Stability analysis of $f(Q)$ gravity models using Dynamical Systems Pooja
Vishwakarma1, Parth Shah2 1Department of Mathematics, VIT AP University,
Amaravati 522237, India 2Department of Mathematics, VIT AP University,
Amaravati 522237, India

###### Abstract

In recent years, the modified theory of gravity known as $f(Q)$ gravity has
drawn interest as a potential alternative to general relativity. According to
this theory, the gravitational force is determined by a function of the so-
called “non-metricity” tensor $Q$, which expresses how far a particle space-
time is from the metric geometry. In contrast to general relativity, which
describes the gravitational field using the curvature tensor, $f(Q)$ gravity
builds a theory of gravity using the non-metricity tensor. For this class of
theories, dynamical system analysis of the background and perturbation
equations has been carried out in this work to determine how various models
behave cosmologically. Here, the critical points are determined for two $f(Q)$
models from the literature: the power law, $f(Q)=Q+mQ^{n}$, and the
logarithmic, $f(Q)=\alpha+\beta logQ$ models. The stability behaviour and
corresponding cosmology are displayed for each critical point. For the power
law model, we achieve a matter-dominated saddle point with the right matter
perturbation growth rate. For the logarithmic model, we get a saddle point
dominated by the geometric component of the $f(Q)$ model with perturbations in
the decomposition of matter. For both models, we later achieved a stable and
accelerating Universe with constant matter perturbations.

Keywords: General relativity $\bullet$ Dynamical System analysis $\bullet$
Center Manifold Theory

##  I Introduction

The Universe’s present acceleration has been a hotly debated issue for the
past 20 years. The Universe’s expansion has begun to accelerate, according to
recent observations [1, 2, 3, 4, 5, 6, 7] of type Ia supernovae (SNIa), Large-
Scale Structure (LSS), Wilkinson Microwave Anisotropy Probe (WMAP) data,
Cosmic Microwave Background (CMB), and Baryonic Acoustic Oscillations (BAO).
The same observational data also shows that everything we currently see makes
up only $4$ percent of the Universe’s total mass, with the other $96$ percent
consisting of mysterious substances named Dark Energy (DE), which makes up
about $70$ percent and Dark Matter (DM), which makes up about $26$ percent.
The outcomes of these recent observations go counter to the standard Friedmann
equations of general relativity (GR), which are part of GR’s large-scale
applications to a homogeneous and isotropic Universe. GR may be a specific
instance of a more general theory rather than the final theory of gravitation,
as we would have imagined [8]. The major problem surfaces when one is trying
to address the Universe’s acceleration theoretically.

Two theories in the literature aim to explain why the Universe is expanding so
rapidly right now [9]. The first strategy in the GR framework is to alter the
composition of the Universe by introducing new elements of matter and energy,
such as DE of an as-of-yet unidentified type that has a significant negative
pressure. The popular candidate for DE is the well-known cosmological constant
$\Lambda$, which Einstein included in his field equations in GR and which
closely matches the previously described evidence. This is the incredibly
straightforward theory known as the $\Lambda CDM$ model, which can account for
the Universe’s cosmic acceleration in the late cosmos. Modifying the Einstein-
Hilbert action in GR theory is the second strategy for explaining the current
acceleration of the Universe’s expansion. The so-called $f(R)$ gravity [10,
11, 12], which generalizes the Hilbert-Einstein action to a generic function
of the Ricci Scalar $R$, is the simplest form of modification of GR. The Ricci
scalar $R$, which describes the curvature imported from Riemannian geometry,
is the essential idea in general relativity (GR). This modified theory of
gravity is renowned for accurately representing cosmic acceleration. It can
also accurately recreate the entire history of the cosmos and the behaviour of
the cosmological constant ($\Lambda$). Other modified theories include the
$f(R,T)$ theory [13, 14], $f(R,L_{m})$ theory [15, 16], where $L_{m}$ is the
Lagrangian density, and the $f(R,G)$ theory [17, 18], where $G$ denotes the
Gauss-Bonnet invariant, among many more.

There are further alternatives to general relativity (GR), such as the
teleparallel equivalent of GR (TEGR), which uses the concept of torsion $T$ to
describe gravitational interactions. Although the Weitzenbock link in
teleparallelism is related to torsion with zero curvature[19], the Levi-Civita
connection in general relativity is associated with curvature with zero
torsion. The $f(T)$ gravity is also the most straightforward adaptation of
TEGR [20, 21]. The symmetric teleparallel equivalent of general relativity
(STEGR), a novel theory of gravity, has recently been put out. In this theory,
the gravitational interactions are defined by the idea of non-metricity $Q$
with zero torsion and curvature [22, 23]. Weyl geometry (a generalization of
Riemannian geometry non-metricity) characterizes the variation in a vector’s
length in parallel transit and serves as the main geometrical factor
explaining the characteristics of gravitational interaction. In Weyl geometry,
$Q_{\gamma\mu\nu}=-\bigtriangledown_{\gamma}g_{\mu\nu}$ [24], is used to
mathematically derive the covariant derivative of the metric tensor, which is
not equal to zero. The $f(Q)$ gravity is additionally STEGR’s most
straightforward adaptation. Torsion and non-metricity, respectively, provide
fascinating geometric frameworks on which teleparallel and symmetric
teleparallel gravity can be constructed [25, 26, 27, 28, 29, 30].

Due to non-linear components in the field equations, one of the main issues
with theories of gravity is the difficulty in obtaining solutions (analytical
or numerical), making it impossible to compare them with observations. The
method known as “Dynamical System Analysis” is used to resolve such equations
or regulate overall dynamical behaviour. This strategy seeks out numerical
answers to better comprehend the qualitative behaviour of a particular
physical system [10, 31, 32, 33, 34, 35, 36, 37, 38, 39]. Finding the critical
points of a collection of first-order ordinary differential equations is the
most crucial idea in dynamical system analysis. By computing the Jacobian
matrix at critical points and determining its eigenvalues, the stability
criteria are determined. After locating the critical points and eigenvalues,
we can linearize the system around each point to determine the flow around
that point. Studying stability characteristics close to a specific critical
point is what this is about [40, 41, 42, 43, 44].

