# Scalar Dark Energy Models and Scalar-Tensor Gravity: Theoretical
Explanations for the Accelerated Expansion of Present Universe

Peixiang Ji1,2 and Lijing Shao2,3 1 Department of Astronomy, School of
Physics, Peking University, Beijing 100871, China  2 Kavli Institute for
Astronomy and Astrophysics, Peking University, Beijing 100871, China  3
National Astronomical Observatories, Chinese Academy of Sciences, Beijing
100012, China<EMAIL_ADDRESS>
###### Abstract

The reason for the present accelerated expansion of the Universe stands as one
of the most profound questions in the realm of science, with deep connections
to both cosmology and fundamental physics. From a cosmological point of view,
physical models aimed at elucidating the observed expansion can be categorized
into two major classes: dark energy and modified gravity. We review various
major approaches that employ a single scalar field to account for the
accelerating phase of our present Universe. Dynamical system analysis is
employed in several important models to seek for cosmological solutions that
exhibit an accelerating phase as an attractor. For scalar field models of dark
energy, we consistently focus on addressing challenges related to the fine-
tuning and coincidence problems in cosmology, as well as exploring potential
solutions to them. For scalar-tensor theories and their generalizations, we
emphasize the importance of constraints on theoretical parameters to ensure
overall consistency with experimental tests. Models or theories that could
potentially explain the Hubble tension are also emphasized throughout this
review.

###### type:

Review Article

††:

Keywords: dark energy, modified gravity, quintessence, dynamical system,
cosmology

##  1 Introduction

In 1998, two independent research groups [1, 2] studying distant Type Ia
Supernovae (SN Ia) presented compelling evidence that the expansion of the
Universe is not just expanding but actually accelerating. Subsequent
observations, which included more in-depth studies of supernovae [3, 4] and
independent data from the cosmic microwave background (CMB) radiation [5, 6,
7, 8, 9, 10, 11], the baryon acoustic oscillation (BAO) peak length scale [12,
13, 14, 15, 16, 17, 18], clusters of galaxies [19, 20], and the large-scale
structure (LSS) of the Universe [21, 22, 23, 24], confirmed this remarkable
acceleration.

According to Einstein’s general theory of relativity without a cosmological
constant, if the Universe was primarily filled with ordinary matter and/or
radiation, which are the known constituents of the Universe, gravitational
interactions would act to slow down the expansion. However, since the Universe
is instead found to be accelerating, we are confronted with two intriguing
possibilities, dark energy and modified gravity, both of which would have
profound implications for our understanding of the Universe and the laws of
physics.

The first possibility is that approximately 70% of the energy density of the
Universe exists in a new form characterized by negative pressure, referred to
as dark energy. When attributing it to the matter sector, dark energy would
modify the stress-energy tensor of matters in Einstein’s field equations,
leading to the observed acceleration. This novel dark energy component has
become central to cosmology, and understanding its nature remains one of the
most significant challenges in modern cosmology and fundamental physics.

The second possibility is that general relativity, as formulated by Einstein,
breaks down when applied to cosmological scales. In this scenario, a more
comprehensive theory, modifying the Einstein-Hilbert action, would be needed
to accurately describe the behavior of the Universe, which would lead to
changes in the gravitational sector of the field equations.

Extra scalar fields besides the known Higgs field in the Standard Model of
particle physics may play a critical role in fundamental physics and provide
valuable insights into various aspects of the Universe [25]. Scalar fields
naturally arise in unified theories, e.g. the string theory, and contribute to
the overall cosmic energy density, offering a plausible explanation for the
observed acceleration of the Universe [26, 27]. In the realm of gravitational
theories, scalar fields are pivotal in various models of modified gravity,
such as in scalar-tensor theories [28, 29, 30, 31]. The scalar field, in
addition to the metric, describes gravitational interactions, resulting in
predictions that can deviate from those of Einstein’s theory of relativity.
These deviations may manifest as changes in the behavior of the gravitational
force at different scales and under extreme conditions, like those found near
black holes, inside neutron stars, or during the early Universe [32, 33, 34,
35, 36].

The mystery of cosmic acceleration is connected to several pivotal questions
in cosmology and fundamental physics [37]. The cosmic acceleration may hold
the key to uncovering a successor to Einstein’s theory of gravity. The
relatively small energy density of the quantum vacuum could yield insights
into concepts such as supersymmetry and superstring. The mechanism governing
the ongoing cosmic acceleration might relate to the primordial inflation in
cosmology, while the quest to unravel the cause of cosmic acceleration could
potentially introduce novel long-range forces or shed light on the enigmatic
smallness of neutrino masses in particle physics.

This review basically focuses on models and theories involving a single scalar
field. The aim of this review is to theoretically clarify how these models
drive the present Universe to undergo an accelerated expansion, with a wide
utilization of dynamical system analysis. We typically do not delve into
whether these models conform to various observations from SN Ia, CMB and LSS
since it usually requires perturbation calculations and further analysis. For
such discussions, the readers are directed to specific articles or reviews on
observational constraints, such as Refs. [38, 39]. In Section 2, we provide an
overview of the historical context of the cosmological constant, tracing its
birth from Einstein to its ultimate confirmation with supernova discoveries.
The cosmological constant problem is also overviewed in this section. Sections
3 and 4 introduce scalar field models of dark energy and modified gravity
theories incorporating a single scalar field, respectively, which offer
explanations for the current expansion of the Universe. We list a series of
scalar dark energy models in Section 3 to show that it is challenging for them
to solve fine-tuning and coincidence problems simultaneously without getting
into other troubles. Interacting dark energy models, which have received much
attention recently, are also introduced in this section. In Section 4, we
additionally provide constraints from terrestrial and Solar System tests of a
general class of scalar-tensor gravity. Screening mechanisms are briefly
included as well to illustrate how to maintain consistency with experimental
results when there is a stronger coupling between the scalar field and matter.
A brief summary is provided in the final section.

Throughout the review, we adopt natural units $c=\hbar=1$, and have a metric
signature $(-,+,+,+)$. We denote the Planck mass as
$m_{\mathrm{Pl}}=G^{-1/2}\simeq 1.22\times 10^{19}\;\mathrm{GeV}$ and the
reduced Planck mass as $M_{\mathrm{Pl}}=(8\pi G)^{-1/2}\simeq 2.44\times
10^{18}\;\mathrm{GeV}$, where $G$ is the Newton’s gravitational constant. When
the unit $\kappa\equiv 8\pi G=1$ is employed, we will explain it in advance.

##  2 Cosmological Constant

The field equation in general relativity with the cosmological constant
$\Lambda$ is

$G_{\mu\nu}+\Lambda g_{\mu\nu}=\kappa T_{\mu\nu}^{\mathrm{m}},$ (2.1)

where $G_{\mu\nu}$, $g_{\mu\nu}$ and $T_{\mu\nu}^{\mathrm{m}}$ are the
Einstein tensor, metric tensor, and energy-momentum tensor for matter in the
standard model of particle physics, respectively. The theory that adds a
cosmological constant term can be considered as the simplest dark energy
model. Introducing a cosmological constant to Einstein’s field equations is a
reasonable improvement in the framework of general relativity. Since we all
know that the metric itself and the Einstein tensor are the only two tensors
constructed from the metric and its derivatives up to the second order, that
possess vanishing covariant divergences and lead to the local conservation of
the energy-momentum tensor.

If one moves the cosmological constant to the right-hand side, Equation (2.1)
becomes

$G_{\mu\nu}=\kappa(T_{\mu\nu}^{\mathrm{m}}+T^{\Lambda}_{\mu\nu}),$ (2.2)

where

$T_{\mu\nu}^{\Lambda}=-\frac{\Lambda}{\kappa}g_{\mu\nu}$ (2.3)

represents the energy-momentum of a new component (dark energy) outside the
standard model of particle physics. Compared with the energy-momentum tensor
of a perfect fluid

$T^{\mathrm{pf}}_{\mu\nu}=(\rho_{\mathrm{pf}}+{p_{\mathrm{pf}}})u_{\mu}u_{\nu}+p_{\mathrm{pf}}g_{\mu\nu},$
(2.4)

one has

$\rho_{\Lambda}=-{p_{\Lambda}}=\frac{\Lambda}{\kappa},$ (2.5)

which indicates that this new component has a constant energy density and a
negative pressure. As a result, any form of matter with constant energy
density will behave as a cosmological constant (and vice versa), because they
enter the field equation with the same mathematical structure.

###  2.1 A Brief History of Cosmological Constant

When the field equation of general relativity was proposed, most physicists,
including Einstein himself, believed that the Universe was in a static state.
However, the original field equation was not able to describe a non-evolving
Universe. To address this, Einstein introduced a new constant of nature and
referred to it as the cosmological constant [40]. Incorporating this constant,
Einstein obtained his static Universe, which he at first failed to notice that
it is unstable [41]. Einstein also believed that the existence of the
cosmological constant does not result in any difference on small scales, such
as within the Solar system, as long as it is sufficiently small.

In 1929, Hubble made the groundbreaking discovery that distant galaxies are
indeed moving away from us, and he presented strong observational evidence for
the expansion of the Universe [42]. It indicated that the Universe is dynamic
rather than static. Hence, concerning the original motivation, the
cosmological constant is no longer needed.

At the end of the last century, the cosmological constant was reintroduced
several times to explain problems related to the relatively young Universe age
[43], as well as the peaking number counts of quasars at $z\sim 2$ [44],
despite it is well-known that the peak has an explanation of astrophysical
origin rather than cosmological nowadays. A non-zero cosmological constant was
also invoked in order to reconcile the flat Universe predicted by inflation
and some other estimations from observations [45, 46].

The final confirmation of a non-zero value of the cosmological constant is
from the accurate measurement of the luminosity distance of SN Ia as a
function of redshift [1, 2]. To fit the results in the framework of general
relativity, the simplest way is to assume that there exists a new cosmic
component, besides radiation and matter (including both ordinary matter and
dark matter). The new component has a negative pressure, and dominates the
Universe’s present energy budget.

###  2.2 The Cosmological Constant Problem

Although the confrontation of Equation (2.1) with the Solar system and
galactic observations have already given an upper bound of the cosmological
constant, a tighter constraint comes from larger scale observations [47, 48]
that $\Lambda$ is of the same order of the present value of the Hubble
parameter $H_{0}$. The corresponding energy density is calculated directly
from Equation (2.5),

$\rho_{\Lambda}\approx\frac{H^{2}_{0}}{8\pi G}\sim
10^{-47}\,\mathrm{GeV}^{4}.$ (2.6)

Theoretically, the constant energy density associated with quantum vacuum
energy is regarded as the source of the cosmological constant. Similar to the
ground state of a quantum harmonic oscillator, which possesses a non-zero
zero-point energy proportional to the oscillation frequency of its classical
counterpart, vacuum-to-vacuum fluctuations arise from the collective
contributions of an infinite number of oscillators capable of vibrating at all
possible frequencies. A cutoff frequency is required to prevent the divergence
of vacuum energy density. The cutoff wavenumber
$k_{\mathrm{c}}=m_{\mathrm{Pl}}$ is considered generally, for the belief of
the validity of the quantum field theory and general relativity below the
Planck scale. Then vacuum energy density has an extremely large value

$\rho_{\mathrm{vac}}\approx\frac{k^{4}_{\mathrm{c}}}{16\pi^{2}}\sim
10^{74}\,\mathrm{GeV}^{4},$ (2.7)

which is more than 120 orders of magnitude larger than the observational bound
of $\rho_{\Lambda}$. This huge discrepancy is known as the cosmological
constant problem, or the vacuum catastrophe.

Hints that the cosmological constant might be non-zero (see Section 2.1) led
Zel’dovich to consider the cosmological constant in terms of vacuum energy in
1967 [49], and since then, numerous solutions have been proposed to address
this mismatch of energy density [47, 50, 51, 52]. Apart from anthropic
principles [47, 53], the solutions can be broadly categorized into three
groups. The first one is to establish a unified framework for quantum gravity,
specifically in exploring the interplay between quantum vacuum and gravity
[54, 55, 56]. The second category is to “eliminate” the vacuum energy. In the
context of supersymmetry—an extension of the standard model—the vacuum energy
is exactly zero due to the equal number of fermionic and bosonic degrees of
freedom [57]. In addition, an alternative approach, called Schwinger’s source
theory [58, 59, 60], changes the interpretation of the quantum field theory
formalism, and the vacuum energy is no longer needed. The last is to modify
the gravity [37, 61], aiming to acquire an effective cosmological constant,
which is actrually not an exact resolution of the vacuum catastrophe. However,
the modified gravity seems to become more crucial because of the discovery of
a tiny but non-zero cosmological constant. If a modified theory of gravity
accurately describes the evolution of the Universe and well explains the
observations, it is plausible that at least one of the following two
statements holds. (I) The vacuum energy of the Universe is either zero or
extremely small. It is possible that the quantum zero-point energy of each
field is offset by its higher order corrections, or there exists a
cancellation mechanism—maybe a hidden symmetry—of vacuum energy among
different fields. In such a scenario, one would need to further scrutinize the
explanations of experimental evidence for vacuum energy like the Casimir
effect, and the vacuum concept associated with quantum chromodynamics and the
spontaneous symmetry breaking in the electroweak force. (II) The vacuum
energy, being a purely quantum concept, does not function as a source within
classical gravity theory that exerts influence on the geometry of spacetime
and other degrees of freedom in modified gravity theories, e.g. scalar fields.
In this scenario, the connection between quantum field theory and gravity
theory poses significant challenges, while the establishment of a unified
framework for describing all fundamental forces remains a central pursuit in
modern theoretical physics.

##  3 Dark Energy

In the framework of general relativity, the radiation and barotropic matter,
as the only contents of the Universe, cannot induce an accelerated expansion.
The concept of dark energy, termed by Turner [62, 63] as a potential
constituent of the Universe, propels the ongoing studies of the accelerated
expansion. In the standard Lambda Cold Dark Matter ($\Lambda$CDM) model [38],
the dark energy is characterized by the presence of a cosmological constant
term, which is not dynamical but exerts influence on the evolution of the
Universe. Despite being the standard model in cosmology, it is confronted with
various theoretical and observational challenges.

  * •

Fine-tuning problem: The current energy density of the estimated cosmological
constant in Equation (2.6) is relatively small based on observations, while
the theoretical value of the vacuum energy density in Equation (2.7) is
significantly larger. In other words, the observed value of energy density
conflicts with the possible energy scales and requires fine-tuning if the
cosmological constant originates from the vacuum energy density.

  * •

Coincidence problem: The $\Lambda$CDM model suggests a comparable energy
density ratio between the matter and the dark energy [11], implying that the
expansion of the Universe transitioned from a decelerated to an accelerated
phase at a redshift of approximately $z\sim 0.67$ [27]. The problem of why an
accelerated expansion should occur now (actually not long ago) in the very
long history of the Universe is called the coincidence problem.

