the same for most of the evolution. Only at the
later phase EoS increases w.r.t the unperturbed dynamics.
In Fig.(2),Fig.(3) we present slow climb parameter $|\epsilon|$ and $|\eta|$
as a function of time with different initial conditions and we clearly observe
that value of $|\eta|$,$|\epsilon|<<1$ indicate inflation.

####  IV.0.2 Exponential potential, $V(\phi)=V_{0}e^{-k\phi}$

In this subsection, we study the GUP-modified background dynamics for
exponential potential. We determine the slow roll parameters and equation of
state and subsequently present them for different initial conditions.
The slow climb parameters are

$\epsilon=\frac{3}{2}\left(\frac{\dot{\phi}^{2}}{6H^{2}}+\frac{V_{0}e^{-k\phi}}{3H^{2}}-1\right)+4\alpha
a^{3}H^{2}+\frac{\alpha a^{3}\dot{\phi}^{4}}{3H^{2}}+\frac{8}{3}\alpha
a^{3}\dot{\phi}^{2},\\\ $ (73)

and

$\eta=\frac{V_{0}k^{2}e^{-k\phi}}{3H^{2}}.$ (74)

The effective equation of state (EoS) for $V(\phi)=V_{0}e^{-k\phi}$ is

$w=\frac{\frac{-\dot{\phi}^{2}}{2}-V_{0}e^{-k\phi}-\frac{2\alpha
a^{3}\dot{\phi}^{4}}{3}-\frac{4\alpha
a^{3}\dot{\phi}^{2}e^{-k\phi}}{3}}{-\frac{\dot{\phi}^{2}}{2}+V_{0}e^{-k\phi}-\frac{\alpha
a^{3}\dot{\phi}^{4}}{3}}.$ (75)

Figure 11: Left panel: For $V(\phi)=V_{0}e^{-k\phi}$ with initial condition as
$a(0)=1$, $\dot{a}(0)=1$ and $\dot{\phi}(0)=0.1$ plot of $\epsilon(t)$ along
Y-axis vs time(t) along X-axis for three different values of $\alpha$ where,
$\alpha=0$ with no GUP fluctuation. Right panel: For initial condition as
$a(0)=0.1$, $\dot{a}(0)=0.1$ and $\dot{\phi}(0)=0.1$.

Figure 12: $\eta(t)$ vs $t$ for exponential potential.

Figure 13: Comparison of EoS vs time graph for potential $V=e^{-k\phi}$ for
different values of $\alpha$.

Figure 14: Left panel : Plot of $a(t)$ along Y-axis vs time($t$) along X-axis
for three values of $\alpha=0,0.01,0.9$ respectively in $V=e^{-k\phi}$
potential with initial condition as $a(0)=1$ and $\phi(0)=1$. Right panel :
Magnified view.

Figure 15: Left panel : Plot of $a(t)$ along Y-axis vs time($t$) along X-axis
for three values of $\alpha=0,0.01,0.9$ respectively in $V=e^{-k\phi}$
potential with initial condition as $a(0)=0.1$ and $\phi(0)=0.1$. Right panel
: Magnified view.

Figure 16: Left panel : Plot of logarithm of $a(t)$ along Y-axis vs time($t$)
along X-axis for three values of $\alpha=0,0.01,0.9$ respectively in
$V=e^{-k\phi}$ potential with initial condition as $a(0)=1$ and $\phi(0)=1$.
Right panel : Magnified view.

Figure 17: Left panel : Plot of logarithm of $a(t)$ along Y-axis vs time($t$)
along X-axis for three values of $\alpha=0,0.01,0.9$ respectively in
$V=e^{-k\phi}$ potential with initial condition as $a(0)=0.1$ and
$\phi(0)=0.1$. Right panel : Magnified view.

Figure 18:  Left panel : Plot of $\phi(t)$ along Y-axis vs time($t$) along
X-axis for three values of $\alpha=0,0.01,0.9$ respectively in $V=e^{-k\phi}$
potential with initial condition as $a(0)=1$ and $\phi(0)=1$ . Right panel :
Magnified view.

Figure 19: Left panel : Plot of $\phi(t)$ along Y-axis vs time($t$) along
X-axis for three values of $\alpha=0,0.01,0.9$ respectively in $V=e^{-k\phi}$
potential with initial condition as $a(0)=0.1$ and $\phi(0)=0.1$. Right panel
: Magnified view.