In this work, we examine the cosmic dynamics of $f(Q)$ gravity models at both
background and perturbation levels using the potent mathematical tool of
dynamical system analysis, which is motivated by the fascinating properties of
$f(Q)$ gravity. Such research can be used to strengthen the conclusions drawn
from the observational analysis. We point out that the dynamical system
technique is typically used at the background level, but it can also be used
at the perturbation level [45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57,
58, 59, 60]. This realization was made recently [61, 62, 64, 65]. So,
regardless of the precise initial conditions, we may determine the background
stable late-time solutions as well as the growth of the structure creation
using this combined approach. Additionally, we can look at the manner in which
the matter perturbations respond to the background solutions. The fact that
$f(Q)$ gravity automatically incorporates a preferred frame into the theory is
one of the main reasons to take it into consideration. The work is structured
as follows: We state the $f(Q)$ gravity field equations in Section II, from
which the background and perturbed cosmological equations can be derived. The
dynamical study of the combined system for the power-law and exponential
models is covered in Section III. In Section IV, the final results are
summarised.

##  II $f(Q)$ COSMOLOGY

Jimenez et al. [22, 23, 66, 67] introduced the symmetric teleparallel gravity,
often known as the $f(Q)$ gravity. This modified gravity’s action is provided
by

$S=\int[-\frac{1}{16\pi G}f(Q)+{\mathcal{L}}_{m}]\sqrt{-g}\,d^{4}x\ ,$ (1)

where $g$ is the determinant of the metric tensor $g_{\mu\nu}$, and
${\mathcal{L}}_{m}$ is Lagrangian density of matter. $f(Q)$ is an arbitrary
function of $Q$, the non-metricity scalar which is responsible for the
gravitational interaction.

The non-metricity tensor is given by

$Q_{\lambda\mu\nu}=\nabla_{\lambda}g_{\mu\nu}=\partial_{\lambda}g_{\mu\nu}-\Gamma^{\beta}_{\lambda\mu}g_{\beta\nu}-\Gamma^{\beta}_{\lambda\nu}g_{\mu\beta},$

where $\Gamma^{\beta}_{\mu\nu}$ is the affine connection and it can be
decomposed into following components:

$\Gamma^{\beta}_{\mu\nu}=\\{^{\beta}_{\mu\nu}\\}+K^{\beta}_{\mu\nu}+L^{\beta}_{\mu\nu}$
,

where the Levi-Civita connection $\\{^{\beta}_{\mu\nu}\\}$ is defined by

$\\{^{\beta}_{\mu\nu}\\}=\frac{1}{2}g^{\beta\sigma}(\partial_{\mu}g_{\sigma\nu}+\partial_{\nu}g_{\sigma\mu}-\partial_{\sigma}g_{\mu\nu})$
,

$L^{\beta}_{\mu\nu}$ is the deformation defined by

$L^{\beta}_{\mu\nu}=\frac{1}{2}Q^{\beta}_{\mu\nu}-Q_{(\mu^{\beta}\nu)}$ ,

and $K^{\beta}_{\mu\nu}$ is the contortion defined by

$K^{\beta}_{\mu\nu}=\frac{1}{2}T^{\beta}_{\mu\nu}+T_{(\mu^{\beta}\nu)}$ ,

with the torsion tensor $T^{\beta}_{\mu\nu}$ defined as the anti-symmetric
part of the affine connection,

$T^{\beta}_{\mu\nu}=2\Gamma^{\lambda}_{[\mu\nu]}$ .

The non-metricity scalar $Q$ is written as [67, 68]

$\begin{split}&Q=-\frac{1}{4}Q_{\alpha\beta\gamma}Q^{\alpha\beta\gamma}+\frac{1}{2}Q_{\alpha\beta\gamma}Q^{\gamma\beta\alpha}\\\
&+\frac{1}{4}Q_{\alpha}Q^{\alpha}-\frac{1}{2}Q_{\alpha}\bar{Q}^{\alpha},\end{split}$
(2)

where

$Q_{\alpha}\equiv Q_{\alpha^{\mu}\mu}$ and $\bar{Q}^{\alpha}\equiv
Q_{\mu}^{\mu\alpha}$

consist of two independence traces that were obtained by condensing the non-
metricity tensor

$Q_{\alpha\mu\nu}\equiv\bigtriangledown_{\alpha}g_{\mu\nu}$ ,

which enables us to define the non-metricity scalar as $Q=-Q_{\alpha\mu\nu}Q$.

To find the field equations, we set that action in (1) is constant with
respect to variations over the metric tensor $g_{\mu\nu}$ and setting $8\pi
G=1$ for simplicity, results in [68, 69]

$\begin{split}&\frac{2}{\sqrt{-g}}\bigtriangledown_{\alpha}\sqrt{-g}g_{\beta\nu}f_{Q}[-\frac{1}{2}L^{\alpha\mu\beta}+\frac{1}{4}g^{\mu\beta}(Q^{\alpha}-\bar{Q}^{\alpha})\\\
&-\frac{1}{8}(g^{\alpha\mu}Q^{\beta}+g^{\alpha\beta}Q^{\mu})]+f_{Q}[-\frac{1}{2}L^{\mu\alpha\beta}-\frac{1}{8}(g^{\mu\alpha}Q^{\beta}\\\
&+g^{\mu\beta}Q^{\alpha})+\frac{1}{4}g^{\alpha\beta}(Q^{\alpha}-\bar{Q}^{\alpha})]Q_{\nu\alpha\beta}+\frac{1}{2}\delta_{\nu}^{\mu}f=T^{\mu}_{\nu},\end{split}$
(3)

where $T_{\mu\nu}$ is the matter energy-momentum tensor , whose form is

$T_{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\delta(\sqrt{-g}{\mathcal{L}}_{m})}{\delta
g^{\mu\nu}}$ ,

and $f_{Q}=\frac{\partial f}{\partial Q}$.