  * •

Hubble tension and more: Recent observations have revealed that within the
framework of $\Lambda$CDM model, there exists certain tensions between
observations from early Universe and late Universe [64], among which the
$H_{0}$ and $S_{8}$ tensions are the most well-known ones. The latest
observations with the James Webb Space Telescope further confirm the existence
of Hubble tension at $8\sigma$ confidence [65].

###  3.1 Quintessence

A dynamical scalar field as dark energy—the quintessence model—is one of the
most popular models for the dark energy. The audiences may be familiar with
the similar inflation model that describes the very early stage of the
Universe [66, 67]. However, except for the huge discrepancy in the energy
scale, the scalar field is the only component in inflation models, while it is
just the fifth element of the Universe in the quintessence model. As the first
and the simplest model in this review, we will look at the quintessence in
more detail. The evolution of the Universe in this model is displayed in
Section 3.1.1, and the introduction of the classification and dynamical system
analysis of the quintessence model are in Section 3.1.2 and Section 3.1.3,
respectively.

The action for the quintessence model is given by [68]

$S[g^{\mu\nu},\phi,\Psi_{\mathrm{m}}]=S_{\mathrm{EH}}[g^{\mu\nu}]+S_{\mathrm{q}}[g^{\mu\nu},\phi]+S_{\mathrm{m}}[g^{\mu\nu},\Psi_{\mathrm{m}}],$
(3.1)

where $S_{\mathrm{EH}}=\int\rmd^{4}x\sqrt{-g}R/2\kappa$ is the Einstein-
Hilbert action,
$S_{\mathrm{m}}=\int\rmd^{4}x\sqrt{-g}\mathcal{L}_{\mathrm{m}}(\Psi_{\mathrm{m}})$
is the action of all possible matter fields, collectively denoted as
$\Psi_{\mathrm{m}}$, and

$S_{\mathrm{q}}=\int\rmd^{4}x\sqrt{-g}\mathcal{L}_{\mathrm{q}}=\int\rmd^{4}x\sqrt{-g}\left[-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-V(\phi)\right],$
(3.2)

is the action for the quintessence. $V(\phi)$ is the potential of the scalar
field and is always assumed to be positive. By variating the action (3.1),
equations of motion for the tensor field $g_{\mu\nu}$ and the scalar field
$\phi$ are

$\displaystyle
G_{\mu\nu}=\kappa\big{(}T_{\mu\nu}^{\mathrm{m}}+T_{\mu\nu}^{\mathrm{q}}\big{)},$
(3.3) $\displaystyle\square\phi=\frac{dV}{d\phi},$ (3.4)

where $\square=g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}$, and
$T_{\mu\nu}^{\mathrm{m}}$ is the energy-momentum tensor of ordinary matter
including dust and radiation,

$T_{\mu\nu}^{\mathrm{m}}\equiv-\frac{2}{\sqrt{-g}}\frac{\delta(\sqrt{-g}\mathcal{L}_{\mathrm{m}})}{\delta
g^{\mu\nu}},$ (3.5)

and the energy-momentum tensor of quintessence is

$T_{\mu\nu}^{\mathrm{q}}=\partial_{\mu}\phi\partial_{\nu}\phi-
g_{\mu\nu}\left[\frac{1}{2}g^{\alpha\beta}\partial_{\alpha}\phi\partial_{\beta}\phi+V(\phi)\right].$
(3.6)

Since the scalar field is minimally coupled to the metric, the action (3.1)
can be rewritten as

$S[g^{\mu\nu},\phi,\Psi_{\mathrm{m}}]=S_{\mathrm{EH}}[g^{\mu\nu}]+S^{\prime}_{\mathrm{m}}[g^{\mu\nu},\Psi_{\mathrm{m}},\phi],$
(3.7)

where $S_{\mathrm{m}}^{\prime}=S_{\mathrm{m}}+S_{\mathrm{q}}$. In such a case,
the scalar field in action (3.7) represents the fifth element of the Universe,
other than baryons, photons, neutrinos and dark matter that are usually
considered in modern cosmology. This dynamical scalar field as a component of
the Universe has first been studied by Ratra & Peebles [69] and Wetterich [70]
in 1988, and was first called quintessence in a 1998 paper by Caldwell et al.
[68].

####  3.1.1 Evolution of the Universe in the Quintessence Model.

Based upon the assumptions of homogeneity and isotropy of the Universe, which
is approximately true on large scales, the Friedmann-Lemaître-Robertson-Walker
(FLRW) metric,

$\rmd s^{2}=-\rmd t^{2}+a^{2}(t)\left(\frac{\rmd
r^{2}}{1-Kr^{2}}+r^{2}\rmd\Omega^{2}_{2}\right),$ (3.8)

is used to describe the geometry of the Universe. Here, $a(t)$ is the scale
factor with the cosmic time $t$, and
$\rmd\Omega_{2}^{2}=\rmd\theta^{2}+\sin^{2}\theta\rmd\phi^{2}$111Here $\phi$
is the azimuthal angle, not to be confused with the scalar field. is the line
element of a 2-sphere. The constant $K$ describes the geometry of the spatial
part of the spacetime, with closed, flat, and open Universes corresponding to
$K=+1,0,-1$, respectively. In this review, we always assume a flat Universe,
which is consistent with current observations [11].

The Friedmann equations are a set of equations that govern the expansion of
space in homogeneous and isotropic models of the Universe. Using the FLRW
metric, one has

$\displaystyle 3M_{\mathrm{Pl}}^{2}H^{2}=\rho_{\mathrm{m}}+\rho_{\mathrm{q}},$
(3.9)
$\displaystyle-2M_{\mathrm{Pl}}^{2}\dot{H}=\rho_{\mathrm{m}}+\rho_{\mathrm{q}}+p_{\mathrm{m}}+p_{\mathrm{q}},$
(3.10)

where $H=\dot{a}/a$ is the Hubble parameter. The subscripts “$\mathrm{q}$” and
“$\mathrm{m}$” represent quintessence and ordinary matter, respectively.
Ordinary matter comprises relativistic radiation and non-relativistic dust,
denoted by subscripts “$\mathrm{r}$” and “$\mathrm{d}$” respectively in the
following. Assuming that the scalar field only depends on the cosmic time, the
energy density and pressure of the quintessence field are

$\displaystyle\rho_{\mathrm{q}}=\frac{1}{2}\dot{\phi}^{2}+V(\phi),$ (3.11)
$\displaystyle p_{\mathrm{q}}=\frac{1}{2}\dot{\phi}^{2}-V(\phi),$ (3.12)

where the overdot denotes the derivative with respect to the cosmic time $t$.
Furthermore, local conservation of the energy-momentum tensor gives

$\displaystyle\dot{\rho}_{\mathrm{m}}+3H(1+w_{\mathrm{m}})\rho_{\mathrm{m}}=0,$
(3.13)
$\displaystyle\dot{\rho}_{\mathrm{q}}+3H(1+w_{\mathrm{q}})\rho_{\mathrm{q}}=0,$
(3.14)

where $w_{\mathrm{m}}=p_{\mathrm{m}}/\rho_{\mathrm{m}}$ and

$w_{\mathrm{q}}=\frac{p_{\mathrm{q}}}{\rho_{\mathrm{q}}}=\frac{\dot{\phi}^{2}-2V(\phi)}{\dot{\phi}^{2}+2V(\phi)},$
(3.15)

are parameters in equations of state (EOS). In the slow-roll limit,
$\dot{\phi}^{2}\ll V(\phi)$, the parameter $w_{\mathrm{q}}$ goes to $-1$ and
the quintessence acts just like the cosmological constant.

For a constant $w_{\mathrm{m}}$, Equation (3.13) gives

$\rho_{\mathrm{m}}=\bar{\rho}_{\mathrm{m}}a^{-3(1+w_{\mathrm{m}})},$ (3.16)

while the solution to Equation (3.14) is

$\rho_{\mathrm{q}}=\bar{\rho}_{\mathrm{q}}\exp\left[-3\int(1+w_{\mathrm{q}})\frac{\rmd
a}{a}\right].$ (3.17)

Here, $\bar{\rho}_{\mathrm{m}}$ and $\bar{\rho}_{\mathrm{q}}$ are integration
constants. Figure 1 depicts the evolution of dust and quintessence energy
density parameters, as well as the effective EOS parameter of the quintessence
model with an exponential potential, over time. The definitions of the
quantities in the plot are provided in Section 3.1.3. As observed from Figure
1, to satisfy $\Omega_{\mathrm{d}}\simeq 0.3$ and $\Omega_{\mathrm{q}}\simeq
0.7$ at present and to experience a period of matter dominance, the initial
conditions must be chosen carefully.

Figure 1: Evolution of the effective EOS parameter $w_{\mathrm{eff}}$, the
dust $\Omega_{\mathrm{d}}$ and quintessence $\Omega_{\mathrm{q}}$ relative
energy densities with an exponential potential. The vertical dashed line
denotes the present cosmological time. One has $\Omega_{\mathrm{d}}\simeq 0.3$
and $\Omega_{\mathrm{q}}\simeq 0.7$ today.

We are currently in a period characterized by the dominance of dark energy. In
the quintessence domination, Equations (3.9) and (3.10) become

$\displaystyle 6M_{\mathrm{Pl}}^{2}H^{2}=\dot{\phi}^{2}+2V(\phi),$ (3.18)
$\displaystyle-2M_{\mathrm{Pl}}^{2}\dot{H}=\dot{\phi}^{2}.$ (3.19)

The acceleration equation comes out of a combination of the Friedmann
equations, as

$-3M_{\mathrm{Pl}}^{2}\frac{\ddot{a}}{a}=\frac{1}{2}(\rho_{\mathrm{q}}+3p_{\mathrm{q}})=\dot{\phi}^{2}-V(\phi),$
(3.20)

which shows that the Universe expands in an increasing rate when
$w_{\mathrm{q}}<-1/3$, or $\dot{\phi}^{2}<V(\phi)$. To see how the scalar
field evolves, we assume a power-law expansion, $a(t)\propto t^{s}$, where the
accelerated expansion occurs for $s>1$. According to Equation (3.19), one
obtains

$\phi=\int\rmd t\sqrt{-2M_{\mathrm{Pl}}^{2}\dot{H}}\propto\ln t.$ (3.21)

The potential can also be expressed in terms of $H$ and $\dot{H}$ as

$V=3M_{\mathrm{Pl}}^{2}H^{2}\left(1+\frac{\dot{H}}{3H^{2}}\right)\propto
t^{-2}.$ (3.22)

Combining Equations (3.21) and (3.22), the potential giving the power-law
expansion is ought to be an exponential form

$V(\phi)=\bar{V}\exp\left(-\frac{\lambda\phi}{M_{\mathrm{Pl}}}\right),$ (3.23)

where $\bar{V}$ is a constant, and $\lambda=\sqrt{2/s}$.

####  3.1.2 Thawing and Freezing Models.

In the flat FLRW background, Equation (3.4) gives the evolution equation of
the quintessence,

$\ddot{\phi}+3H\dot{\phi}+\frac{\rmd V}{\rmd\phi}=0.$ (3.24)

Although the presence of ordinary matter is not explicitly accounted for in
Equation (3.24), its influence on the Hubble parameter will impact the
evolution of the quintessence field. In Equation (3.24), the first term
represents the acceleration of the scalar field, while the second term
corresponds to a frictional effect, known as Hubble friction or Hubble drag,
resulting from the expansion of the Universe, and the third denotes a driving
force arising from the steepness of the potential.

The potential function of the scalar field is the only adjustable component in
the quintessence model, whose specific shape exerts a significant influence on
the cosmic evolution. In order to reproduce the cosmic acceleration today we
require that the potential is flat enough to satisfy the condition

$\left|\frac{M^{2}_{\mathrm{Pl}}}{V}\frac{\rmd^{2}V}{\rmd\phi^{2}}\right|\lesssim
1.$ (3.25)

Hence, the quintessence mass squared
$m^{2}_{\phi}\equiv\rmd^{2}V/\rmd\phi^{2}$ needs to satisfy

$|m^{2}_{\phi}|\lesssim\frac{V_{0}}{M^{2}_{\mathrm{Pl}}}\sim H_{0}^{2},$
(3.26)

where $V_{0}$ and $H_{0}$ are the present values of the potential and Hubble
parameter, respectively. As a result, the mass of the scalar field has to be
extremely small, say $m_{\phi}\lesssim 10^{-33}\;\mathrm{eV}$, to be
compatible with the present cosmic acceleration. Apart from this requirement,
we also hope that the potential appears in some particle physics models [27].

Caldwell and Linder [71] divided quintessence models into two categories
according to whether the scalar field accelerates ($\ddot{\phi}>0$) or
decelerates ($\ddot{\phi}<0$). A coasting in the scalar field dynamics,
$\ddot{\phi}=0$, is nongeneric, as the field would need to be finely tuned to
be perfectly balanced, neither accelerate due to the slope of the potential
nor decelerate due to the Hubble drag.

In the accelerating region, the field that has been frozen by the Hubble
friction into a cosmological-constant-like state until recently and then
begins to evolve gives the thawing model. The current accelerated expansion is
driven by the dominance of quintessence and the scalar field in its early
stages of “thawing”, implying that $w_{q}$, while increasing, remains smaller
than $-1/3$. A representative potential of this class is the one arising from
the pseudo Nambu-Goldstone boson [72],

$V(\phi)=\mu^{4}\left[1+\cos\left(\frac{\phi}{f}\right)\right],$ (3.27)

where $\mu$ and $f$ are constants characterizing the energy scale and the mass
scale of spontaneous symmetry breaking, respectively. If the field mass
squared $|m^{2}_{\phi}|\simeq\mu^{4}/f^{2}$ around the potential maximum is
smaller than $H^{2}$, the field is stuck there with $w_{\mathrm{q}}\simeq-1$.
Once $H^{2}$ drops below $|m^{2}_{\phi}|$, the scalar field starts to move and
$w_{\mathrm{q}}$ starts to increase. Likelihood analysis using CMB shift
parameters measured by WMAP7 [8] combined with SN Ia and BAO data placed a
bound, $w_{\mathrm{q}}|_{z=0}<-0.695$, at the 95 % confidence level, when the
quintessence prior is assumed to be $w_{\mathrm{q}}>-1$ [73].

In the decelerating region, the field that initially rolls due to the
steepness of the potential but later approaches the cosmological constant
state gives the freezing model. A representative potential of this class is
the Ratra-Peebles potential [69, 74],

$V(\phi)=M^{4+n}\phi^{-n},$ (3.28)

where $M$ and $n$ are positive constants. The potential does not possess a
minimum and hence the field rolls down the potential towards infinity. The
movement of the field gradually slows down when $\phi$ is large enough and the
system enters the phase of cosmic acceleration. Under the quintessence prior
$w_{\mathrm{q}}^{\rmi}>-1$ for tracking solutions (see Section 3.1.3),
likelihood analysis using the iterative solution in Ref. [75] resulted in
bounds, $w_{q}^{\rmi}<-0.923$ and $0.675<\Omega_{\mathrm{q}}<0.703$, at the 95
% confidence level, from the observational data of Planck 2015, SN Ia, and CMB
[76]. For the Ratra-Peebles potential, this bound translates to $n<0.17$,
which means that the tracker solutions arising from the inverse power-law
potential with positive integer powers are observationally excluded.