We analyze the effect of GUP modification on the background dynamics for the
exponential potential from Fig.(14),(15),(16) and (17). The figures depict the
line representing the small strength of $\alpha$, $\alpha=0.01$ is closer to
the original dynamics. However, as we increase the strength of the GUP
modification, the dynamics deviate further from the original dynamics.
The behavior of the EoS parameter in the phantom field with and without the
effect of GUP is represented in Fig.(13). The EoS starts with a value of -1,
indicating proximity to the Cosmological constant era, and transitions toward
the phantom-dominated era for both GUP and without GUP dynamics.
The behavior of $\phi$ and t for exponential potentials with different initial
conditions are shown in Fig.(18) and Fig.(19). In addition, a magnified view
of the GUP deformation for different values of $\alpha$ is shown on the right-
hand side of the Fig.(18) and Fig.(19). On the left side of the scalar field,
it starts from the highest value and then decreases linearly downward for
different initial conditions. We observe from the graph of $\phi$ vs t that
the inclusion of a comparable level of GUP in the dynamics causes the line to
distort downward from the original blue line, which represents the dynamics
without GUP. Furthermore, the plots of $|\epsilon|$ and $|\eta|$ in Fig.(11)
and Fig.(12) indicates inflation in the case of exponential potential.

##  V  Conclusion

In this paper, we comprehensively review the construction of the GUP-corrected
effective Hamiltonian from the scratch, ie. Einstein-Hilbert action. To this
effect, we consider a minimally coupled phantom scalar field with the
cosmological constant as the toy model. Following this, we perform the same
exercise with arbitrary potentials. In particular, we focus on introducing
momentum deformation to our dynamics due to the generalized uncertainty
principle. Having derived the effective Hamiltonian, we obtain all the
background equations of motion in terms of the Raychaudhuri, Friedmann, and
Klien-Gordon equations. Interestingly, we show that the Klien-Gordon equation
is free from any explicit quantum correction due to GUP. On the other hand,
the Raychaudhuri and Friedmann equation receives quantum correction
explicitly. This situation is consistent with the cosmological models
constructed using the framework of loop quantum gravity.

The system of equations obtained is highly nonlinear. This demands qualitative
analysis using the tools of dynamical system analysis to extract information
about the system before actually solving it.

We achieve this by performing a detailed dynamical system analysis using the
tools of linear stability analysis. We observe that the introduction of GUP
affects the local behavior of the system, although the overall dynamics remain
similar by and large. This is confirmed from Fig.(1) and from table(1,2,4,3).
In the case of quadratic potential, we observe from table(1,2) that after
introducing GUP distortion to the dynamics, certain fixed points disappear
from the scenario. In the case of exponential potential, table(4,3) indicates
that after introducing GUP corrections, only one saddle point disappeared.
Consequently, this leaves us with two stable points in the GUP modified
scenario.

As our final goal, we return to the question of the cosmological implications
of the considered model in Sec.(IV). We discuss inflationary scenarios after
GUP correction through plots of scale factor and the logarithm of scale factor
for different sets of initial conditions in Fig.(5,6,7,8)for square potential
and in Fig.(14,15,16,17) for exponential potential. We observe nearly
exponential expansion for both potentials, indicating inflation. Finally, we
calculated the slow climb parameters for both potentials and plotted them in
Fig.(2,3,11,12) which clearly shows inflation because of the values
$|\eta|$,$|\epsilon|<<1$. This confirms the inflationary scenario in this
study.

Furthermore, we calculated the GUP-induced EoS parameter and plotted it in
Fig.(4,13) starting nearly from -1 and then decreasing to more negative values
for phantom scenarios. To be precise in the case of quadratic potential, when
we incorporate the GUP correction term, the graph of the Equation of Sate
(EoS) parameter rapidly approaches the value $-1$ compared to the case without
GUP corrections. We also observe a deviation from the original dynamics.

Our analysis is general. In the future, we propose to extend our analysis to
more viable models of inflationary scenarios. Also, the effects of GUP
correction have not been studied in the context of linearized gravity yet. We
leave it as our future project.