At the background level, we assume a Friedmann-Lemaitre-Robertson-Walker
(FLRW) spacetime that is homogeneous, isotropic, and spatially flat, whose
metric is of the form

$ds^{2}=-N^{2}(t)dt^{2}+a^{2}(t)(dx^{2}+dy^{2}+dz^{2}),$ (4)

where $N(t)$ is a Lapse function and $t$ is the cosmic time. Because $Q$
theories retain a residual time reparameterization invariant, they allow for
the selection of a specific Lapse function. As a result, symmetry is used to
set $N(t)=1$ [22, 23, 70, 71]. The function $a(t)$ is known as the scale
factor, is a measure of the size of the Universe at time $t$ and $x,y,z$ are
the cartesian coordinates. Note that for the non-metricity scalar, we obtain
$Q=6H^{2}$ in FLRW metric, where $H=\frac{\dot{a}}{a}$ is the Hubble function
which is a measure of the Universe’s expansion rate at time $t$ and ‘.’
denotes derivative with respect to $t$. Imposing the splitting $f(Q)=Q+F(Q)$,
and applying the FLRW metric, the corresponding field equations are [67, 68,
69, 70, 71, 72]

$3H^{2}=\rho+\frac{F}{2}-QF_{Q},$ (5)
$(2QF_{QQ}+F_{Q}+1)\dot{H}+\frac{1}{4}(Q+2QF_{Q}-F)=-2p,$ (6)

where $F_{Q}=\frac{dF}{dQ}$ , $F_{QQ}=\frac{d^{2}F}{dQ^{2}}$ , $\rho$ and $p$
are the matter fluid’s energy density and pressure respectively. They satisfy
the energy conservation equation or the fluid equation in the absence of
interaction which is given by

$\dot{\rho}+3H(\rho+p)=0$ (7)

with a linear equation of state parameter between them given by
$p=\rho\omega$, where the equation of state parameter $\omega\in[-1,1]$. From
the Friedmann’s equation (5), we have

$1=\frac{\rho}{3H^{2}}+\frac{\frac{F}{2}-QF_{Q}}{3H^{2}}$.

For an accelerating Universe, one requires $\omega_{eff}<-\frac{1}{3}$ [10,
72] and therefore, it is convenient to introduce the energy density parameters
$\Omega_{m}$ and $\Omega_{Q}$ as

$\Omega_{m}=\frac{\rho}{3H^{2}},\
\Omega_{Q}=\frac{\frac{F}{2}-QF_{Q}}{3H^{2}}.$ (8)

Hence the Friedmann’s equation(5) can be simply written as

$\Omega_{m}+\Omega_{Q}=1.$ (9)

Introducing the effective total energy density $\rho_{eff}$ and total energy
pressure $p_{eff}$, as [71]

$\rho_{eff}\equiv\rho+\frac{F}{2}-QF_{Q},$ (10)
$p_{eff}\equiv\frac{\rho(1+\omega)}{2QF_{QQ}+F_{Q}+1}-\frac{Q}{2},$ (11)

the corresponding total equation of state $\omega_{eff}$ can be written as

$\omega_{eff}=\frac{p_{eff}}{\rho_{eff}}=-1+\frac{\Omega_{m}(1+\omega)}{2QF_{QQ}+F_{Q}+1}.$
(12)

For the investigation of the linear perturbation level, we focus on the matter
density contrast $\delta=\frac{\delta_{\rho}}{\rho}$, where $\delta_{\rho}$ is
the perturbation of the matter energy density. The evolution equation for
matter over density at the quasi-static limit [68, 69, 70] is

$\ddot{\delta}+2H\dot{\delta}=\frac{\rho\delta}{2(1+F_{Q})},$ (13)

where the right hand side’s denominator explains how an apparent Newton’s
constant appears. The terms involving temporal derivatives in the perturbed
equations are omitted in the quasi-static limit at tiny scales, well within
the cosmic horizon, leaving just spatial derivative terms [10, 72, 73].

##  III DYNAMICAL SYSTEM ANALYSIS OF THE MODELS

For a generic function $F(Q)$, we build the dynamical system of the background
and perturbed equations in this section. In this regard, we transform the
equations (5)-(7) and (13) into first order autonomous system by taking into
account the following dynamical variables:

$x=\frac{F}{6H^{2}},\ y=-2F_{Q},\ u=\frac{d(\ln\delta)}{d(\ln a)}.$ (14)

As a result, while variables $x$ and $y$ are related to Universe’s background
evolution, variable $u$ measures the expansion of matter disturbances. Hence,
at any point of time the matter density contrast is positive. $u>0$ denotes
the increase of matter perturbations and $u<0$ denotes their decay.

The cosmic background parameters $\Omega_{m}$ , $\Omega_{Q}$, and
$\omega_{\text{eff }}$ can be represented as [66]

$\Omega_{m}=1-x-y$ ,

$\Omega_{Q}=x+y$ ,

$\omega_{eff}=-1+\frac{(1-x-y)(1+\omega)}{2QF_{QQ}-\frac{y}{2}+1}.$ (15)

The cosmological equations can now be expressed as the following dynamical
system in terms of variables (14):

$x^{\prime}=-\frac{\dot{H}}{H^{2}}(y+2x),$ (16)
$y^{\prime}=2y\frac{\dot{H}}{H^{2}}\frac{QF_{QQ}}{F_{Q}},$ (17)
$u^{\prime}=-u(u+2)+\frac{3(1-x-y)}{2-y}-\frac{\dot{H}}{H^{2}}u,$ (18)

where $(^{\prime})$ denotes the differentiation with respect to $lna$ and
$(.)$ stands for differentiation with respect to $t$ and

$\frac{\dot{H}}{H^{2}}=-\frac{3-3(1-x-y)}{4QF_{QQ}-y+2}.$ (19)

The physical system is a product space of the perturbed space $\mathbb{P}$,
which contains the variable $u$ , and the background phase space $\mathbb{B}$,
which contains the variable $x$ and $y$. The combined system’s phase space
under the physical condition $0\leq\Omega_{m}\leq 1$ is

$\Psi=\mathbb{B}\times\mathbb{P}=(x,y,u)\in\mathbb{R}^{2}\times\mathbb{R}:0\leq
x+y\leq 1.$

Note that the projection of orbits of the product space $\Psi$ on space
$\mathbb{B}$ reduces to the corresponding background orbits.