####  3.1.3 Dynamical System Approach.

The Universe comprises multiple components, and each of these exerts its own
influence on the dynamic evolution of the Universe. The dynamical system
approach is a powerful method that helps illustrate the global dynamics and
full evolution of the Universe [77, 78]. Through the careful choice of the
dynamical variables, a given cosmological model can be written as an
autonomous system of some differential equations, which further provides a
quick answer to whether it can reproduce the observed expansion of the
Universe.

An autonomous system is a system of ordinary differential equations which does
not explicitly depend on the independent variable,

$\frac{\rmd}{\rmd t}\mathbf{x}(t)=f(\mathbf{x}(t)),$

where $\mathbf{x}$ takes values in $n$-dimensional Euclidean space. Many laws
in physics, where $t$ is often interpreted as time, are expressed as
autonomous systems because it is always assumed that the laws of nature which
hold now are identical to those for any point in the past or future. The time
$t$ is usually replaced by the scale factor $a$ or its logarithm in cosmology
because the Universe expands monotonously in most of cases so that the scale
can be used as a measure of time.

In order to describe the quintessence model, the following dimensionless
variables

$x_{\mathrm{K}}\equiv\frac{\dot{\phi}}{\sqrt{6}M_{\mathrm{Pl}}H},\quad
x_{\mathrm{V}}\equiv\frac{\sqrt{V}}{\sqrt{3}M_{\mathrm{Pl}}H},\quad
x_{\mathrm{r}}\equiv\frac{\sqrt{\rho_{\mathrm{r}}}}{\sqrt{3}M_{\mathrm{Pl}}H},$
(3.29)

were introduced in the autonomous system [79, 80]. These above variables are
generally called the expansion normalised variables [81] that are widely
employed in quintessence models and their generalizations, for the sake of
easily closing the dynamical system. Furthermore, one defines density
parameters for the kinetic term and potential of the scalar field,

$\Omega_{\mathrm{K}}\equiv\frac{\dot{\phi}^{2}/2}{3M_{\mathrm{Pl}}^{2}H^{2}}=x_{\mathrm{K}}^{2},\quad\Omega_{\mathrm{V}}\equiv\frac{V}{3M_{\mathrm{Pl}}^{2}H^{2}}=x_{\mathrm{V}}^{2},$
(3.30)

which give

$\Omega_{\mathrm{q}}\equiv\frac{\rho_{\mathrm{q}}}{3M_{\mathrm{Pl}}^{2}H^{2}}=x_{\mathrm{K}}^{2}+x_{\mathrm{V}}^{2}.$
(3.31)

Density parameters of radiation, dust, and the total ordinary fluid are given
by

$\Omega_{\mathrm{r}}\equiv\frac{\rho_{\mathrm{r}}}{3M_{\mathrm{Pl}}^{2}H^{2}},\quad\Omega_{\mathrm{d}}\equiv\frac{\rho_{\mathrm{d}}}{3M_{\mathrm{Pl}}^{2}H^{2}},\quad\Omega_{\mathrm{m}}\equiv\frac{\rho_{\mathrm{m}}}{3M_{\mathrm{Pl}}^{2}H^{2}}=\Omega_{\mathrm{r}}+\Omega_{\mathrm{d}}.$
(3.32)

From Equation (3.9), one has

$\Omega_{\mathrm{q}}+\Omega_{\mathrm{m}}=\Omega_{\mathrm{K}}+\Omega_{\mathrm{V}}+\Omega_{\mathrm{d}}+\Omega_{\mathrm{r}}=1,$
(3.33)

which implies the flatness of the Universe. The EOS parameter $w_{\mathrm{q}}$
and the effective EOS parameter of the Universe are expressed by
$x_{\mathrm{K}}$ and $x_{\mathrm{V}}$, via [80]

$w_{\mathrm{q}}=\frac{x_{\mathrm{K}}^{2}-x_{\mathrm{V}}^{2}}{x_{\mathrm{K}}^{2}+x_{\mathrm{V}}^{2}},\quad\quad
w_{\mathrm{eff}}\equiv\frac{p_{\mathrm{m}}+p_{\mathrm{q}}}{\rho_{\mathrm{m}}+\rho_{\mathrm{q}}}=x_{\mathrm{K}}^{2}-x_{\mathrm{V}}^{2}+\frac{1}{3}x_{\mathrm{r}}^{2}.$
(3.34)

Differentiating the variables $x_{\mathrm{K}}$, $x_{\mathrm{V}}$ and
$x_{\mathrm{r}}$ with respect to the number of $\mathrm{e}$-foldings, $N=\ln
a$, as well as using Equations (3.9), (3.10), and (3.24), one obtains the
following first-order differential equations [80]

$\displaystyle\frac{{\rmd}x_{\mathrm{K}}}{\rmd
N}=-3x_{\mathrm{K}}+\frac{\sqrt{6}}{2}\lambda
x_{\mathrm{V}}^{2}-x_{\mathrm{K}}\frac{1}{H}\frac{\rmd H}{\rmd N},$ (3.35a)
$\displaystyle\frac{\rmd x_{\mathrm{V}}}{\rmd N}=-\frac{\sqrt{6}}{2}\lambda
x_{\mathrm{K}}x_{\mathrm{V}}-x_{\mathrm{V}}\frac{1}{H}\frac{\rmd H}{\rmd N},$
(3.35b) $\displaystyle\frac{\rmd x_{\mathrm{r}}}{\rmd
N}=-2x_{\mathrm{r}}-x_{\mathrm{r}}\frac{1}{H}\frac{\rmd H}{\rmd N},$ (3.35c)

where

$\frac{1}{H}\frac{\rmd H}{\rmd
N}=\frac{\dot{H}}{H^{2}}=-\frac{1}{2}(3+3x_{\mathrm{K}}^{2}-3x^{2}_{\mathrm{V}}+x_{\mathrm{r}}^{2}),$
(3.35aj)

and $\lambda$ is defined by

$\lambda\equiv-\frac{M_{\mathrm{Pl}}}{V}\frac{\rmd V}{\rmd\phi}.$ (3.35ak)

If $\lambda$ is constant, the differential equations are closed to become an
autonomous system, and fortunately, the potential is exactly the form of
Equation (3.23). In this way the analysis of the features of the phase space
of this system, i.e. the analysis of the fixed points and the determination of
the general behaviour of the orbits, provides insight on the global behaviour
of the Universe.

Let us first clarify some mathematical concepts and their relation to cosmic
evolution. A point $\mathbf{x}_{\mathrm{c}}$ is said to be a fixed point or a
critical point of the autonomous system if $f(\mathbf{x}_{\mathrm{c}})=0$. The
fixed points are divided into three categories based on their stability. The
first class is the stable node and stable spirals, distinguished by whether
the eigenvalues of Jacobian matrix are pure real or complex. A stable fixed
point $\mathbf{x}_{\mathrm{c}}$, no matter a stable node or a stable spiral,
is an attractor. It represents the position where the system converges towards
in the infinite future, i.e. $\mathbf{x}\to\mathbf{x}_{\mathrm{c}}$ for
$t\to\infty$. In the context of cosmic evolution, the Universe eventually
moves to an attractor in phase space. Point that indicates matter-dominated or
radiation-dominated epoch should not be an attractor generally, since the
Universe has already entered an accelerating stage. On the contrary, models
with a stable fixed point representing the accelerating expansion are
theoretically favored, especially those give the current radio of background
fluid and dark energy for solving the coincidence problem. The second class
refers to the unstable node, often considered as the initial point of
evolution, and is thus sometimes termed the “past attractor”. The radiation
dominated or matter dominated stage is encouraged to be unstable in order to
initiate the observational acceleration. The last class is the saddle point,
neither serving as a past attractor nor a future one. It may potentially
represent a certain stage of cosmic evolution, but this stage is only
temporary, as the system cannot actually reach the saddle point and therefore
cannot remain there for an extended duration, unless the system initially
starts there.

By setting $\rmd x_{\mathrm{K}}/\rmd N=\rmd x_{\mathrm{V}}/\rmd N=\rmd
x_{\mathrm{r}}/\rmd N=0$ in Equations (3.35a), (3.35b) and (3.35c), the fixed
points are acquired and those satisfying condition

$x_{\mathrm{V}}>0,\quad x_{\mathrm{r}}>0$

are listed in Table 1 and plotted in Figure 2.

Table 1: Properties of fixed points for the quintessence in the exponential
potential (3.23) [79, 80].

  * Fixed point |  $(x_{\mathrm{K}},x_{\mathrm{V}},x_{\mathrm{r}}$) | $\Omega_{\mathrm{r}}$ | $\Omega_{\mathrm{d}}$ | $\Omega_{\mathrm{q}}$ | $w_{\mathrm{q}}$ | $w_{\mathrm{eff}}$
---|---|---|---|---|---|---
R | $(0,0,1)$ | $1$ | $0$ | $0$ | — | $\frac{1}{3}$
D | $(0,0,0)$ | $0$ | $1$ | $0$ | — | $0$
$\mathrm{R}_{\mathrm{sc}}$ | $\Big{(}\frac{2\sqrt{6}}{3\lambda},\frac{2\sqrt{3}}{3\lambda},\frac{\sqrt{\lambda^{2}-4}}{\lambda}\Big{)}$ | $1-\frac{4}{\lambda^{2}}$ | $0$ | $\frac{4}{\lambda^{2}}$ | $\frac{1}{3}$ | $\frac{1}{3}$
$\mathrm{D}_{\mathrm{sc}}$ | $\Big{(}\frac{\sqrt{6}}{2\lambda},\frac{\sqrt{6}}{2\lambda},0\Big{)}$ | $0$ | $1-\frac{3}{\lambda^{2}}$ | $\frac{3}{\lambda^{2}}$ | $0$ | $0$
$\mathrm{K}_{\pm}$ | $(\pm 1,0,0)$ | $0$ | $0$ | $1$ | $1$ | $1$
Q |  $\Big{(}\frac{\lambda}{\sqrt{6}},\frac{\sqrt{6-\lambda^{2}}}{\sqrt{6}},0$) | $0$ | $0$ | $1$ | $-1+\frac{\lambda^{2}}{3}$ | $-1+\frac{\lambda^{2}}{3}$

Figure 2: The phase space of $x_{\mathrm{K}}$ and $x_{\mathrm{V}}$ with
$\lambda=1$ (the upper subfigure), $\lambda=2$ (the middle subfigure), and
$\lambda=3$ (the bottom subfigure). Here we assume $x_{\mathrm{r}}=0$ to
reduce the phase space to two dimensions. The fixed points listed in Table 1
are plotted, with yellow, magenta, and black representing stable, saddle, and
unstable fixed points, respectively. Each streamline represents the evolution
of the Universe under different initial conditions. The red region identifies
accelerated expansion. It is clear that the scaling solution
$\mathrm{D}_{\mathrm{sc}}$ is not in the accelerated expansion region.

Point R (D) is a radiation (dust) dominated solution and is able to describe
the corresponding dominating epoch. Point $\mathrm{R}_{\mathrm{sc}}$
($\mathrm{D}_{\mathrm{sc}}$) is the scaling solution [79, 82] for the ratio
$\Omega_{\mathrm{q}}/\Omega_{\mathrm{r}}$
($\Omega_{\mathrm{q}}/\Omega_{\mathrm{d}}$) is a non-zero constant, which can
be used to hide the presence of the scalar field in the cosmic evolution, at
least at the background level. The scaling solution must satisfy the condition
$\lambda^{2}>3(1+w_{\mathrm{m}})$ in order to ensure that
$\Omega_{\mathrm{q}}<1$. Since $w_{\mathrm{eff}}=w_{\mathrm{m}}$ and usually
$0<w_{\mathrm{m}}<1$, scaling solution usually does not describe the cosmic
acceleration (unless for a double exponential potential [83] and some related
others [84, 85, 86]). Notice that the scaling solution returns to the
corresponding dominated solution in the limit $\lambda\to\infty$.

Points $\mathrm{K}_{\pm}$ represent solutions dominated by a scalar field but
are unable to account for the accelerating expansion due to the wrong EOS
parameter. Hence, the only fixed point that represents the present cosmic
epoch is the point Q, and the cosmic acceleration is realized if
$\lambda^{2}<2$. As a result, the evolution of the Universe is consistent with
the trajectory $\mathrm{R}\to\mathrm{D}\to\mathrm{Q}$.

Figure 3: Evolution of $\Omega_{\mathrm{r}}$, $\Omega_{\mathrm{d}}$,
$\Omega_{\mathrm{q}}$, and $w_{\mathrm{q}}$ for the potential
$V(\phi)=M^{5}/\phi$ versus the $\rme$-folding time. The vertical dashed line
denotes the present cosmological time, and the present values
$\Omega_{\mathrm{d}}\simeq 0.3$ and $\Omega_{\mathrm{q}}\simeq 0.7$ are
achieved. Here the complete curve of $w_{\mathrm{q}}$ is plotted, though it is
literally no longer considered as quintessence when $w_{\mathrm{q}}>-1/3$. The
solution enters the tracking regime in which the field energy density tracks
the background fluid density. The initial conditions are chosen from [87].

If $\lambda$ is not a constant, we have to consider an additional equation

$\frac{\rmd\lambda}{\rmd N}=-\sqrt{6}\lambda^{2}(\Gamma-1)x_{\mathrm{K}},$
(3.35al)

where

$\Gamma\equiv V\frac{\rmd^{2}V}{\rmd\phi^{2}}\left(\frac{\rmd
V}{\rmd\phi}\right)^{-2}.$ (3.35am)

In this scenario, the fixed points obtained in the case of a constant
$\lambda$ can be considered as “instantaneous” fixed points that change over
time [88, 89], if assuming that the time scale for the variation of $\lambda$
is significantly shorter than $H^{-1}$, the time scale that produces
significant changes in the scale of the Universe.