## References

  * [1] N. C. Rana, ”An investigation of the motions of the node and perihelion of Mercury”, _A & A_, 181, 195(1987)
  * [2] Genov et al., ”Mimicking celestial mechanics in metamaterials”, _Nat. Phys._ , 5, 687(2009)
  * [3] R. Wojtak, S. H. Hansen, J. Hjorth, ”Gravitational redshift of galaxies in clusters as predicted by general relativity”, _Nature_ , 477, 567(2011)
  * [4] A. Albrecht, P. J. Steinhardt, ”Cosmology for grand unified theories with radiatively induced symmetry breaking”, _Phys. Rev. Lett._ , 48, 1220(1982)
  * [5] A. H. Guth, ”Inflationary universe: A possible solution to the horizon and flatness problems”, _Phys. Rev. D_ , 23, 347(1981)
  * [6] A. R. Liddle , D. H. Lyth, ”COBE, gravitational waves, inflation and extended inflation”, _Phys. Rev. B_ , 291, 391(1992)
  * [7] W. H. Kinney, ”TASI lectures on inflation”, _arXiv preprint arXiv:0902.1529_ , 1992
  * [8] A. Riotto, ”Inflation and the theory of cosmological perturbations”, _arXiv preprint hep-ph/0210162_ , 2002
  * [9] A.A.Starobinsky, ”A new type of isotropic cosmological models without singularity”, _Phys. Rev. B_ , 91, 99(1980)
  * [10] A.H. Guth, S.Y. Pi, ”Fluctuations in the new inflationary universe”, _Phys. Rev. Lett._ , 49, 1110(1982)
  * [11] A.D. Linde, ”A new inflationary universe scenario: a possible solution of the horizon, flatness, homogeneity, isotropy and primordial monopole problems”, _Phys. Rev. B_ , 108, 389(1982)
  * [12] J.M. Bardeen, P.J. Steinhardt, M.S. Turner, ”Spontaneous creation of almost scale-free density perturbations in an inflationary universe”, _Phys. Rev. D_ , 28, 679(1983)
  * [13] A. Ashtekar and M. Bojowald, ”Quantum geometry and the Schwarzschild singularity”, _Classical and Quantum Gravity_ , 23, 391(2005)
  * [14] A. Ashtekar, ”Loop quantum cosmology: an overview”, _Gen. Relativ. Gravit._ , 41, 707(2009)
  * [15] X. Calmet, M. Graesser, and Stephen D. Hsu, ”Minimum length from quantum mechanics and classical general relativity”, _Phys. rev. lett._ , 93, 211101(2004)
  * [16] D. Savickas, ”Relations between Newtonian Mechanics, general relativity, and quantum mechanics”, _Am. J. Phys._ , 70, 798(2002)
  * [17] B.G. Sidharth, ”Quantum mechanical black holes: Towards a unification of quantum mechanics and general relativity”, _arXiv preprint quant-ph/9808020_
  * [18] D. Savickas, ”The quantum state of the universe”, _Nucl. Phys. B._ , 239, 257(1984)
  * [19] A. Ashtekar and E. Bianchi, ”A short review of loop quantum gravity”, _Reports on Progress in Physics_ , 84, 042001(2021)
  * [20] C. Rovelli, ”Loop quantum gravity”, _Living rev. relativ._ , 11, 1(2008)
  * [21] C. Rovelli, ”Strings, loops and others: a critical survey of the present approaches to quantum gravity”, _arXiv preprint gr-qc/9803024_ , (1998)
  * [22] M. Sharma, T. Zhu and A Wang, ”Background dynamics of pre-inflationary scenario in Brans-Dicke loop quantum cosmology”, _Commun. Theor. Phys._ , 71, 1205(2019)
  * [23] L.J. Garay, ”Quantum gravity and minimum length”, _Int. J. Mod. Phys._ , 10, 145(1995)
  * [24] A. Kempf, ”Quantum groups and quantum field theory with nonzero minimal uncertainties in positions and momenta”, _Czechoslovak Journal of Physics_ , 44, 1041(1994)
  * [25] A. kempf, ”On quantum field theory with nonzero minimal uncertainties in positions and momenta”, _J. Math. Phys._ , 38, 1347(1997)