The system’s dynamical evolution will be described in the following stage by
isolating its critical points and evaluating their stability. In terms of
physics, a stable point with $u>0$ denotes an unbounded expansion of matter
perturbations, and as a result, the system is not stable with respect to
matter perturbations. The system is asymptotically stable with respect to
perturbations because a stable point with $u<0$ denotes the decay of mater
disturbances. Finally, it is implied that matter perturbations remain constant
when $u=0$ at a stable position. In conclusion, the perturbations of matter
expand but do not last forever in an unstable or saddle point with $u>0$,
which is required to explain the matter epoch of the universe. This unstable
or saddle point must be followed by a stable late-time attractor with $u=0$,
which corresponds to acceleration [66].

We must define the function $F$ in order to move on to the specific analysis
and, as a result, establish the term $\frac{QF_{QQ}}{F_{Q}}$. In the
subsections that follow, we discuss two specific models that are known to
provide an intriguing cosmic phenomenology.

### MODEL I : $F(Q)=mQ^{n}$

We begin by taking into account a power-law model with

$F(Q)=mQ^{n}$ (20)

having two parameters, $m$ and $n$. This model fits the BBN requirement [74]
and can explain the acceleration of the Universe in the late Universe. We
point out that this model reduces to the symmetric teleparallel equivalent of
general relativity for $n=1$ and that for $n=0$ it is equivalent to the
concordance $\Lambda CDM$ scenario [68, 75, 76]. As

$QF_{QQ}=\frac{(1-n)}{2}y$

in this instance, the system (16)-(18) becomes,

$x^{\prime}=\frac{3(1-x-y)}{y(1-2n)+2}(y+2x),$ (21)
$y^{\prime}=\frac{3(1-x-y)}{y(1-2n)+2}2(1-n)y,$ (22)
$u^{\prime}=\frac{3(1-x-y)}{2-y}-u(u+2)+\frac{3(3-x-y)}{y(1-2n)+2}u.$ (23)

Four critical points can be found in the relevant dynamical system:

Critical Points |  Eigen-values |  $\mathbf{\Omega_{m}}$ |  $\mathbf{\omega_{eff}}$ |  $\mathbf{u}$ |  Stability condition
---|---|---|---|---|---
$(1-y,y,0)$ |  $(0,-2,-3)$ |  $0$ |  $-1$ |  $0$ |  Stable
$(1-y,y,-2)$ |  $(0,2,-3)$ |  $0$ |  $-1$ |  $-2$ |  Saddle
$(0,0,-\frac{3}{2})$ |  $(3,\frac{5}{2},-3(1-n))$ |  $1$ |  $0$ |  $-\frac{3}{2}$ |  Unstable for $n<1$, Saddle for $n>1$
$(0,0,1)$ |  $(3,-\frac{5}{2},-3(1-n))$ |  $1$ |  $0$ |  $1$ |  Saddle
Table 1: Critical points, Stability conditions, EoS parameter and matter
perturbation

Figure 1:  Phase Portrait for the dynamical system of Model I, (i)left
panel($u=0$); (ii)middle panel($y=0$); (iii)right panel($x=0$ and $n=0.5$)

* Critical Point $(1-y,y,0)$ : This point’s curve represents a solution in which the effective dark energy component dominates, or $\Omega_{Q}=1$, and the Universe accelerates with $\omega_{eff}=-1$, which is similar to the behaviour of a cosmological constant. In this case, it indicates that the critical point is dominated by the geometric component of $f(Q)$ model. Moreover, $u=0$ at the perturbation level means that the matter perturbation is still constant. The curve is one-dimensional with one vanishing eigenvalue, and other eigenvalues are $-2$ and $-3$. One may assess the stability of the curve by looking at the signature of the remaining non-vanishing eigenvalues. Hence, we draw the conclusion that it is consistently stable. In addition, there is one vanishing eigenvalue in the corresponding Jacobian matrix, which is the non-hyperbolic point. As a result, we discover that the system of equations is asymptotically stable at its origin using the Center Manifold Theory.

In this case $x$ is the central variable and $y,u$ are the stable variables.
The corresponding matrices are $A=0$ and $B=\begin{pmatrix}-2&0\\\
0&-3\end{pmatrix}$. The center manifold has now the form $y=h_{1}(x)$ and
$u=h_{2}(x)$; the approximation $N$ has two components:

$N_{1}(h_{1}(x))=h_{1}^{\prime}(x)\frac{3(1-x-y)}{y(1-2n)+2}(y+2x)-\frac{3(1-x-y)}{y(1-2n)+2}2(1-n)y,$

$N_{2}(h_{2}(x))=h_{2}^{\prime}(x)\frac{3(1-x-y)}{y(1-2n)+2}(y+2x)-\frac{3(1-x-y)}{2-y}+u(u+2)-\frac{3(1-x-y)}{y(1-2n)+2}u.$

For zeroth approximation:
$N_{1}(h_{1}(x))=0+\mathcal{O}(x^{2})$ and
$N_{2}(h_{2}(x))=-\frac{1}{2}(1-x)+\mathcal{O}(x^{2})$.
Therefore the reduced equation gives us

$u^{\prime}=-6+6x+O(x^{2})$

This gives us the negative constant part. Therefore, the system of equations
(21)-(23) is asymptotically stable at zero as per the Central Manifold Theory.

In conclusion, the argument outlines the dark-energy-dominated Universe of the
late Universe at background and perturbation levels.

* Critical Point $(1-y,y,-2)$: The curve of this critical point $(1-y,y,-2)$, like the curve of critical point $(1-y,y,0)$, corresponds to a solution with $\omega_{eff}=-1$, which is dominated by the geometric component of $f(Q)$ model. Also, $u=-2$ implies that it is characterised by the decay of matter disturbances. However, it is saddle with eigenvalues $(0,2,-3)$. As a result, unlike the curve of the critical point $(1-y,y,0)$, the critical point $(1-y,y,-2)$ cannot, at the perturbation level, reflect a late-time, dark-energy-dominated Universe. This point depicts the inflationary era of the Universe due to its saddle nature and negative $\omega_{eff}$ value.