The possibility of a transition from the fixed point
$\mathrm{D}_{\mathrm{sc}}$ to Q arises when $\lambda$ decreases over time. If
the condition $\Gamma>1$ is satisfied, the absolute value of $\lambda$
decreases towards $0$. This means that the solutions finally approach the
accelerated instantaneous point Q even if $\lambda^{2}>2$ during radiation and
matter domination. The condition $\Gamma>1$ is the so-called tracking
condition under which the field density eventually catches up with that of the
background fluid, and a tracking solution characterized by

$\Omega_{\mathrm{q}}\simeq\frac{3(1+w_{\mathrm{q}})}{\lambda^{2}}$ (3.35an)

is called a tracker [90]. Figure 3 shows the evolution of each components of a
tracking solution. If $\Gamma$ varies slowly in time, the EOS parameter of
quintessence is nearly constant during the matter domination and radiation
domination periods [90, 91],

$w_{\mathrm{q}}\simeq
w_{\mathrm{q}}^{\rmi}=\frac{w_{\mathrm{m}}-2(\Gamma-1)}{2\Gamma-1}.$ (3.35ao)

The fact that $w_{\mathrm{q}}<w$ results in a slower decrease of the
quintessence energy density compared to the fluid energy density, and
ultimately it leads to an attainable value of $\Omega_{\mathrm{q}}=1$. From
Equation (3.35an), $w_{\mathrm{q}}$ is $-1+\lambda^{2}/3$ in the scalar field
dominated epoch, which corresponds to the stable fixed point Q as long as
$\lambda<3(1+w_{\mathrm{m}})$. The tracker fields correspond to attractor-like
solutions, wherein the field energy density tracks the background fluid
density across a wide range of initial conditions [74]. This implies that the
energy density of quintessence does not necessarily need to be significantly
smaller than that of radiation or matter in the early Universe, unlike in the
cosmological constant scenario. Consequently, this offers a potential solution
to alleviate the coincidence problem. Another benefit of the tracker solution
is that it does not require the introduction of a new mass hierarchy in
fundamental parameters [74].

In contrast to the tracking solution where $\Gamma>1$, if $\lambda$ gradually
increases over time, the accelerated phase of the Universe is limited at late
times because the energy density of the scalar field becomes negligible
compared to that of the background fluid.

###  3.2 Cosmological Boundary

####  3.2.1 Phantom Dark Energy.

The EOS parameter for the cosmological constant, $w_{\Lambda}=-1$, is referred
to as the cosmological boundary, for the reason not only distinguishing
between the rates of accelerating expansion, whether slower or faster than
exponential, but also marking the point where perturbation divergence occurs
[92]. A canonical scalar field, when treated as a perfect fluid, is bound by
an EOS parameter within the range $(-1,1)$. However, current observations
suggest the possibility of the equation of state parameter of dark energy
being smaller than $w_{\Lambda}$, a scenario even favored by the data [93, 94,
95, 10].

The phantom dark energy model was proposed as a complement to the canonical
quintessence model [96], offering a super-accelerating phase, the expansion
faster then exponential. This can be achieved by substituting the quintessence
action with an alternative one:

$S_{\mathrm{p}}[g^{\mu\nu},\phi]=\int\rmd^{4}x\sqrt{-g}\mathcal{L}_{\mathrm{p}}=\int
d^{4}x\sqrt{-g}\left[\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-V(\phi)\right].$
(3.35ap)

The EOS parameter of the phantom field is

$w_{\mathrm{p}}\equiv\frac{p_{\mathrm{p}}}{\rho_{\mathrm{p}}}=\frac{\dot{\phi}^{2}+2V(\phi)}{\dot{\phi}^{2}-2V(\phi)},$
(3.35aq)

then one has $w_{\mathrm{p}}>1$ for the kinetic dominated regime
($\dot{\phi}^{2}/2>V$) and $w_{\mathrm{p}}<-1$ for the potential dominated
regime ($V>\dot{\phi}^{2}/2$).

Figure 4: The phase space of $x_{\mathrm{K}}$ and $x_{\mathrm{V}}$ with value
$x_{\mathrm{r}}=0$ and $\lambda=1$. The magenta saddle fixed point D
represents the matter domination, while the black stable point P represents
the phantom dominated stage. Each streamline represents the evolution of the
Universe under different initial conditions. The light red region shows where
Universe undergoes a standard accelerated expansion
$-1<w_{\mathrm{eff}}<-1/3$, while the dark red region identifies expansion
faster then exponential. The two green dashed lines correspond to
$x_{\mathrm{V}}=\pm x_{\mathrm{K}}$ where the phantom EOS parameter diverges.

The choice of the same expansion normalised variables (3.29) also closes the
system if the exponential potential (3.23) is considered. Hence, the analogous
analysis can be transplanted for phantom models. Figure 4 shows the phase
space of a phantom dark energy Universe with $w_{\mathrm{m}}=0$ (a different
$w_{\mathrm{m}}\in[0,\frac{1}{3}]$ doesn’t alter qualitative descriptions
below). Only two fixed points exist in the system: one is the saddle point D,
where the Universe contains only matter, and the other is the stable state P,
representing a phantom dominated attractor. A heteroclinic orbit connecting
point D to point P would represent the late time translation from matter to
phantom domination. On the way to the attractor, the expanding solution for
the scalar field will be

$a(t)\propto(t_{\infty}-t)^{-2/\lambda^{2}},$ (3.35ar)

ultimately leading to the occurrence of the “Big Rip” singularity within a
finite proper time $t_{\infty}$ [97, 98]. A potential bound from above is
necessary to suppress this singularity, since the phantom field rolls up the
potential and subsequently converge towards $w_{\mathrm{p}}=-1$ as the field
stabilizes at the maximum of potential [99, 100].

Quantum instability always arises in phantom models due to the flipped sign of
the kinetic term. Once the phantom quanta interact with other fields, even
though gravity, there will be an instability of the vacuum because energy is
no longer bound from below [99, 101]. Moreover, phantom models are problematic
at the classical level because a strongly anisotropic CMB background is
expected unless the Lorentz symmetry is broken for the phantom at an energy
scale lower than 3 MeV [101]. Last but not least, the phantom model with an
exponential potential fails to address the coincidence problem and suffers
from the fine-tuning of initial condition. Consequently, the inclusion of
sophisticated potentials and matter fluids beyond the range of
$[0,\frac{1}{3}]$ have to be considered [102, 103].

####  3.2.2 Cross the Cosmological Boundary.

The EOS parameters for dark energy in both the quintessence and phantom models
are confined to opposite sides of the cosmological boundary, preventing them
from crossing over into each other’s domain. A no-go theorem indeed confirms
this, stating as follows [104]: for the theory of dark energy in the FLRW
Universe described by a single fluid or a single scalar field $\phi$ with a
Lagrangian
$\mathcal{L}=\mathcal{L}(\phi,\partial_{\mu}\phi\partial^{\mu}\phi)$, which
minimally coupled to Einstein gravity, its EOS $w_{\mathrm{de}}$ cannot across
the cosmological constant boundary.

In a scenario where gravity is still governed by general relativity,
additional degrees of freedom are necessary to enable dark energy to
effectively traverse the boundary. The quintom model was proposed as a multi-
field model incorporating a quintessence-like field and a phantom-like field
[105, 106],

$S_{\mathrm{qp}}[g^{\mu\nu},\phi_{\mathrm{q}},\phi_{\mathrm{p}}]=\int\rmd^{4}x\sqrt{-g}\mathcal{L}_{\mathrm{qp}}\left[-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi_{\mathrm{q}}\partial_{\nu}\phi_{\mathrm{q}}+\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi_{\mathrm{p}}\partial_{\nu}\phi_{\mathrm{p}}-V(\phi_{\mathrm{q}},\phi_{\mathrm{p}})\right].$
(3.35as)

The energy density, pressure and the corresponding EOS parameter of the
quintom model are

$\displaystyle\rho_{\mathrm{de}}=\frac{1}{2}\dot{\phi}_{\mathrm{q}}^{2}-\frac{1}{2}\dot{\phi}_{\mathrm{p}}^{2}+V(\phi_{\mathrm{q}},\phi_{\mathrm{p}}),$
(3.35at) $\displaystyle
p_{\mathrm{de}}=\frac{1}{2}\dot{\phi}_{\mathrm{q}}^{2}-\frac{1}{2}\dot{\phi}_{\mathrm{p}}^{2}-V(\phi_{\mathrm{q}},\phi_{\mathrm{p}}),$
(3.35au) $\displaystyle
w_{\mathrm{de}}\equiv\frac{p_{\mathrm{qp}}}{\rho_{\mathrm{qp}}}=\frac{\dot{\phi}_{\mathrm{q}}^{2}-\dot{\phi}_{\mathrm{p}}^{2}-2V(\phi_{\mathrm{q}},\phi_{\mathrm{p}})}{\dot{\phi}_{\mathrm{q}}^{2}-\dot{\phi}_{\mathrm{p}}^{2}+2V(\phi_{\mathrm{q}},\phi_{\mathrm{p}})}.$
(3.35av)

It is clear that $w_{\mathrm{qp}}\gtrless-1$ if
$\dot{\phi}_{\mathrm{q}}\gtrless\dot{\phi}_{\mathrm{p}}$ and the cosmological
constant scenario is recovered when
$\dot{\phi}_{\mathrm{q}}=\dot{\phi}_{\mathrm{p}}$.

The potential $V(\phi_{\mathrm{q}},\phi_{\mathrm{p}})$ for non-interacting
scalar fields can be generally written as

$V(\phi_{\mathrm{q}},\phi_{\mathrm{p}})=V_{\mathrm{q}}(\phi_{\mathrm{q}})+V_{\mathrm{p}}(\phi_{\mathrm{p}}),$
(3.35aw)

where $V_{\mathrm{q}}(\phi_{\mathrm{q}})$ and
$V_{\mathrm{p}}(\phi_{\mathrm{p}})$ are arbitrary self-interacting potentials
for the quintessence and phantom respectively. In such case, the two scalar
field can be treated exactly as if they were single models. Due to the fact
that $w_{\mathrm{p}}<w_{\mathrm{q}}$, the Universe will eventually evolve to a
stage where phantom dominates $w_{\mathrm{eff}}=w_{\mathrm{de}}\leq-1$. The
$w_{\mathrm{de}}$ across the cosmological boundary from above to below if the
matter dominated epoch is followed by a temporary quintessence dominated stage
[106, 107, 108]. If future observations will constrain the EOS parameter of
dark energy to be below $-1$ today but above $-1$ in the past, then the
quintom scenario of is arguably the simplest framework where such a situation
can arise.

Unfortunately, neither the coupled nor the uncoupled quintom models seem to
solve the cosmic coincidence problem and the fine-tuning of initial conditions
simultaneously, not to mention the fundamental problems associated with the
ghost field. Furthermore, couplings between kinetic terms and scalar fields,
like $\partial_{\mu}\phi_{\mathrm{q}}\partial^{\mu}\phi_{\mathrm{p}}$ [109]
and the so-called hessence dark energy model [110, 111, 112] are investigated
as well. Finally, theories beyond general relativity, such as those
incorporating higher curvature terms and non-minimally coupled scalar fields
between gravity or matter, can also cross the cosmological boundary without
encountering negative kinetic energy and the associated quantum instability,
as discussed in 4.3.3.

###  3.3 Non-canonical Kinetic Term

In the preceding sections, some simplest dynamical dark energy models are
introduced, providing as much detail as possible regarding their evolution.
Here we briefly review dark energy models from scalar fields with modified
kinetic terms, which hold significant potential for addressing those problems
of $\Lambda$CDM model at once.

####  3.3.1 k-essence.

A scalar field model of dark energy with a modified kinetic term is the so-
called kinetically driven essence, or k-essence for simplicity. The initial
focus of the k-essence was on its application to inflation [113], and numerous
theoretical models, such as low energy effective string theory [114], ghost
condensate [115], tachyon [116, 117], and Dirac-Born-Infeld (DBI) theory,
[118, 119] can be classified under this dark energy model.

The action of the k-essence dark energy model is given by [113]

$S[g^{\mu\nu},\phi,\Psi_{\mathrm{m}}]=S_{\mathrm{EH}}[g^{\mu\nu}]+\int\rmd^{4}x\sqrt{-g}P(\phi,X)+S_{\mathrm{m}}[g^{\mu\nu},\Psi_{\mathrm{m}}],$
(3.35ax)

where $P(\phi,X)$ is an arbitrary function of a scalar field $\phi$ and its
kinetic energy

$X\equiv-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi.$ (3.35ay)

The k-essence model can achieve accelerated expansion solely through kinetic
energy, even in the absence of a potential [120, 121], distinguishing it from
models with canonical kinetic energy.

The energy-momentum tensor of the k-essence is given by

$T^{\mathrm{k}}_{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\delta(\sqrt{-g}P)}{\delta
g^{\mu\nu}}=P_{X}\partial_{\mu}\phi\partial_{\nu}\phi+g_{\mu\nu}P,$ (3.35az)

where the subscript $X$ of function $P$ represents the partial derivative with
respect to $X$. Comparing with the energy-momentum tensor of a perfect fluid
(2.4), we have the pressure and energy density of the k-essence,

$\displaystyle p_{\mathrm{k}}=P,$ (3.35ba)
$\displaystyle\rho_{\mathrm{k}}=2XP_{X}-P,$ (3.35bb)

with the 4-velocity $u_{\mu}=\partial_{\mu}\phi/\sqrt{2X}$. If $P$ only
depends on $\phi$, we obtain $p_{\mathrm{k}}=-\rho_{\mathrm{k}}$, which
corresponds to general relativity with the cosmological constant. If $P$
solely depends on $X$, we have $\rho_{\mathrm{k}}=\rho_{\mathrm{k}}(X)$.
Hence, $\rho_{\mathrm{k}}$ can be rearranged to give
$p_{\mathrm{k}}=p_{\mathrm{k}}(\rho_{\mathrm{k}})$, which is the EOS for an
isentropic fluid. In the general case where $p_{\mathrm{k}}=P(\phi,X)$, the
hydrodynamical analogy is still useful, and the EOS parameter is acquired
directly,

$w_{\mathrm{k}}\equiv\frac{p_{\mathrm{k}}}{\rho_{\mathrm{k}}}=\frac{P}{2XP_{X}-P},$
(3.35bc)

and $w_{\mathrm{k}}$ approaches to $-1$ if the condition $|XP_{X}|\ll|P|$ is
satisfied.

A non-canonical scalar field Lagrangian will in general introduce theoretical
problems at both the quantum and classical levels. Hence, the selection of
function $P$ has limitations. The propagation sound speed
$c_{\mathrm{s}}^{\mathrm{k}}$ of the k-essence field is given by [122],

$\big{(}c^{\mathrm{k}}_{\mathrm{s}}\big{)}^{2}=\frac{\partial p_{k}/\partial
X}{\partial\rho_{k}/\partial X}=\frac{P_{X}}{P_{X}+2XP_{XX}},$ (3.35bd)

which should be a positive number. Furthermore, due to the positive
definiteness of the Hamiltonian [123], both the denominator $\xi_{1}$ and the
numerator $\xi_{2}$ of $\big{(}c^{\mathrm{k}}_{\mathrm{s}}\big{)}^{2}$ in
Equation (3.35bd) should be positive. To be in accordance with the principle
of causality, $c_{\mathrm{s}}^{\mathrm{k}}$ is considered favorable if it does
not exceed the speed of light which gives $P_{XX}\geq 0$; however, see Ref.
[124].