  * [26] G. Veneziano, ”A stringy nature needs just two constants”, _Europhys. Lett._ , 2, 199(1986)
  * [27] E. Witten, ”Reflections on the fate of spacetime”, _Phys. Today_ , 49, 24(1996)
  * [28] F. Scardigli, ”Generalized uncertainty principle in quantum gravity from micro-black hole gedanken experiment”, _Phys. Lett. B_ , 452, 39(1999)
  * [29] D. J. Gross, P. F. Mende, ”String theory beyond the Planck scale”, _Nucl. Phys. B._ , 303, 407(1988)
  * [30] D. Amati, M. Ciafaloni, G. Veneziano, ”Can spacetime be probed below the string size?”, _Phys. Lett. B_ , 216, 41(1989)
  * [31] T. Yoneya, ”ON THE INTERPRETATION OF MINIMAL LENGTH IN STRING THEORIES”, _Gen. Relativ. Gravit._ , 4, 16(1989)
  * [32] A. Ashtekar, S. Fairhurst and J.L. Willis, ”Quantum gravity, shadow states and quantum mechanics”, _Classical and Quantum Gravity_ , 20, 1031(2003)
  * [33] G.M. Hossain, V. Husain and S.S. Seahra, ”Background-independent quantization and the uncertainty principle”, _Gen. Relativ. Gravit._ , 27, 165013(2010)
  * [34] F. Scardigli et al., ”Modified Unruh effect from generalized uncertainty principle”, _Eur. Phys. J. C_ , 78, 1(2018)
  * [35] A.N. Tawfik and A.M. Diab, ”A review of the generalized uncertainty principle”, _Rep. Prog. Phys._ , 78, 1326001(2015)
  * [36] M. J. Lake, ”Generalised uncertainty relations and the problem of dark energy”, _arXiv preprint arXiv:2112.13938_ , (2021)
  * [37] C. A. Mead, ”Possible connection between gravitation and fundamental length”, _Phys. Rev._ , 135, B849(1964)
  * [38] C. A. Mead, ”String theory, supersymmetry, unification, and all that”, _Rev. Mod. Phys._ , 71, S112(1999)
  * [39] P. Bosso, ”Deformed algebra and the effective dynamics of the interior of black holes”, _Classical and Quantum Gravity_ , 38, 145006(2021)
  * [40] K. Nozari and A. Etemadi, ”Minimal length, maximal momentum, and Hilbert space representation of quantum mechanics”, _Phys. Rev. D_ , 85, 104029(2012)
  * [41] R. J. Adler and D. I. SantiagoA, ”On gravity and the uncertainty principleN GRAVITY AND THE UNCERTAINTY PRINCIPLE”, _Mod. Phys. Lett. A_ , 14, 1371(1999)
  * [42] V. Sahni, A. Starobinsky, ”The case for a positive cosmological $\Lambda$-term”, _Int. J. Mod. Phys. D_ , 9, 373(2000)
  * [43] L. P. Chimento and R. Lazkoz, ”On big rip singularities”, _Mod. Phys. Lett. A_ , 19, 2479(2004)
  * [44] Y.S. Piao and E. Zhou, ”Nearly scale-invariant spectrum of adiabatic fluctuations may be from a very slowly expanding phase of the Universe”, _Phys. Rev. D_ , 68, 083515(2003)
  * [45] P. F. Gonzalez-Diaz, ”Wormholes and ringholes in a dark-energy universe”, _Phys. Rev. D_ , 68, 084016(2003)
  * [46] B. Ratra and P. J. E. Peebles, ”Loop quantum cosmology: an overview”, _Phys.Rev. D_ , 37, 3406(1988)
  * [47] P. J. E. Peebles and B. Ratra, ”The cosmological constant and dark energy”, _Rev. Mod. Phys._ , 75, 559(2003)
  * [48] L. P. Chimento, A. S. Jakubi, ”Scalar field cosmologies with perfect fluid in Robertson-Walker metric”, _Int. J. Mod. Phys. D._ , 5, 71(1996)
  * [49] E. J. Copeland, M. Sami, S. Tsujikawa, ”Dynamics of dark energy”, _Int. J. Mod. Phys. D._ , 15, 1753(2006)
  * [50] T. Zhu et al., ”Pre-inflationary universe in loop quantum cosmology”, _Phys. Rev. D_ , 96, 083520(2017)
  * [51] B. F. Li, P. Singh, A. Wang, ”Qualitative dynamics and inflationary attractors in loop cosmology”, _Phys. Rev. D_ , 98, 066016(2018)