* Critical Point $(0,0,-\frac{3}{2})$: This position, where $\Omega_{m}=1$ and $\omega_{eff}=0$, corresponds to the matter supremacy at the background level. This statement however, could not explain the development of structures at the perturbation level because the matter over density fluctuates as $\rho\propto a^{-\frac{3}{2}}$ when $u=-\frac{3}{2}$. The Jacobian matrix’s eigenvalues are $(3,\frac{5}{2},-3(1-n))$, making this point unstable for $n\geq 1$ and saddle for $n<1$.

* Critical Point $(0,0,1)$: The background parameters are $\Omega_{m}=1$ and $\omega_{eff}=0$, and this point corresponds to a matter-dominated critical solution. We have $u=1$ at the perturbation level, which means that the matter over density $\delta$ varies as the scale factor $\rho\propto a$ and hence rises with the expansion of the Universe. Given that the appropriate Jacobian matrix has the eigenvalues $(3,-\frac{5}{2},-3(1-n))$, the point $(0,0,1)$ is always a saddle point for any value of $n$. In light of this, the trajectories pass through this point and then depart from it after being drawn to a late-time stable point. As a result, we come to the conclusion that this point may be the best choice to explain how structures were formed during the matter-dominant period at both the background and perturbation levels.

### MODEL II : $F(Q)=\alpha+\beta logQ-Q$

We consider the logarithmic model in this subsection

$F(Q)=\alpha+\beta logQ-Q,$ (24)

which has two parameters, $\alpha$ and $\beta$. This model also fits in the
BBN requirements [74] and can explain the acceleration of the Universe in the
late Universe. As,

$QF_{QQ}=\frac{(y-2)}{2}$

in this instance, the system (16)-(18) becomes.

$x^{\prime}=\frac{3(1-x-y)}{y-2}(y+2x),$ (25)
$y^{\prime}=\frac{3(1-x-y)}{y-2}2(y-2),$ (26)
$u^{\prime}=\frac{3(1-x-y)}{2-y}-u(u+2)+\frac{3(1-x-y)}{y-2}u.$ (27)

Two critical points can be found in this dynamical system:

Critical Points |  Eigen-values |  $\mathbf{\Omega_{m}}$ |  $\mathbf{\omega_{eff}}$ |  $\mathbf{u}$ |  Stability condition
---|---|---|---|---|---
$(1-y,y,0)$ |  $(0,-3,-2)$ |  $0$ |  $-1$ |  $0$ |  Stable
$(1-y,y,-2)$ |  $(0,-3,2)$ |  $0$ |  $-1$ |  $-2$ |  Saddle
Table 2: Critical points, Stability conditions, EoS parameter and matter
perturbation

Figure 2:  Phase Portrait for the dynamical system of Model II, (i)left
panel($u=0$); (ii)middle panel($y=0$); (iii)right panel($x=0$)

* Critical Point $(1-y,y,0)$:  The curves at this time depict a solution in which the Universe accelerates with $\omega_{eff}=-1$ and $\omega_{Q}=1$, which is consistent with the behaviour of a cosmological constant dominant Universe. Moreover, the matter perturbation is still constant because $u=0$ at the perturbation level. The corresponding eigenvalues of the one-dimensional curve are $(0,-2,-3)$. By examining the signature of the non-vanishing eigenvalues one can judge the stability of the curve. So, we infer that it is consistently stable. Since it is a non-hyperbolic point due to one vanishing eigenvalue, we use the Center Manifold Theory to study the system of equations.

Here $x$ is the central variable and $y,u$ are the stable variables. The
corresponding matrices are $A=0$ and $B=\begin{pmatrix}-3&0\\\
0&-2\end{pmatrix}$. The center manifold has now the form $y=h_{1}(x)$ and
$u=h_{2}(x)$; the approximation $N$ has two components:

$N_{1}(h_{1}(x))=h_{1}^{\prime}(x)\frac{3(1-x-y)}{y-2}(y+2x)-\frac{3(1-x-y)}{y-2}2(y-2),$

$N_{2}(h_{2}(x))=h_{2}^{\prime}(x)\frac{3(1-x-y)}{y-2}(y+2x)-\frac{3(1-x-y)}{2-y}+u(u+2)-\frac{3(1-x-y)}{y-2}u.$

For zeroth approximation:
$N_{1}(h_{1}(x))=6(x-1)+\mathcal{O}(x^{2})$ and
$N_{2}(h_{2}(x))=-\frac{3}{2}(1-x)+\mathcal{O}(x^{2})$.
Therefore the reduced equation gives us

$u^{\prime}=-\frac{9}{4}\frac{(x-1)}{(3x-4)}(3x^{2}-10x+1)+\mathcal{O}(x^{2}).$

This gives us the negative constant part. Therefore, the equation (25)-(27) is
asymptotically stable at zero.

This point describes late Universe’s dark-energy-dominated background and
perturbation levels.

* Critical Point $(1-y,y,-2)$:  The curve of this critical point $(1-y,y,-2)$ is similar to the curve of the critical point $(1-y,y,0)$, and both curves corresponds to a solution with $\omega_{eff}=-1$, which is dominated by the geometric component of $f(Q)$ model. In addition, it exhibits perturbations in the decomposition of matter because $u=-2$. Eigenvalues $(0,2,-3)$ are present, though. Thus, this critical point $(1-y,y,-2)$ cannot, at the perturbation level, depict a late-time dark-energy-dominated Universe, in contrast to the curve of critical points $(1-y,y,0)$. This point depicts the inflationary era of the Universe due to its saddle nature and negative $\omega_{eff}$ value.