On account of the unknown dependence upon the arbitrary function $P(\phi,X)$,
it is too complicated to directly study with dynamical systems techniques.
Therefore, to study their dynamics and evolution, k-essence are typically
divided into several classes [78]:

$\displaystyle\mathrm{Class\ I:}\quad$ $\displaystyle P(\phi,X)=XG(Y),\quad
Y=\frac{X}{V(\phi)},$ (3.35be) $\displaystyle\mathrm{Class\ II:}\quad$
$\displaystyle P(\phi,X)=K(\phi)\tilde{P}(X),$ (3.35bf)
$\displaystyle\mathrm{Class\ III:}\quad$ $\displaystyle
P(\phi,X)=F(X)-V(\phi),$ (3.35bg)

where $G$, $K$, $\tilde{P}$, $F$ and $V$ are functions of their own arguments.
It is worth noting that a model can be classified into one of two classes, as
seen with the quintessence model, or it may not fit into either category, as
exemplified by the DBI model.

The first class of k-essence exhibits a deep connection with scaling solutions
that are of great relevance for the cosmic coincidence problem. It has been
proved that [123] scaling solutions appear in k-essence models only if a
Lagrangian of the type (3.35be) is assumed with
$V(\phi)\propto\rme^{-\lambda\phi/M_{\mathrm{Pl}}}$. Further research
indicated that dark energy late-time solutions are invariably unstable in the
presence of a scaling solution, unless they belong to the phantom type, even
with a coupling to the matter sector [125, 126]. Note that the quintessence
exhibits scaling solutions because it corresponds to $G(Y)=1-1/Y$.

Unlike the Class I, the second class is intriguing for its tracking solutions.
It has been shown that tracking solutions naturally appear for the models
(3.35bf) during the radiation dominated era, and a cosmological-constant-like
behavior shortly after the transition to matter domination were put forward,
which claims to have solved the coincidence problem without fine-tuning the
parameters [127, 128]. However, a singularity associated with a diverging
sound speed is present in such models [129], so that it cannot arise as a low-
energy effective field theory of a causal, consistent, high-energy theory.
Moreover, the Lagrangians in these models are hard to construct in the
framework of particle physics.

The third class (3.35bg) has a non-canonical kinetic term appears together
with a standard self-interacting potential for the scalar field. The famous
(dilatonic) ghost condensate is usually considered a subclass of this kind,
which is simply recommended in next Section 3.3.2. One interesting thing is
that some special functions $F(X)$ and $V(\phi)$ produce scaling solutions,
even if the Lagrangian cannot be written in the form of Class I [130].

####  3.3.2 (Dilatonic) Ghost Condensate.

The quintessence and phantom models, as special cases of k-essence, have

$P(\phi,X)=\epsilon X-V(\phi),$ (3.35bh)

where $\epsilon=\pm 1$ represents quintessence and phantom respectively.

The denominator and numerator of propagation sound speed are
$\xi_{1}=\xi_{2}=\epsilon$, indicating the phantom model is unstable. The
instabilities come form the perturbation level, where the gradient energy
plays an important role. Therefore, one approach is to introduce a background
action resembling that of a ghost, allowing the spatial gradient terms to
address the instability issue. To implement the idea, Arkani-Hamed et al.
[115] proposed an effective field theory of a rolling ghost, which can be
conceptualized as an infrared modification of gravity. They termed this
modification as ghost condensate with the arbitrary function taking the form

$P(\phi,X)=-X+\frac{X^{2}}{M^{4}},$ (3.35bi)

where $M$ is a constant having a dimension of mass. This model was later
generalized to a modified version as

$P(\phi,X)=-X+\frac{X^{2}}{M^{4}}\exp{\left(\frac{\lambda\phi}{M_{\mathrm{Pl}}}\right)},$
(3.35bj)

which is called dilatonic ghost condensate model [123].

Here we briefly introduce the cosmic evolution of dilatonic ghost condensate
dark energy model. The pressure, energy density and the corresponding EOS
parameter are

$\displaystyle p_{\mathrm{dGC}}=-X+\frac{X^{2}}{M^{4}}\rme^{\lambda\phi},$
(3.35bk)
$\displaystyle\rho_{\mathrm{dGC}}=-X+\frac{3X^{2}}{M^{4}}\rme^{\lambda\phi},$
(3.35bl) $\displaystyle
w_{\mathrm{dGC}}\equiv\frac{p_{\mathrm{dGC}}}{\rho_{\mathrm{dGC}}}=\frac{1-\rme^{\lambda\phi}X/M^{4}}{1-3\rme^{\lambda\phi}X/M^{4}},$
(3.35bm)

where $M_{\mathrm{Pl}}=1$ is used for briefness. Quantum stability
necessitates $\rme^{\lambda\phi}X/M^{4}\geq 1/2$, resulting in the EOS
parameter lying within the range $-1\leq w_{\mathrm{dGC}}<1/3$. In particular
the de Sitter solution $w_{\mathrm{dGC}}=-1$ is realized at
$X=M^{4}e^{-\lambda\phi}/2$. Then it is possible to explain the present cosmic
acceleration for $M\sim 10^{-3}\;\mathrm{eV}$ in ghost condensate model as
$\rho_{\mathrm{GC}}=-p_{\mathrm{dGC}}=M^{4}/4$ at the de Sitter point. Hence,
the (dilatonic) ghost condensate is another example of accelerated expansion
that can be generated without the assistance of a potential.

By adjusting the selection of dynamic variables to [87]

$x_{\mathrm{K}}\equiv\frac{\dot{\phi}}{\sqrt{6}H},\quad
x_{\mathrm{M}}\equiv\frac{\dot{\phi}^{2}\rme^{\lambda\phi}}{2M^{4}},\quad
x_{\mathrm{r}}\equiv\frac{\sqrt{\rho_{\mathrm{r}}}}{\sqrt{3}H}.$ (3.35bn)

we can get the autonomous system describing the dilatonic ghost condensate
model. There always exists a stable point describing accelerated expansion
toward which the Universe evolves, as long as $0\leq\lambda<\sqrt{6}/3$. The
corresponding sound speed is in the range $0\leq
c_{\mathrm{s}}^{\mathrm{dGC}}<1/3$, which means this model does not violate
causality. The EOS parameter of this future attractor is [87]

$w_{\mathrm{eff}}=w_{\mathrm{dGC}}=\frac{\lambda^{2}f(\lambda)-8}{3\lambda^{2}f(\lambda)+8},\quad
f(\lambda)=1+\sqrt{1+\frac{16}{3\lambda^{2}}},$ (3.35bo)

which suggests that the ghost condensate behaves like a cosmological constant
in dynamics during the final stage of cosmic evolution.

####  3.3.3 Unified Dark Matter and Dark Energy.

The temptation to unify dark matter and dark energy into a single entity has
occurred to many cosmologists almost from the beginning, in order to solve two
big dark sector problems at once. Both a fluid and a scalar field are
attempted to represent dark matter and dark energy at the same times [131,
132, 133, 134]. Here we simply introduce a kind of k-essence model that is
regarded as dark matter and dark energy simultaneously.

The Lagrangian of a such k-essence model with only a kinetic term is [132]

$P(\phi,X)=-F_{0}+F_{2}(X-X_{0})^{2},$ (3.35bp)

where $F_{0}$ and $F_{2}$ are constants. Notice that there is an extremum
$X_{0}$ of the kinetic energy, around which the pressure and energy density of
k-essence are approximately

$\displaystyle p_{\mathrm{k}}\simeq-F_{0},$ (3.35bq)
$\displaystyle\rho_{\mathrm{k}}\simeq F_{0}+4F_{2}X_{0}(X-X_{0}).$ (3.35br)

Furthermore, by substituting $X=X_{0}(1+\varepsilon a^{-3})$ with
$0<\varepsilon a^{-3}\ll 1$ ($\varepsilon$ being a constant), the solution to
the continuity equation around $X_{0}$ yields the EOS parameter

$w_{\mathrm{k}}=-\left(1+\frac{4F_{2}}{F_{0}}X_{0}^{2}\varepsilon
a^{-3}\right)^{-1}.$ (3.35bs)

It is possible to realize $w_{\mathrm{k}}\simeq 0$ during the matter
domination provided that the condition $4F_{2}X_{0}^{2}/F_{0}\gg 1$ is
satisfied, while $w_{\mathrm{k}}$ approaches the de Sitter value $-1$ at late
time due to $\varepsilon a^{-3}\to 0$. This specific model does not suffer
from the problem of stability or causality if $X>X_{0}$, rendering it a more
reliable unified model.

###  3.4 Interacting Dark Energy Model

Typically, the dark energy models are based on scalar fields minimally coupled
to the gravity, and do not implement the explicit coupling of the field to
background matter, as illustrated by the models introduced above. However,
there is no fundamental reason for this assumption in the absence of an
underlying symmetry supposed to suppress the coupling, in particular the dark
sector that we barely no nothing about from a microscopic and dynamical field
theory perspective. Although the interaction between dark energy and normal
matter particles are heavily restricted by observations [135, 38], this is not
the case for dark matter particles, and the neglect of this potential
interaction of dark components may result in misinterpretations of
observational data. In fact, the study of the interaction between dark energy
and dark matter is getting more promising recently, for the possible solution
to the coincident problem and tensions in $\Lambda$CDM models.

####  3.4.1 Phenomenological Description.

Since there is no established approach from first principles that can specify
the form of coupling between dark energy and dark matter, dark matter could
manifest as either bosonic or fermionic particles, adhering to either the
standard model or extending beyond it. Likewise, dark energy could be
conceptualized as a fluid, a field, or in other forms. Constructing
phenomenological models initially, grounded in intuition and experience, is
consistently beneficial. These models can then be rigorously tested against
diverse observational data.

The interaction between dark matter and dark energy is described by following
modified conservation equations,

$\displaystyle\dot{\rho}_{\mathrm{dm}}+3H\rho_{\mathrm{dm}}=Q,$ (3.35bt)
$\displaystyle\dot{\rho}_{\mathrm{de}}+3H(1+w_{\mathrm{de}})\rho_{\mathrm{de}}=-Q,$
(3.35bu)

where $Q$ is termed the interacting kernel and its sign determines the
direction of the energy flux. Specifically, energy transfers from dark energy
to dark matter when $Q>0$ and the transmission reverses when $Q<0$. The
original model is restored if the interaction is “turned off”.

In order to illustrate how interaction between the dark components acts on
cosmological dynamics, consider the time evolution of the radio
$r\equiv\rho_{\mathrm{dm}}/\rho_{\mathrm{de}}$,

$\dot{r}=\frac{\rho_{\mathrm{dm}}}{\rho_{\mathrm{de}}}\left(\frac{\dot{\rho}_{\mathrm{dm}}}{\rho_{\mathrm{dm}}}-\frac{\dot{\rho}_{\mathrm{de}}}{\rho_{\mathrm{de}}}\right)=-3\Gamma
Hr,\quad\Gamma=-w_{\mathrm{de}}-\frac{1+r}{\rho_{\mathrm{dm}}}\frac{Q}{3H}.$
(3.35bv)

It is not difficult to observe that the interaction kernel should be the
product of an energy density and a term with an inverse time dimension.
Additionally, it is natural to assume that the kernel depends solely on
relevant dynamical quantities (such as $\rho_{\mathrm{dm}}$,
$\rho_{\mathrm{de}}$ and $a$) and possibly their derivatives. The simplest
cases can be the following kernels

$Q\supset\xi_{1}\rho_{\mathrm{dm}}H,\xi_{2}\rho_{\mathrm{de}}H,\xi_{3}\dot{\rho}_{\mathrm{dm}},\xi_{4}\dot{\rho}_{\mathrm{de}},\cdots$

and their linear combinations. Other forms of interaction kernel including
nonlinearity and a lot of complexity are summarized in review papers like
[136, 137].

Now, let us consider the case
$Q=3H(\xi_{1}\rho_{\mathrm{dm}}+\xi_{2}\rho_{\mathrm{de}})$ and examine how
the interacting dark energy model can resolve the coincidence problem. For
this particular kernel,

$\Gamma=-w_{\mathrm{de}}-\frac{(1+r)(\xi_{1}r+\xi_{2})}{r},$ (3.35bw)

and the stationary solutions are obtained imposing
$r_{\mathrm{s}}\Gamma(r_{\mathrm{s}})=0$, which gives ($\xi_{1}\neq 0$)

$r_{\mathrm{s}}^{\pm}=-\frac{w_{\mathrm{de}}+\xi_{1}+\xi_{2}\pm\sqrt{(w_{\mathrm{de}}+\xi_{1}+\xi_{2})^{2}-4\xi_{1}\xi_{2}}}{2\xi_{1}}.$
(3.35bx)

The coincidence problem is completely solved if one of the above roots
$r_{\mathrm{s}}^{\pm}$ is the future attractor of the system. Moreover, the
evolution of the Universe should gradually approach the attractor from above,
implying that the ratio of matter to dark energy decreases over time. In the
special case $\xi_{1}=\xi_{2}=\xi$, the larger root $r_{\mathrm{s}}^{-}$ is a
past attractor while the smaller root $r_{\mathrm{s}}^{+}$ is a future
attractor [138, 139]. As Universe expands, $r(t)$ will evolve from
$r_{\mathrm{s}}^{-}$ to the stable solution $r_{\mathrm{s}}^{+}$ avoiding the
coincidence problem if $r_{\mathrm{s}}^{+}\sim 3/7$. If $\xi_{2}=0$, the
system still retains a past attractor $r_{s}^{+}$ with no future attractor
($r_{\mathrm{s}}^{-}=0$ in this case). However, the variation is
$|(\dot{r}/r)_{0}|<H_{0}$, slower than in $\Lambda$CDM, which mitigates the
coincidence problem [140]. As for the $\xi_{1}=0$ case, it is fortunate that
the only fixed point $r_{\mathrm{s}}=-\xi_{2}/(w_{\mathrm{de}}+\xi_{2})$ is
stable. Consequently, there is no more coincidence problem if we select
$\xi_{2}\sim 0.3$ under the assumption of $w_{\mathrm{de}}=-1$. Figure 5
showing the phase space of this example will help understand the conclusion.

Figure 5: The diagram of $\Omega_{\mathrm{m}}$-$\Omega_{\mathrm{de}}$ with
values $w_{\mathrm{de}}=-1$ and $\xi_{2}=0.3$ in the case where
$Q=3\xi_{2}H\rho_{\mathrm{de}}$. The black point A is the only stable fixed
point in the system representing the future attractor with
$\Omega_{\mathrm{m}}=0.3$ and $\Omega_{\mathrm{de}}=0.7$. The black point
$\Lambda$ is the future attractor of $\Lambda$CDM model with
$\Omega_{\mathrm{m}}=0$ and $\Omega_{\mathrm{de}}=1$. The orange point R and
magenta point D are unstable radiation domination and saddle matter domination
respectively. Each streamline represents the evolution of the Universe under
different initial conditions. The red region identifies accelerated expansion.

While the initial drive behind interacting dark energy models was to solve or
alleviate the coincidence problem, their focus has recently shifted towards
elucidating the disparity between the observed value of the Hubble constant
derived from CMB data and the local measurements. The interacting dark energy
model increases the proportion of late-time dark energy, directly boosting the
Hubble expansion rate. Moreover, the reduction in dark matter entails an
increase in the Hubble constant to uphold the physical ratio of dark matter
energy density in accordance with CMB constraints. In addition, dark matter
decaying into dark energy also reduces structure growth of matter at late
time, thereby alleviating the $S_{8}$ problem. Consequently, the interacting
dark energy model emerges as a multifaceted potential solution to the Hubble
tension.