  * [52] B. Novosyadlyj, ”Quintessence versus phantom dark energy: the arbitrating power of current and future observations”, _J. Cosmol. Astropart. Phys._ , 2013, 042(2013)
  * [53] J. L. Lopez, M. Sabido, C. Yee-Romero, ”Phase space deformations in phantom cosmology”, _Phys. Dark Universe_ , 19, 104(2018)
  * [54] O. Lopez-Aguayo, J. C. Lopez-DomÃnguez and M. Sabido, ”On the Generalized Uncertainty Principle and Cosmology”, _arXiv preprint arXiv:2307.04667_ ,
  * [55] G. N. Remmen, S. M. Carroll, ”Attractor solutions in scalar-field cosmology”, _Phys. Rev. D_ , 88, 083518(2013)
  * [56] M. Shahalam et al., ”Preinflationary dynamics in loop quantum cosmology: Power-law potentials”, _Phys. Rev. D_ , 96, 123533(2017)
  * [57] A. Ashtekar, ”Preinflationary dynamics in loop quantum cosmology: Monodromy Potential”, _J. Cosmol. Astropart. Phys._ , 2018, 003(2018)
  * [58] M. Sharma et al., ”Quantum geometric formulation of Brans-Dicke theory for Bianchi I spacetime”, _arXiv preprint arXiv:2309.01080_ , (2023)
  * [59] M. Shahalam et al., ”Preinflationary dynamics in loop quantum cosmology: Power-law potentials”, _Phys. Rev. D_ , 96, 123533(2017)
  * [60] M. Bojowald, ”Loop quantum cosmology”, _Living Rev. Relativ._ , 11, 1(2008)
  * [61] A. Ashtekar and B. Gupt, ”Generalized effective description of loop quantum cosmology”, _Phys. Rev. D_ , 92, 024035(2015)
  * [62] A. Ashtekar et al., ”Loop quantum cosmology of k= 1 FRW models”, _Phys. Rev. D_ , 75, 024035(2007)
  * [63] M. G. Brown, K. Freese, W. H. Kinney, ”The phantom bounce: a new oscillating cosmology”, _J. Cosmol. Astropart.Phys._ , 2008, 2(2008)
  * [64] T. Zhu et al., ”Universal features of quantum bounce in loop quantum cosmology”, _Phys. Lett. B_ , 773, 196(2017)
  * [65] M. Sharma, S. D. Pathak and S. Li, ”Nonsingular bouncing model in closed and open universe”, _Phys. Sci. Rev._ , 7, 49(2023)
  * [66] D. Battefeld and P. Peter b, ”A critical review of classical bouncing cosmologies”, _Phys. Rep._ , 571, 1(2015)
  * [67] Y. F. Cai, D. A. Easson and R. Brandenberger, ”Towards a nonsingular bouncing cosmology”, _J. Cosmol. Astropart. Phys._ , 2012, 020(2012)
  * [68] A Alho, WC Lim and C Uggla, ”Cosmological global dynamical systems analysis”, _Classical and Quantum Gravity_ , 39, 145010(2022)
  * [69] M.Shahalam et al., ”Dynamics of coupled phantom and tachyon fields”, _Eur. Phys. J. C_ , 77, 1(2017)
  * [70] M.Shahalam et al., ”Dynamics of interacting quintessence”, _Eur. Phys. J. C_ , 75, 1(2015)
  * [71] J. Kaur et al., ”Inflection point of coupled quintessence”, _Astropart. Phys._ , 157, 102926(2024)
  * [72] Y. Gong, ”The general property of dynamical quintessence field”, _Phys. Lett. D_ , 731, 342(2014)
  * [73] S. Bahamonde et al., ”Dynamical systems applied to cosmology: dark energy and modified gravity”, _Phys. Rep._ , 775, 1(2018)
  * [74] S. C. C. Ng, N. J. Nunes, and F. Rosati, ”Applications of scalar attractor solutions to cosmology”, _Phys. Rev. D_ , 64 , 083510(2001)
  * [75] N. Roy and N. Banerjee, ”Quintessence scalar field: A dynamical systems study”, _Eur. Phys. J. Plus_ , 129, 1(2014)
  * [76] Yun-Song Piao and Yuan-Zhong Zhang, ”Phantom inflation and primordial perturbation spectrum”, _Phys. Rev. D_ , 70, 063513(2004)
  * [77] M. Sami and A. Toporensky, ”Phantom field and the fate of the universe”, _Mod. Phys. Lett. A_ , 19, 1509(2004)