##  IV Discussion and Conclusion

In the current work, we performed a combined dynamical system analysis of both
background and perturbation equations in order to examine the validity of each
result using an independent method. The Universe is assumed to be described by
the flat FLRW metric. This work was motivated by the fact that cosmological
models based on $f(Q)$ gravity are very efficient in fitting observational
datasets at both background and perturbation levels.

After converting the background and perturbation equations into an autonomous
system, we concentrated on the power-law and exponential $f(Q)$ models that
have been examined in the literature. Due to the additional variable
associated with matter perturbations, each background critical point splits
into two points, each with a distinct behaviour of matter perturbations and
stability.

In the power-law model $f(Q)=Q+mQ^{n}$, there were four critical points, Out
of which $(1-y,y,0)$ point was found to be asymptotically stable at origin by
using Center Manifold Theory. The points $(1-y,y,-2)$ and $(0,0,1)$ were
saddle points where matter predominated and the growth rate of matter
perturbations was accurate. This was followed by a change to a stable,
accelerated Universe where dark energy predominated and the growth rate of
matter perturbations was constant, as shown through the critical points in
Table 1. The point $(0,0,-\frac{3}{2})$ was unstable for $n\geq 1$ and saddle
for $n<1$. We also plotted phase portraits in Figure 1, where the behaviour at
those points is more clearly represented. Our study shows that various
critical points correspond to various kinds of matter disturbances.
Furthermore, similar background critical points react in various ways at the
perturbation level. The points $(0,0,1)$ and $(0,0,-\frac{3}{2})$ describe the
background level of the decelerated matter-dominated era, but only point
$(0,0,1)$ exhibits the proper growth of matter structure. Also, Point
$(0,0,1)$ is a saddle, therefore it provides a natural exit to a late-time
accelerated period. However, at late times, defining the accelerated dark-
energy dominant epoch, the curves of critical points $(1-y,y,0)$ and
$(1-y,y,-2)$ are identical at the background level. However, only curve
$(1-y,y,0)$ displays continuous matter perturbations and is stable, making it
the only curve at the perturbation level that is physically and observably
interesting. The model describes the transition from matter domination towards
an accelerated dark-energy-dominated epoch. In conclusion, the power-law model
is capable of capturing both the background and perturbation levels of the
required thermal history of the universe. According to our investigation, the
aforementioned is true for any value of n.

In the logarithmic model $f(Q)=\alpha+\beta logQ$, there were two critical
points $(1-y,y,0)$ and $(1-y,y,-2)$. Using the Center Manifold Theory, the
point $(1-y,y,0)$ was found to be asymptotically stable at the origin. We also
obtained curves of points that correspond to matter dominance, the geometric
component of $f(Q)$ model dominance, and matter perturbation growth. The fact
that these curves are saddle-shaped allows a smooth transition to the stable
solution with constant matter perturbations, as demonstrated by the critical
points in Table 2. In Figure 2, we also showed phase portraits that more
accurately depict the behaviour at those points. Similar to Model I, we
observe that the distinction between critical points that are similar at the
background level is made by the presence of disturbances. Because of this, we
discover through the combined background and perturbation analysis that only
curve $(1-y,y,0)$ corresponds to late-time dark-energy dominance, with a
stable history of matter perturbations. As a final remark, in this article,
stability and acceleration phases are achieved without adding an “unobserved”
quantity, i.e., dark energy.

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Preliminaries sec:preliminaries This section serves as an introduction to some
essential concepts. These concepts lay the foundation for understanding the
backdrop of this survey report. Blockchain sec:blockchain A blockchain is a
distributed ledger with growing lists of records (blocks) that are securely
linked together via cryptographic hashes. enwiki:1220180266:blockchain. This
information is stored in blocks, where each block’s information is verifiable
by every participating computer. It’s designed to have decentralized
management instead of the traditional hierarchical systems we’re familiar
with. A dispersed structure like the blockchain helps to ensure trust,
validity, and usability.

A block in a blockchain primarily consists of:

  * •

List of transactions - Valid transactions signed by the sender are accepted
once they become a part of the accepted block.

  * •

Hash of the previous block - allows the block producer to commit to the
blockchain history it considers valid inductively. This renders blockchains
immutable.

A consensus mechanism dictates which proposed blocks are considered valid.
Blockchain participants broadcast valid blocks and proposed transactions they
hear to other connected network participants. We discuss these in detail in
the following sections.

###  Transaction

A ledger is a collection of transactions from one account to another.
Similarly, a blockchain is a collection of publicly accepted transactions. A
transaction can be a token ownership transfer, as in Bitcoin 111Apart from
transferring an owner’s UTXOs to a public key, other types of transactions are
also possible on Bitcoin using Script. In the case of a smart contract
transaction on Ethereum, SegWit, and Taproot updates further offer a wider
variety of transactions., or even a valid state change. In most blockchains,
transaction creators, aka users, whom we refer to as _users_ , can pay
transaction fees to speed up their transaction’s inclusion.

###  Transaction Mempool

A transaction signed by a user is broadcasted to the network for inclusion in
a future block. Every miner/block producer validates the transactions it
receives and, if valid, adds them to a list of transactions to be added to the
blocks. This list of transactions waiting to be added is referred to as
_transaction mempool_. It is a staging area for transactions, and some
blockchains allow transactions to be canceled or replaced while in the
mempool.

###  Publishing Blocks

The success of blockchain relies on the number of participants volunteering to
pick valid transactions up from the mempool and produce a valid block. Most
blockchains offer a pre-defined reward to block publishers for their efforts.
Each blockchain specifies a mechanism to decide a participant’s eligibility to
publish a block. Such participants are called _miners_ in PoW-like consensus
mechanisms and _validators_ in PoS-like consensus mechanisms. Blockchains also
specify the desired block size and block publishing rate of the network. In
Bitcoin, one block is published in around $10$ minutes. In Ethereum, one block
is published every $12$ seconds. There is a trade-off between block size,
block publishing rate, and the network’s security, limiting the throughput
blockchains can achieve.