####  3.4.2 Coupled Scalar Fields.

If focusing our attention on dark energy in the form of quintessence, one has
the coupled quintessence model with the modified Klein-Gordon equation

$\ddot{\phi}+3H\dot{\phi}+\frac{\rmd V}{\rmd\phi}=-\frac{Q}{\dot{\phi}}.$
(3.35by)

The basic kernel $Q\propto\rho_{\mathrm{dm}}\dot{\phi}$ naturally emerges in
certain modified gravity theories (such as $f(R)$ gravity [141, 142]) in the
Einstein frame, through which dynamics are introduced in Section 4.2.1.

In general, all aforementioned dark energy models can be generalized by
introducing a coupling in the dark sector, following the same approach as with
quintessence. The coupled phantom models with exponential potentials and
frequently analyzed couplings demonstrate that scaling solutions cannot remain
stable [27, 143], as is the case with quintessence. Therefore, they are unable
to resolve the cosmic coincidence problem.

The situation is different for more general k-essence models. Scaling
solutions typically exist in Lagrangians of the form
$P(\phi,X)=Q^{2}(\phi)XG(XQ^{2}(\phi)\rme^{\lambda\phi/M_{\mathrm{Pl}}})$,
where $Q(\phi)$ is the coupling function and $G$ an arbitrary function [126].
However, it has been demonstrated that an evolutionary trajectory capable of
resolving the coincidence problem is prohibited due to the singularity
associated with both dynamical variables and the sound speed [126]. As
additional resources, the study of coupled tachyonic models can be found in
[144, 145, 146], while coupled DBI scalar field models are discussed in [147,
148].

##  4 Scalar-Tensor Gravity

Physicists introduced dark energy to explain the accelerating phase of the
Universe, but the nature of this bizarre substance is unknown and the validity
has also been called into questions. Since the evidence for dark energy comes
entirely from its gravitational effects, which are inferred assuming the
validity of general relativity, the consideration of modifications to general
relativity, such that the necessity of dark energy is obviated, is another
rational approach [37]. In this section, we focus on the scalar-tensor
gravity, one of the most studied modified theories of gravity.

###  4.1 Introduction to Scalar-Tensor Gravity

The central idea behind scalar-tensor gravity theory is the incorporation of a
scalar field alongside the metric tensor field to participate in the
gravitational interaction. This addition of the scalar field allows for
variations in the strength of gravity over spacetime, opening up new avenues
for understanding the fundamental forces of the Universe. Scalar-tensor theory
has since evolved and been refined in various forms and models, becoming a
focal point for research in cosmology, astrophysics, and fundamental physics
[30].

The origins of scalar-tensor gravity can be traced back to the pioneering work
in the mid-20th century. One of the earliest proponents of this theory was the
German physicist Jordan [149], who in 1955 introduced the concept of a scalar
field coupled to gravity as a means to unify gravity with electromagnetism.
Moreover, it was Brans and Dicke who further developed and formalized the
theory in 1961 [150], now known as the Brans-Dicke (BD) theory. The action for
the theory is,

$S_{\mathrm{BD}}[g^{\mu\nu},\Phi,\Psi_{\mathrm{m}}]=\int\rmd^{4}x\sqrt{-g}\left(\Phi\frac{R}{2\kappa}-\frac{\omega_{\mathrm{BD}}}{2\Phi}g^{\mu\nu}\partial_{\mu}\Phi\partial_{\nu}\Phi\right)+S_{\mathrm{m}}[g^{\mu\nu},\Psi_{\mathrm{m}}],$
(3.35a)

where $w_{\mathrm{BD}}$ is the only parameter in the theory. Here we add a
coefficient of $\frac{1}{2}$ to the Lagrangian of the gravitational part of
the traditional BD theory. Shortly afterward, Bergmann [151] and Wagoner [152]
separately generalized the action to

$S_{\mathrm{J}}[g^{\mu\nu},\Phi,\Psi_{\mathrm{m}}]=\int\rmd^{4}x\sqrt{-g}\left[\Phi\frac{R}{2\kappa}-\frac{W(\Phi)}{2\Phi}g^{\mu\nu}\partial_{\mu}\Phi\partial_{\nu}\Phi-U(\Phi)\right]+S_{\mathrm{m}}[g^{\mu\nu},\Psi_{\mathrm{m}}]\,,$
(3.35b)

by letting the coupling parameter $w_{\mathrm{BD}}$ be a function of the
scalar field $W(\Phi)$ and adding a potential term $U(\Phi)$. The subscript
“J” here represents the Jordan frame.

A suitable conformal transformation

$g_{\mu\nu}\to g_{\mu\nu}^{*}=A^{-2}(\phi)g_{\mu\nu}$ (3.35c)

is able to take us from the Jordan frame to the Einstein frame [153]. We
denote the scalar field in the Einstein frame as $\phi$ instead of the field
$\Phi$ in the Jordan frame. The action in the Einstein frame is

$S_{\mathrm{E}}[g^{\mu\nu}_{*},\phi,\Psi_{\mathrm{m}}]=\int\rmd^{4}x\sqrt{-g_{*}}\left[\frac{R_{*}}{2\kappa_{*}}-\frac{1}{2}g_{*}^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-V(\phi)\right]+S_{\mathrm{m}}[A^{-2}(\phi)g^{\mu\nu}_{*},\Psi_{\mathrm{m}}],$
(3.35d)

where $\kappa_{*}=8\pi G_{*}$, and $G_{*}$ is a bare gravitational constant.
The physical quantity marked with a star indicates that it is in the Einstein
frame. The potential $V(\phi)$ in the Einstein frame is related to the
potential $U(\Phi)$ in the Jordan frame by the rescaling function $A(\phi)$ as

$V(\phi)=A^{4}(\phi)U(\phi).$ (3.35e)

The gravity theory is usually not invariant under conformal transformation,
including Einstein’s general relativity. A natural question arises: which
conformal frame is physical? Unfortunately, as of now, the answer remains
unknown. In the subsequent Section 4.2 and Section 4.3, the Einstein frame and
Jordan frame are both considered, respectively. Cosmologies within these two
frames are discussed as well in corresponding sections.

###  4.2 Einstein Frame: Coupled Quintessence

The canonical form of action (3.35d) is acquired if $A(\phi)$ satisfies

$\displaystyle A^{2}(\phi)=\frac{\kappa}{\kappa_{*}\Phi},$ (3.35f)
$\displaystyle\alpha^{2}(\phi)=\frac{1}{2[2\kappa W(\Phi)+3]},$ (3.35g)

where $\alpha(\phi)\equiv{\rmd\ln A}/{\rmd\phi}$. The relation between scalar
fields $\Phi$ and $\phi$ in Jordan and Einstein frames is easily acquired from
Equations (3.35f) and (3.35g), as

$\frac{\rmd\phi}{\rmd\Phi}=\pm\frac{\sqrt{\kappa W(\Phi)+3/2}}{\Phi}.$ (3.35h)

From now on, we will use the unit $\kappa=1$ for simplicity.

The equations of motion derived from action (3.35d) read

$\displaystyle
G_{\mu\nu}^{*}=\kappa_{*}(T^{*\mathrm{m}}_{\mu\nu}+T_{\mu\nu}^{*\phi}),$
(3.35i) $\displaystyle\square_{*}\phi=\frac{\rmd V}{\rmd\phi}-\alpha
T_{\mathrm{m}}^{*},$ (3.35j)

where $T^{*}_{\mathrm{m}}\equiv g^{\mu\nu}_{*}T^{*\mathrm{m}}_{\mu\nu}$ is the
trace of $T_{\mu\nu}^{*\mathrm{m}}$, and the energy-momentum tensors of the
background fluid and the scalar field are

$\displaystyle
T^{*\mathrm{m}}_{\mu\nu}=-\frac{2}{\sqrt{-g_{*}}}\frac{\delta(\sqrt{-g_{*}}\mathcal{L}_{\mathrm{m}})}{\delta
g_{*}^{\mu\nu}},$ (3.35k) $\displaystyle
T^{*\phi}_{\mu\nu}=\partial_{\mu}\phi\partial_{\nu}\phi-g^{*}_{\mu\nu}\left[\frac{1}{2}g^{\alpha\beta}_{*}\partial_{\alpha}\phi\partial_{\beta}\phi+V(\phi)\right].$
(3.35l)

When $\alpha=0$, the field equations of scalar-tensor gravity in the Einstein
frame, Equations (3.35i) and (3.35j), along with energy-momentum tensors in
Equations (3.35k) and (3.35l), are in the same form of the field equations
(3.3) and (3.4) and the energy-momentum tensors in Equations (3.5) and (3.6)
for the quintessence model, respectively, so that the gravity is entirely
described by the metric tensor. The modification of gravity comes from the
direct coupling term $\alpha(\phi)T_{\mathrm{m}}^{*}$ in Equation (3.35j)
between the fluid and the scalar field $\phi$. Hence, the local conservation
of the energy-momentum tensor no longer holds in the Einstein frame, and it
changes to

$\nabla^{*}_{\mu}T^{\mu\nu}_{*\mathrm{m}}=\alpha(\phi)T_{\mathrm{m}}^{*}\nabla^{\nu}_{*}\phi.$
(3.35m)

As a result, the test particle does not move along geodesics of
$g^{*}_{\mu\nu}$ anymore, as if it suffers from a kind of “fifth force”.
According to Equation (3.35m), the original quintessence model is recovered as
$\alpha(\phi)$ vanishes, which corresponds to the case where the function
$W(\Phi)$ in the Jordan frame diverges. In this subsection we discuss the
results in the Einstein frame, and the star notation of physical quantity is
omitted in the following part of this subsection for simplicity. But keep in
mind that we are in the Einstein frame.

####  4.2.1 Dynamics of Coupled Quintessence.

The fact that the energy density of dark energy is of the same order as that
of dark matter in the present Universe suggests that there may be some
relation between them. Among various coupled dark energy models [80], the
coupled quintessence includes an interaction between a scalar field $\phi$ and
the dust matter—nonrelativistic matter whose pressure is zero—in the form
[154, 155],

$\nabla^{\mu}T_{\mu\nu}^{\phi}=-\alpha(\phi)T_{\mathrm{d}}\nabla^{\nu}\phi,\quad\nabla^{\mu}T^{\mathrm{d}}_{\mu\nu}=\alpha(\phi)T_{\mathrm{d}}\nabla^{\nu}\phi,$
(3.35n)

where $T_{\mu\nu}^{\phi}$ and $T^{\mathrm{d}}_{\mu\nu}$ are the energy-
momentum tensors of the scalar field and the dust. The coupling of the scalar
field and radiation vanishes due to the vanishing trace of
$T_{\mu\nu}^{\mathrm{r}}$, the energy-momentum tensor of the radiation.
Generally speaking, the coupling strength $\alpha(\phi)$ between the scalar
field and various components of the Universe is not necessarily uniform [156,
157, 158, 159], and the baryons are usually treated uncoupled to the scalar
field to avoid an extra long-range force besides gravity.

We only consider the case where the coupling $\alpha$ is a constant and the
potential is exponential for simplicity,222In the general scalar-tensor theory
with action (3.35ac), the constant coupling condition is approximately
satisfied, according to Equation (3.35bbbd), if  $\left(\frac{\rmd
F}{\rmd\varphi}\right)^{2}\gg\frac{2}{3}\zeta F.$ (3.35o) Then, one gets the
function $F(\phi)=\exp(\pm\sqrt{2/3}\phi)$ and the constant coupling
$\alpha\approx\pm 1/\sqrt{6}$. Furthermore, a simple way to construct the
exponential potential $V(\phi)$ is assuming that the coupling function
$F(\varphi)$ and the potential function $U(\varphi)$ satisfy the relation
$U(\varphi)\propto F^{k}(\varphi).$ (3.35p) which holds, for instance, when
both $U$ and $F$ are power-law or exponential, but is also valid for much more
complicated functions, like products of power-law and exponential forms. The
relation (3.35p) was first used by Amendola et al. [160] to reveal the
relation between the scalar-tensor theory and the coupled quintessence, and to
study the non-minimal coupling gravity in the strong coupling limit. However,
the solutions of cosmic evolution are ruled out by the present constraints on
the variability of the gravitational coupling, and they only allow for an
energy density $\Omega_{\phi}\simeq 0.04$ in the form of quintessence [161].
then the Friedmann equations are Equations (3.9) and (3.10), which are the
same as for the quintessence. However, the evolution of the dust and the
scalar field obeys different equations

$\displaystyle\dot{\rho}_{\mathrm{d}}+3H\rho_{\mathrm{d}}=\alpha\rho_{\mathrm{m}}\dot{\phi},$
(3.35q)
$\displaystyle\dot{\rho}_{\phi}+3H(\rho_{\phi}+p_{\phi})=-\alpha\rho_{\mathrm{m}}\dot{\phi},$
(3.35r)

where $\rho_{\phi}$ and $p_{\phi}$ have already been shown in Equations (3.11)
and (3.12).

Introducing the same expansion normalised variables $x_{\mathrm{K}}$,
$x_{\mathrm{V}}$ and $x_{\mathrm{r}}$ defined in Equation (3.29), we have an
altered differential equation

$\frac{{\rmd}x_{\mathrm{K}}}{\rmd
N}=-3x_{\mathrm{K}}+\frac{\sqrt{6}}{2}\lambda
x_{\mathrm{V}}^{2}-x_{\mathrm{K}}\frac{1}{H}\frac{\rmd H}{\rmd
N}-\frac{\sqrt{6}}{2}\alpha(1-x_{\mathrm{K}}^{2}-x_{\mathrm{V}}^{2}-x^{2}_{\mathrm{r}})$
(3.35s)

for $x_{\mathrm{K}}$, while the equations for $x_{\mathrm{V}}$ and
$x_{\mathrm{r}}$ do not change, so as the EOS parameters and density
parameters defined in Section 3.1.3. Except for the fixed points in Table 1,
additional ones of the coupled quintessence are listed in Table 2, and Figure
6 shows fixed points in the phase space with $\lambda=1$ and different
coupling constant $\alpha$.