###  Consensus

Participants of a distributed system need a mechanism to achieve consensus on
updates proposed in the network. Bitcoin [nakamoto] uses _proof-of-work_ (PoW)
(also known as Nakamoto consensus) as its consensus mechanism. Ethereum
recently shifted from proof-of-work to _proof-of-stake_ (PoS). We discuss both
PoW and PoS in the following sections. Proof-of-Work One must solve a puzzle
to publish a block in PoW, proving they have done computationally intensive
work. This renders Sybil attacks useless. While any participant can spin up
new identities, they are limited in their total computational power. As
discussed in [bitcoin-book], the following properties are essential for a
puzzle to be used to achieve consensus:

  * •

Easy-verifiability - The solution to the puzzle must be easily verifiable so
that everyone can verify it.

  * •

Proportional advantage - The probability of mining a block should be
proportional to the fraction of the participant’s computational power. Thus,
unlike traditional voting-like mechanisms, there’s no advantage in generating
more identities.

  * •

Adjustable difficulty - The difficulty of finding a solution should be
adjustable to maintain a pre-defined block publishing rate.

  * •

Non-reusable - A solution for a block shouldn’t be usable for any other block.

Miner There are three types of participants in the blockchain network:

  * •

_Light nodes_ : Participants who do not invest a lot of computational power
and only store and retrieve (from full nodes) the information they need to
verify their own transaction status.

  * •

_Full nodes_ : Participants store the entire blockchain and verify the
validity of all transactions and blockchain state changes.

  * •

_Miner_ : Full nodes that deploy their computational resources to mine new
blocks according to the PoW consensus mechanism.

Miners collect transactions in the mempool, arrange them in a block, and solve
the PoW puzzle. Miners get to keep the transaction fees offered by transaction
senders. Often, miners keep their transaction mempool sorted and add
transactions in their block in descending order of transaction fees. We see
how this could be leveraged by greedy parties in Section LABEL:ssec:PGA and
what challenges it leads to. Proof-of-Stake In PoS-based blockchains,
participants interested in publishing their block must stake some of their
blockchain’s native token. While their tokens are staked, they can’t be used
in transactions. Though in all PoS protocols, a participant’s probability of
publishing a block is proportional to its fraction of the total stake,
different PoS mechanisms propose their own randomization methods to select
which _staker_ gets to publish the block. We discuss Ethereum’s PoS briefly in
the following section. Proposers, Validators, and Committees in Ethereum A
staker can create multiple validator clients, each of which stakes 16 to 32
ETH. Blocks are published in slots (some of which can be empty) of $12$
seconds. $32$ slots make one epoch. Each slot is assigned a _proposer_ and a
committee of validators. The proposer is responsible for publishing the block
for their assigned slot. Committee members attest to the blocks from the
proposer if they consider them valid.

###  Incentives

In PoW, the block consists of a _coinbase_ transaction in which the miner
claims a mining reward. After the fourth halving on 19th April 2024, the block
reward for Bitcoin was reduced to $3.125$ BTC. If a block mined by a miner is
not accepted, the miner doesn’t receive the reward. In Ethereum, validators
get rewards for attesting blocks that most validators attest to. They are
penalized for failing to attest or attesting blocks that aren’t finalized.
Proposers of the block get larger rewards for publishing an accepted block.
Participants can get slashed (i.e., penalized by losing some staked ETH) for
malicious behavior like double-voting by attestors or double proposal by the
proposer. Slashing rewards are received by the proposer who reports the
malicious behavior.

##  Smart Contract

A transaction in Bitcoin is a small program in scripting language Script.
Using Script, one can write a transaction as a contract. Limited capabilities
of Script limit the number of possible attacks. Using Script, one can execute
different types of contracts on Bitcoin. Vitalik questioned, Why restrict
contracts to only monetary transactions? Why not anything that has value? It
leads to the realization of Nick Szabo’s concept of _smart contracts_
[Szabo2018SmartCB]. Towards this, Vitalik introduced _Ethereum_. Ethereum and
some similar blockchains support a Turing-complete instruction set and allow a
wider variety of contracts to be written and executed on the blockchain. Smart
contracts are programs that reside on blockchain and have their state (balance
and variables) and functions. We discuss Ethereum’s smart contract specifics
in the next section.

###  Ethereum

Ethereum as a blockchain is a virtual machine known as _Ethereum Virtual
Machine_ (). The state of the machine evolves with each transaction, which is
a computational payload. These computations can be arbitrary, and the
transaction sender needs to pay gas fees to the block publisher to run their
transactions. The native currency of Ethereum is _Ether_ , and its ticker is
_ETH_. Ethereum is arguably the most popular chain that allows smart contracts
and has the most DeFi activity. To improve scalability, many chains derive
security from Ethereum by committing their states to Ethereum, such as
Polygon, StarkNet, Arbitrum, etc. Such blockchains are called layer 2 or _L2_
, while Ethereum is considered layer 1 or _L1_. As mentioned above, the
Ethereum blockchain can be imagined as a single computer where users can
request arbitrary computations by sending transaction requests in the mempool.
Transactions list opcodes/instructions to be executed. An Ethereum address
originates the transaction and is signed by the private key corresponding to
the address. The specification is provided by the Ethereum Yellowpaper
[ethereumyellowpaper]. Ethereum execution clients follow this paper’s
specifications. Gas As is Turing complete, users can request arbitrary
computations. A block in Ethereum needs to be finalized every $12$ seconds;
thus, computations on must be limited. The protocol ensures this as follows.
The computation required by each opcode is represented in units of _gas_.
Users need to pay a base fee and optionally a priority fee for their
transactions, calculated as:

$Total\\_fee=gas\\_units\times(base\\_fee+priority\\_fee)$

If the gas fee offered by the user falls short of the total gas fee required
(more appropriately, if the gas is exhausted despite computations being left),
the block creator can stop the execution and include the transaction as a
failed transaction in the block while charging the gas fee offered. We now
briefly explain how DeFi is executed on .