Table 2: Properties of additional fixed points for the coupled quintessence with a constant $\alpha$ and potential (3.23) [155, 87]. Fixed point | $(x_{\mathrm{K}},x_{\mathrm{V}},x_{\mathrm{r}})$ | $\Omega_{\mathrm{r}}$ | $\Omega_{\mathrm{d}}$ | $\Omega_{\phi}$ | $w_{\phi}$ | $w_{\mathrm{eff}}$
---|---|---|---|---|---|---
$\mathrm{R}_{\mathrm{ex}}$ | $\Big{(}-\frac{1}{\sqrt{6}\alpha},0,\frac{\sqrt{2\alpha^{2}-1}}{\sqrt{2}\alpha}\Big{)}$ | $1-\frac{1}{2\alpha^{2}}$ | $\frac{1}{3\alpha^{2}}$ | $\frac{1}{6\alpha^{2}}$ | $1$ | $\frac{1}{3}$
$\mathrm{D}^{\prime}$ | $\Big{(}-\frac{\sqrt{6}\alpha}{3},0,0\Big{)}$ | $0$ | $1-\frac{2\alpha^{2}}{3}$ | $\frac{2\alpha^{2}}{3}$ | $1$ | $\frac{2\alpha^{2}}{3}$
$\mathrm{D}_{\mathrm{sc}}^{\prime}$ | $\Big{(}\frac{\sqrt{6}}{2(\alpha+\lambda)},\frac{\sqrt{2\alpha^{2}+2\alpha\lambda+3}}{2(\alpha+\lambda)},0\Big{)}$ | $0$ | $\frac{\lambda^{2}+\alpha\lambda-3}{(\alpha+\lambda)^{2}}$ | $\frac{\alpha^{2}+\alpha\lambda+3}{(q+\lambda)^{2}}$ | $-\frac{\alpha(\alpha+\lambda)}{\alpha(\alpha+\lambda)+3}$ | $-\frac{\alpha}{\alpha+\lambda}$

Figure 6: The phase space of $x_{\mathrm{K}}$ and $x_{\mathrm{V}}$ with
$\alpha=0$ (the upper subfigure), $\alpha=1$ (the middle subfigure), and
$\alpha=2$ (the bottom subfigure). Here we assume $x_{\mathrm{r}}=0$ and
$\lambda=1$. The fixed points of coupled quintessence are plotted, with
yellow, magenta, and black representing stable, saddle, and unstable fixed
points, respectively. Each streamline represents the evolution of the Universe
under different initial conditions. The red region identifies accelerated
expansion. Each point on the curve $x_{\mathrm{K}}^{2}+x_{\mathrm{V}}^{2}=1$
corresponds to $\Omega_{\mathrm{q}}=1$. Since the future attractor
$\mathrm{D}_{\mathrm{sc}}^{\prime}$ in the bottom subfigure lies in the red
region but not on the semi-circle, it offers a solution to the coincidence
problem.

In the coupled quintessence model, fixed point $\mathrm{D}^{\prime}$
($\mathrm{D}_{\mathrm{sc}}^{\prime}$) replaces D ($\mathrm{D}_{\mathrm{sc}}$)
in the uncoupled case, and $\mathrm{D}^{\prime}$
($\mathrm{D}_{\mathrm{sc}}^{\prime}$) returns to D
($\mathrm{D}_{\mathrm{sc}}$) when the coupling $\alpha$ vanishes. Notice that
both $\mathrm{D}^{\prime}$ and $\mathrm{D}_{\mathrm{sc}}^{\prime}$ are scaling
solutions, but $\mathrm{D}^{\prime}$ returns to a domination solution while
$\mathrm{D}_{\mathrm{sc}}^{\prime}$ returns to a scaling solution as
$\alpha\to 0$. The fixed point $\mathrm{R}_{\mathrm{ex}}$ represents an extra
phase of the radiation domination.

Two fixed points, Q and $\mathrm{D}_{\mathrm{sc}}^{\prime}$, are possible to
represent the present accelerating phase of the Universe, giving two feasible
evolutions. The first is the sequence
$\mathrm{R}\to\mathrm{D}^{\prime}\to\mathrm{D}_{\mathrm{sc}}^{\prime}$, which
is able to give rise to a global attractor with $\Omega_{d}\sim 0.3$ and
$\Omega_{q}\sim 0.7$, so that it can be used for alleviating the coincidence
problem. However, the coupled quintessence with an exponential potential does
not allow for such a cosmological evolution, because the condition
$\alpha^{2}\ll 1$ is required to have point $\mathrm{D}^{\prime}$ compatible
with observations whereas large values of $|\alpha|$ are needed to get the
late-time cosmic acceleration [155]. A solution to this problem is to consider
a step-like function of the coupling $\alpha$ [162].

The other evolution route is $\mathrm{R}\to\mathrm{D}^{\prime}\to\mathrm{Q}$,
where the phase that is intermediate between the radiation dominated epoch and
the accelerated epoch is $\mathrm{D}^{\prime}$, a saddle point in the phase
space which replaces point D in the uncoupled model. The presence of the
$\phi$-matter domination era changes the background expansion history of the
Universe as $a\propto t^{2/(3+2\alpha^{2})}$. Therefore, one gets a smaller
sound horizon at the decoupling epoch, as well as a larger growth rate of
matter perturbations, relative to the uncoupled quintessence. According to
them, the CMB data and the Lyman-$\alpha$ power spectra would put an upper
bound $\alpha\sim 0.1$ on the coupling strength [163, 164].

####  4.2.2 Chameleon Mechanism.

Although the general couplings between background fluid and scalar field are
not necessarily universal, the preferential coupling of dark energy to dark
matter looks like a “conceptual fine-tuning”. If the scalar field couples to
baryons, unless the coupling is weak enough, a long-range fifth force should
be observed. Nevertheless, the experimental bounds on the coupling would not
effectively apply on the large cosmological scale if some kind of screening
mechanism works.

The chameleon mechanism [165, 166] effectively shields the fifth force by
mediating the dynamics of the scalar field with the matter density of the
environment. Consequently, the behavior of the scalar field varies in
different environments, analogous to a chameleon adjusting its color in
response to various surroundings.

Considering the chameleon mechanism, the motion equation of the scalar field
(3.35j) is rewritten as

$\square\phi=\frac{\rmd
V}{\rmd\phi}-\alpha\rme^{\alpha\phi}\hat{\rho}_{\mathrm{d}},$ (3.35t)

where $\hat{\rho}_{\mathrm{d}}\equiv\rme^{-\alpha\phi}\rho_{\mathrm{d}}$,
satisfying

$\dot{\hat{\rho}}_{\mathrm{d}}+3H\hat{\rho}_{\mathrm{d}}=0,$ (3.35u)

thus $\hat{\rho}_{\mathrm{d}}$ is conserved in the Einstein frame. A non-
trivial assumption of the chameleon mechanism is that the matter density
considered in the Klein-Gordon equation of motion is the conserved
$\hat{\rho}_{\mathrm{d}}$ in the Einstein frame, which is independent of
$\phi$. According to Equation (3.35t), the dynamics of the chameleon is not
governed by $V(\phi)$ alone, but by the effective potential,

$V_{\mathrm{eff}}(\phi,\hat{\rho}_{\mathrm{d}})=V(\phi)+\hat{\rho}_{\mathrm{d}}\rme^{\alpha\phi},$
(3.35v)

which is a function also depends on the dust density of the environment.

The value of $\phi$ at the minimal of the potential, $\phi_{\mathrm{min}}$,
and the mass squared of small fluctuations around the minimum,
$m_{\mathrm{eff}}^{2}=\partial_{\phi}^{2}V_{\mathrm{eff}}(\phi_{\mathrm{min}})$
depend on the environment as well, provided such a minimum exists. The denser
the environment, the more massive the chameleon is. The effective mass, in
turn, determines the reach of the Yukawa-type potential for the interaction,
in a form $\propto\rme^{-m_{\mathrm{eff}}r}/r$. The larger the effective mass
is, the weaker the fifth-force associated with the chameleon field is, since
it results in a faster decay of the interaction with the distance. For a
massive object, the fifth force from deep within to the exterior profile is
Yukawa-suppressed. Consequently, only the contribution from within a thin
shell beneath the surface significantly affects the exterior profile, a
phenomenon known as the thin-shell effect. Since the chameleon effectively
couples only to the shell, whereas gravity of course couples to the entire
bulk of the object, the chameleon force on an exterior test mass is suppressed
compared to the gravitational force. To satisfy solar system tests, the Milky
Way galaxy must be screened, which gives the condition [166]

$\ln\frac{A(\phi_{0})}{A(\phi_{\mathrm{MW}})}\lesssim 10^{-6}.$ (3.35w)

Here, $\phi_{0}$ represents the cosmic background scalar field today, while
$\phi_{\mathrm{MW}}$ denotes the ambient field value at the center of the
Milky Way, and $A(\phi)$ is the function in the conformal transformation
(3.35c).

The prototypical chameleon potential considers the Ratra-Peebles potential
(3.28), thereby the effective potential is given by, up to an irrelevant
constant,

$V_{\mathrm{eff}}(\phi)=\frac{M^{4+n}}{\phi^{4}}+\alpha\hat{\rho}_{\mathrm{d}}\phi,$
(3.35x)

where uses the fact that $\phi\ll 1$ over the relevant field range. For a
positive coupling $\alpha$, it displays a minimum at
$\phi_{\mathrm{min}}\propto\hat{\rho}_{\mathrm{d}}^{-1/(n+1)}$ and an
effective mass $m_{\mathrm{eff}}\propto\hat{\rho}_{\mathrm{d}}^{(n+2)/(n+1)}$
that increases when the ambient matter gets denser.

The most stringent constraint on this model arises from laboratory tests of
the inverse-square law, which impose an upper limit of approximately 50
$\mathrm{\mu m}$ on the range of the fifth force, assuming gravitational
strength coupling [167]. The constraints on deviations from general
relativity, including effective principles as well as post-Newtonian tests in
the solar system and observations of binary pulsars, translate to an upper
limit on $M$ of around $10^{-3}\;\mathrm{eV}$, which coincides with the dark
energy scale [165, 168].

An intriguing observation is that the vacuum expectation value of the
chameleon field increases in high-density environments. In other words, the
effective cosmological constant also increases, consequently leading to a
higher local Hubble expansion rate. This suggests the chameleon dark energy
model could potentially serve as a model to elucidate the Hubble tension (and
even the $S_{8}$ tension) [169].

###  4.3 Jordan Frame: Extended Quintessence

Quintessence modeled by a non-minimally coupled scalar field is called the
extended quintessence. The action for the extended quintessence is given by
Perrotta et al. [170],

$S[g^{\mu\nu},\varphi,\Psi_{\mathrm{m}}]=\int\rmd^{4}x\sqrt{-g}\left[\frac{1}{2}f(\varphi,R)-\frac{1}{2}\zeta(\varphi)g^{\mu\nu}\partial_{\mu}\varphi\partial_{\nu}\varphi\right]+S_{\mathrm{m}}[g^{\mu\nu},\Psi_{\mathrm{m}}],$
(3.35y)

where $f$ is a general function of the scalar field $\varphi$ and the Ricci
scalar $R$, and $\zeta$ is a function of $\varphi$ only. The action (3.35y)
includes a wide variety of theories such as the $f(R)$ gravity [141, 171],
where

$f(\varphi,R)=f(R),\quad\zeta(\varphi)=0,$ (3.35z)

the BD gravity, where

$f(\varphi,R)=\varphi
R,\quad\zeta(\phi)=\frac{\omega_{\mathrm{BD}}}{\varphi},$ (3.35aa)

and the dilaton gravity [172], where

$f(\varphi,R)=2\rme^{-\varphi}R-2U(\varphi),\quad\zeta(\varphi)=-2\rme^{-\varphi}.$
(3.35ab)

Here we focus on particular cases when $f(\varphi,R)=F(\varphi)R-2U(\varphi)$,
and the action becomes

$S[g^{\mu\nu},\varphi,\Psi_{\mathrm{m}}]=\int\rmd^{4}x\sqrt{-g}\left[\frac{1}{2}F(\varphi)R-\frac{1}{2}\zeta(\varphi)g^{\mu\nu}\partial_{\mu}\varphi\partial_{\nu}\varphi-U(\varphi)\right]+S_{\mathrm{m}}[g^{\mu\nu},\Psi_{\mathrm{m}}].$
(3.35ac)

The related equations of motion are

$\displaystyle
FG_{\mu\nu}=T_{\mu\nu}^{\mathrm{m}}+\zeta\partial_{\mu}\varphi\partial_{\nu}\varphi-\frac{\zeta}{2}g_{\mu\nu}g^{\alpha\beta}\partial_{\alpha}\varphi\partial_{\beta}\varphi-
g_{\mu\nu}U+(\nabla_{\mu}\nabla_{\nu}-g_{\mu\nu}\square)F,$ (3.35ad)
$\displaystyle\zeta\square\varphi+\partial_{\varphi}\zeta\cdot
g^{\mu\nu}\partial_{\mu}\varphi\partial_{\nu}\varphi+\frac{\partial_{\varphi}F}{2}R=\partial_{\varphi}U.$
(3.35ae)

####  4.3.1 Non-minimal Coupling Theory.

In order to illustrate the dynamics of the extended quintessence scenario, we
consider the non-minimal coupling theroy,

$F(\varphi)=1-\xi\varphi^{2},\quad\zeta(\varphi)=1,$ (3.35af)

where $\xi$ is the non-minimal coupling constant. Special values of the
constant $\xi$ have received particular attention in literature. For example,
$\xi=1/6$ corresponds to the conformal coupling, because the Klein-Gordon
equation (3.35ae) and the physics of $\varphi$ are conformally invariant if
$U=0$ or $U\propto\varphi^{4}$ [173]. Besides that, $\xi=0$ and $|\xi|\gg 1$
are the minimal coupling and strong coupling, respectively.

One defines the energy-momentum tensor of the scalar field as333There are
different possible definitions of the effective energy-momentum tensor for the
scalar field in scalar-tensor theories, and the effective energy density,
pressure, and EOS are unavoidably linked to one of these definitions. See Ref.
[174] for non-minimal coupling theory and Ref. [175] for the BD theory. [176]

$T_{\mu\nu}^{\varphi}=\partial_{\mu}\varphi\partial_{\nu}\varphi-
g_{\mu\nu}\left[\frac{1}{2}g^{\alpha\beta}\partial_{\alpha}\varphi\partial_{\beta}\varphi+U(\varphi)\right]+\xi(g_{\mu\nu}\square-\nabla_{\mu}\nabla_{\nu})(\varphi^{2})+\xi\varphi^{2}G_{\mu\nu}.$
(3.35ag)

The field equation of the tensor field becomes

$G_{\mu\nu}=T_{\mu\nu}^{\mathrm{m}}+T_{\mu\nu}^{\varphi}.$ (3.35ah)

Then the cosmological dynamics is governed by

$\displaystyle 3H^{2}=\rho_{\mathrm{m}}+\rho_{\varphi},$ (3.35ai)
$\displaystyle-2\dot{H}=\rho_{\mathrm{m}}+\rho_{\varphi}+p_{m}+p_{\varphi},$
(3.35aj)

where

$\displaystyle\rho_{\varphi}=\frac{1}{2}\dot{\varphi}^{2}+V+3H\xi\varphi(2\dot{\varphi}+H\varphi),$
(3.35ak) $\displaystyle
p_{\varphi}=\frac{1}{2}\dot{\varphi}^{2}-V-\xi\Big{[}(2\dot{H}+3H^{2})\varphi^{2}+4H\varphi\dot{\varphi}+2\varphi\ddot{\varphi}+2\dot{\varphi}^{2}\Big{]}.$
(3.35al)

The equation of motion for the scalar field in a flat FLRW background is given
by

$\ddot{\varphi}+3H\dot{\varphi}+\frac{\rmd U_{\mathrm{eff}}}{\rmd\varphi}=0,$
(3.35am)

where

$U_{\mathrm{eff}}=U+\frac{\xi}{2}R\varphi^{2}\,,$ (3.35an)

and $R=6(2H^{2}+\dot{H})$ is the Ricci curvature scalar. At a sufficiently
early time when the curvature is high enough, the non-minimal coupling term
dominates over the self-interaction potential $U$. Then the field $\varphi$
settles down to a slow-roll regime where the friction term, $3H\dot{\varphi}$,
balances the term $\xi R\varphi$. After that $\varphi$ starts to roll fast,
then the coupling can be ignored and the field behaves as a minimally coupled
field.