##  Decentralized Finance

With fungible and non-fungible tokens possessing monetary values, ownership
transfers, and exchanges are required. Smart contracts allow these to be
completely decentralized, removing trust assumptions required in traditional
finance. DeFi is the ecosystem that caters to all finances happening on-chain,
governed by smart contract logic, which is auditable by everyone. Some
decentralized exchanges (DEX) maintain an order book. There can be massive
delays as these require two trading parties to have complementary requirements
available simultaneously. _Automated market makers_ (AMMs) bypass this
requirement. We discuss AMMs in the next section.

###  Automated Market Makers

The most commonly used DeFi application is swapping one token for another.
AMMs allow users to swap two tokens (most commonly, native currency like Ether
and one other token) without requiring another party. There are various ways
of defining the swapping mechanism for an AMM. _Constant function market
makers_ (CFMM) achieve this by maintaining a pool of the two tokens such that
the product of their supply remains constant.

$x\times y=k$

where $x$ is the token1’s supply, $y$ is token2’s supply and $k$ is known as
the _invariant_. Let the current supply of token1 be $x_{0}$, and the current
supply of token2 be $y_{0}$. If a user wants to swap $a$ token1 and receive
some token2 (= $b$), the new supplies will be $(x_{0}+a)$ for token1 and
$(y_{0}-b)$ for token2. As their product remains equal to the invariant $k$,
the amount of token2 the user gets is:

$b=y_{0}-\frac{k}{(x_{0}+a)}$

We ignore fees for the above example. Uniswap V1 whitepaper [uniswapV1] delves
into more details.

###  in Decentralized Markets

The curve $x\times y=k$ is a hyperbola symmetric around the line $y=x$. It can
be shown that as the supply of token1 increases, diminishing amounts of token2
are received. If $x\xrightarrow{}\infty$, then $y\xrightarrow{}0$. For such
$x$ and $y$, any added supply $\Delta x$ will give negligible token2, but if
for any finite $\Delta y$, huge amounts of token1 will be swapped. The
unreasonable price discourages parties from swapping. If the latter is the
case, the parties get the system closer to equilibrium. Earning rewards to get
the system back to equilibrium is called _arbitrage_. Arbitrage is considered
essential for the smooth functioning of markets. Block creators can bring the
system back to equilibrium by executing an inverse swap after every AMM swap
by users. Thus, arbitrage is also considered a type of . Suppose the block
publisher doesn’t have enough tokens to perform arbitrage. In that case, they
can buy tokens before the user’s transaction by placing them before the user’s
transaction and then performing arbitrage after the user’s transactions,
placing the block publisher’s transaction before the user’s transaction is
called _frontrunning_ , and placing it after the user’s transaction is called
_back-running_. Doing both together is called _sandwiching_. All these are
also types of . We discuss examples of each in the next section.

###  Example Extraction

Frontrunning Frontrunning involves a miner or trader intercepting and
prioritizing their transaction in front of another user’s transaction to
profit from the anticipated price movement caused by that transaction. Suppose
trader $T_{i}$ decides to buy a large amount of tokenx on DEX1. These will
likely cause a temporary increase in the price of tokenx on DEX1 due to the
sudden decrease in supply. $T_{i}$ places a large market buy order on DEX1.
Trader $T_{j}$, who monitors pending transactions, anticipates this price rise
and aims to profit by executing their transaction to buy tokenx just before
$T_{i}$’s buy order at a lower price. $T_{j}$ submits his own transaction to
buy tokenx on DEX1 with higher gas fees than $T_{i}$’s transaction. Miners
typically prioritize transactions based on gas fees, and as $T_{j}$’s gas fee
is higher than $T_{i}$’s transaction, $T_{j}$’s transaction is included before
$T_{i}$’s transaction. Suppose $T_{j}$’s transaction is successfully included
in a block before $T_{i}$’s. In that case, it will affect the price of tokenx
before $T_{i}$’s transaction is executed, allowing $T_{j}$ to front-run
$T_{i}$ and potentially profit from the price movement. Backrunning
Backrunning is the opposite of frontrunning, where the attacker tries to
anticipate and exploit a price decrease instead of an increase. The attacker
aims to submit their transaction just after the target transaction but before
it gets included in the block. Suppose trader $T_{i}$ decides to sell a large
amount of tokenx on DEX1. These will likely cause a temporary decrease in its
price on DEX1 due to the sudden increase in supply. $T_{i}$ places a large
market sell order on DEX1. Trader $T_{j}$, who monitors pending transactions,
anticipates this price drop and aims to profit by executing their transaction
to buy tokenx just after $T_{i}$’s sell order. $T_{i}$ submits his own
transaction to buy tokenx on DEX1 with slightly lower gas fees than $T_{i}$’s
transaction. Miners typically prioritize transactions based on gas fees, and
as $T_{j}$’s gas fee is only slightly lower than $T_{i}$’s transaction,
$T_{j}$’s transaction is included right after $T_{i}$’s transaction. If
$T_{j}$’s transaction is successfully included in a block right after
$T_{i}$’s, it can sell the tokenx it bought at the higher price after the
rebound, earning a profit from the price discrepancy. Sandwiching Sandwiching
involves placing two transactions around a target transaction, effectively
“sandwiching” it between them. The attacker places a transaction before and
after the target transaction to capture value from both sides of the trade.
Suppose two decentralized exchanges, DEX1 and DEX2, are trading the same
token, tokenx, at prices $A$ and $B$ such that $A<B$. Trader $T_{i}$ decides
to execute the arbitrage trade by buying tokenx on DEX1 and selling it on
DEX2. $T_{i}$ submits its buy order on DEX1, which will land in the mempool
pending and waiting to be confirmed. Trader $T_{j}$ monitors pending
transactions, observes $T_{i}$’s pending transactions and submits its buy
order on DEX1 with a sufficiently high gas fee and sell order transaction with
a slightly lower gas fee than $T_{i}$’s buy order. Miners include $T_{j}$’s
buy transaction before $T_{i}$’s buy order and its sell transaction right
after. Suppose $T_{j}$’s transactions are successfully included as per this
order. In that case, it will affect the price of tokenx before $T_{i}$’s
transactions are executed, allowing $T_{j}$ to front-run and then back-run
$T_{i}$’s transaction and potentially snatching the arbitrage opportunity.