To have an accelerated expansion with $0<\xi<1/6$, one needs a potential
$U(\varphi)$ that does not grow with $\varphi$ faster than the function

$C(\varphi)\equiv\bar{U}\varphi^{k}\exp\left(\frac{\varphi^{2}}{6}\right),$
(3.35ao)

where $\bar{U}$ is a constant, and $k\approx 0.26/\xi$ by assuming
$\Omega_{m}\simeq 0.3$ and $\Omega_{\varphi}\simeq 0.7$ today [29]. If instead
$\xi<0$, then $U(\varphi)$ must grow faster than $C(\varphi)$. In this case,
$k$ is negative and, upon a rescaling of $\varphi$, the function (3.35ao)
reduces to the supergravity potential [177].

When the Ratra-Peebles potential is used with $\xi\neq 0$, the tracking
solution [see Equation (3.35ao)]

$w_{\varphi}=\frac{nw_{\mathrm{m}}-2}{n+2}$ (3.35ap)

is still valid, where $n$ is the parameter in the Ratra-Peebles potential
(3.28). This shows that the scaling solution does not depend on the coupling
$\xi$, and the solution is always stable since the value of $\xi$ only
determines the nature of the stable point [176]. This relation (3.35ap)
generalizes the one found for minimally coupled scalar fields [69, 74, 178].

The value of $\xi$ however, cannot be arbitrary because the scalar field is
ultra-light. It mediates a long-range force that is constrained by Solar
system experiments. From effects induced on photon trajectories [25] and
Equation (3.35av), one has

$\omega_{\mathrm{BD}}=\frac{F_{0}}{F_{0}^{\prime 2}}>4\times 10^{4},$ (3.35aq)

which gives

$|\xi|<\frac{4.3\times 10^{-3}}{\sqrt{n(n+2)}}.$ (3.35ar)

Here we write the abstract function $F(\varphi)$ instead of a specific shape
because the restrictions (3.35aq), as well as Equation (3.35as) below, are
generally valid for any form of $F$. Also, a constraint from the time
variation of the gravitational constant [179]

$\left.\frac{\dot{G}_{\mathrm{eff}}}{G_{\mathrm{eff}}}\right|_{0}=\frac{\dot{F}_{0}}{F_{0}}\leq
10^{-11}\;{\rm yr}^{-1}$ (3.35as)

yields a constraint on $\xi$ [180]

$-10^{-2}\lesssim\xi\lesssim 10^{-1}$ (3.35at)

for the Ratra-Peebles potential.

The scalar field, as an additional source of fluctuations, can cause new and
observable effects in the CMB and in the formation of LSSs [181]. In addition,
the time variation of the potential $U_{\mathrm{eff}}$ between the last
scattering surface and the present time would enhance the integrated Sachs-
Wolfe effect [182]. The non-minimal coupling in extended quintessence models
also modifies the positions of the acoustic peak multipoles, which can be
$\sim 10\%$ to $\sim 30\%$ with respect to the standard quintessence [170].

####  4.3.2 The Brans-Dicke theory.

Before going through the general scalar-tensor theory, it is helpful to
analyze a simple case—the Brans-Dicke theory—to acquire an advance impression
of how non-minimally coupling diverges from the minimal approach in cosmic
evolution. Usually, to meet the constraint (3.35aq), a non-self-interacting BD
scalar cannot be in the form of quintessence [29], and one needs a BD theory
with a potential,

$S[g^{\mu\nu},\Phi,\Psi_{\mathrm{m}}]=\int\rmd^{4}x\sqrt{-g}\left[\frac{\Phi}{2}R-\frac{\omega_{\mathrm{BD}}}{2\Phi}g^{\mu\nu}\partial_{\mu}\Phi\partial_{\nu}\Phi-U(\Phi)\right]+\int\rmd^{4}x\sqrt{-g}\mathcal{L}_{\mathrm{m}}.$
(3.35au)

Notice that the action (3.35au) is equivalent to the action of a non-minimal
coupling theory by applying a redefinition of the scalar field $\Phi\to
F(\varphi)$ that satisfies

$F\left(\frac{\rmd F}{\rmd\varphi}\right)^{-2}={\omega_{\mathrm{BD}}}.$
(3.35av)

To avoid the obvious fraction term in the action, we need the so-called string
frame representation [183] of a dilatonic BD theory by rescaling the scalar
field $\Phi=\rme^{\psi}$, into

$S[g^{\mu\nu},\psi,\Psi_{\mathrm{m}}]=\int\rmd^{4}x\sqrt{-g}\rme^{\psi}\left[\frac{R}{2}-\frac{\omega_{\mathrm{BD}}}{2}g^{\mu\nu}\partial_{\mu}\psi\partial_{\nu}\psi-V(\psi)+\rme^{-\psi}\mathcal{L}_{\mathrm{m}}\right],$
(3.35aw)

where $V(\psi)=U(\Phi)/\Phi$. Choosing

$y_{\mathrm{K}}\equiv\frac{\dot{\psi}}{\sqrt{6}H},\quad
x_{\mathrm{V}}\equiv\frac{\sqrt{V}}{\sqrt{3}H},$ (3.35ax)

and $\lambda$ [defined in Equation (3.35ak)] as variables of the phase space
in the autonomous system, the density parameters are defined as follows,

$\displaystyle\Omega_{\mathrm{K}}^{\mathrm{eff}}\equiv
w_{\mathrm{BD}}y_{\mathrm{K}}^{2}-\sqrt{6}y_{\mathrm{K}},$ (3.35ay)
$\displaystyle\Omega_{\mathrm{V}}\equiv x_{\mathrm{V}}^{2},$ (3.35az)
$\displaystyle\Omega_{\mathrm{m}}^{\mathrm{eff}}\equiv\frac{\rme^{-\psi}\rho_{\mathrm{m}}}{3H^{2}}.$
(3.35ba)

From the mathematical expression, the effective kinetic energy density,
$\Omega^{\mathrm{eff}}_{\mathrm{K}}$—unlike $\Omega_{\mathrm{K}}$ in the
quintessence model—has not to be positive, and it is rational in physics
because the scalar field is a part of gravity in a modified theory rather than
a part of cosmic contents.

To investigate the asymptotic dynamics of the BD cosmological model, it is
useful to consider the vacuum cosmology with
$\Omega_{\mathrm{m}}^{\mathrm{eff}}=0$. The autonomous ordinary differential
equations of BD theory are derived from the equations of motion [184],

$\displaystyle\frac{\rmd y_{\mathrm{K}}}{\rmd
N}=(\omega_{\mathrm{BD}}y_{\mathrm{K}}^{2}-\sqrt{6}y_{\mathrm{K}}-1)\left[3y_{\mathrm{K}}-\frac{(1+\lambda)(3y_{\mathrm{K}}+\sqrt{6})}{2\omega_{\mathrm{BD}}+3}\right],$
(3.35bba) $\displaystyle\frac{\rmd\lambda}{\rmd
N}=-\sqrt{6}\lambda^{2}(\Gamma-1)y_{\mathrm{K}},$ (3.35bbb)

where $N=\ln a$ is the $\mathrm{e}$-folding number and $\Gamma$—assumed to be
written as a function of $\lambda$—is defined in Equation (3.35am). We only
consider a non-negative potential $V(\psi)$, i.e. $\Omega_{\mathrm{V}}\geq 0$,
which restricts the range of $y_{\mathrm{K}}$ in the phase space,

$y_{-}\leq y_{\mathrm{K}}\leq y_{+},\quad
y_{\pm}=\frac{\sqrt{3}\pm\sqrt{2\omega_{\mathrm{BD}}+3}}{\sqrt{2}\omega_{\mathrm{BD}}}.$
(3.35bbbc)

The bounds of $\lambda$, if any, are set by the concrete form of the
potential.

Table 3: Properties of fixed points for the vacuum BD Universe [31].

  * Fixed point | $y_{\mathrm{K}}$ | $\lambda$ | $\Omega_{\mathrm{V}}$ | $q$ | $\Gamma$
---|---|---|---|---|---
GR-de Sitter | $0$ | $-1$ | $1$ | $-1$ | $1$
BD-de Sitter | $\frac{1}{\sqrt{6}(1+\omega_{\mathrm{BD}})}$ | $0$ | $\frac{12+17\omega_{\mathrm{BD}}+6\omega_{\mathrm{BD}}^{2}}{6(1+\omega_{\mathrm{BD}})^{2}}$ | $-1$ | —
Stiff-dilaton | $y_{\pm}$ | $0$ | $0$ | $2+\sqrt{6}y_{\pm}$ | —

The system has four fixed points listed in Table 3. The parameter $q$ in the
table is the deceleration parameter. For these solutions in the table, the GR-
de Sitter solution is a potential-dominated solution, which corresponds to the
cosmological constant domination in general relativity. In contrary, the BD-de
Sitter phase, which does not arise in the minimally coupled model, is a
scaling solution, and it arises even if the matter exists. The last two stiff-
dilaton solutions are dominated by the effective kinetic term.

Further investigation shows that this GR-de Sitter solution is stable.
However, it has been shown by Garcia-Salcedo et al. [184] that the BD
cosmology does not have the $\Lambda$CDM phase as a universal attractor unless
the given potential approaches to the exponential form $V\propto\rme^{\psi}$
as an asymptote, since according to Table 3, the GR-de Sitter solution
corresponds to the exponential potential $V(\psi)\propto\rme^{\psi}$, which
amounts to the quadratic potential $U(\Phi)\propto\Phi^{2}$ in terms of the
standard BD field $\Phi$ in action (3.35au). It has also been shown that very
specific conditions on the coupling function $W(\Phi)$ are to be imposed for
the given scalar-tensor gravity to have the GR-de Sitter limit [185, 186].

####  4.3.3 Dynamics of the Extended Quintessence.

Now let us concentrate on the general scalar-tensor theory. To analyze the
dynamics, it is convenient to use the scalar $\phi$ in the Einstein frame
instead of the original $\varphi$. The suitable conformal transformation
described by the parameter

$\alpha\equiv-\frac{\partial_{\phi}F}{2F}=-\frac{\partial_{\varphi}F}{2F}\left[\frac{3}{2}\left(\frac{\partial_{\varphi}F}{F}\right)^{2}+\frac{\zeta}{F}\right]^{-1/2},$
(3.35bbbd)

takes the action (3.35ac) into

$\displaystyle S[g^{\mu\nu},\phi,\Psi_{\mathrm{m}}]=$
$\displaystyle\int\rmd^{4}x\sqrt{-g}\left[\frac{1}{2}F(\phi)R-\frac{1}{2}(1-6\alpha^{2})F(\phi)g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-U(\phi)\right]$
(3.35bbbe) $\displaystyle+S_{\mathrm{m}}[g^{\mu\nu},\Psi_{\mathrm{m}}].$

The rescaled scalar field of the extended quintessence in the Einstein frame
is

$\phi=\int\rmd\varphi\sqrt{\frac{3}{2}\left(\frac{\partial_{\varphi}F}{F}\right)^{2}+\frac{\zeta}{F}}.$
(3.35bbbf)

In the uncoupled limit $\alpha\equiv\rmd\ln A/\rmd\phi\to 0$, the action
(3.35bbbe) reduces to the action of quintessence model.

For simplicity, we treat $\alpha$ as a constant from now on,444Usually,
$\alpha$ is not a constant. For instance, the coupling $\alpha$ is field-
dependent in the non-minimal coupling theory,
$\alpha(\varphi)=\frac{\xi\varphi}{\sqrt{1-\xi(1-6\xi)\varphi^{2}}}.$
(3.35bbbg) Here, $\alpha=\xi\varphi$ for $|\xi|\ll 1$ and $\alpha=\pm
1/\sqrt{6}$ in the strong coupling limit $|\xi|\gg 1$. Nevertheless, action
(3.35bbbe) with a constant $\alpha$ is able to represent some meaningful
theories. The BD theory with a potential in Equation (3.35au) is equivalent to
the case in which the parameter $\omega_{\mathrm{BD}}$ is related to $\alpha$
via the relation  $2\omega_{\mathrm{BD}}+3=\frac{1}{2\alpha^{2}}.$ (3.35bbbh)
In the general relativistic limit $\alpha\to 0$, we have
$\omega_{\mathrm{BD}}\to\infty$ as expected. In addition, $f(R)$ theory is
acquired in the metric (Palatini) formalism when $\alpha=-1/\sqrt{6}$
($\alpha^{2}\to\infty$), and the dilaton gravity (3.35ab) is recovered when
$\alpha=1/\sqrt{2}$. leading to

$F(\phi)=\rme^{-2\alpha\phi}.$ (3.35bbbi)

In a flat FLRW background, the variation of action (3.35bbbe) with respect to
the metric $g_{\mu\nu}$ and the scalar field $\phi$ leads to

$\displaystyle
3FH^{2}=\frac{1}{2}\left(1-6\alpha^{2}\right)F\dot{\phi}^{2}+U-3H\dot{F}+\rho_{\mathrm{d}}+\rho_{\mathrm{r}},$
(3.35bbbj) $\displaystyle
2F\dot{H}=-\left(1-6\alpha^{2}\right)F\dot{\phi}^{2}-\ddot{F}+H\dot{F}-\rho_{\mathrm{d}}-\frac{4}{3}\rho_{\mathrm{r}},$
(3.35bbbk)
$\displaystyle(1-6\alpha^{2})F\left(\ddot{\phi}+3H\dot{\phi}+\frac{\dot{F}}{2F}\dot{\phi}\right)+\frac{\rmd
U}{\rmd\phi}+\alpha FR=0,$ (3.35bbbl)

where the overdot denotes the derivative of the cosmic time $t$, and
$\rho_{\mathrm{d}}$ ($\rho_{\mathrm{r}}$) is the energy density of dust
(radiation). We introduce the following variables of the autonomous system

$x_{\mathrm{K}}\equiv\frac{\dot{\phi}}{\sqrt{6}H},\quad
y_{\mathrm{V}}\equiv\frac{1}{H}\sqrt{\frac{U}{3F}},\quad
y_{\mathrm{r}}\equiv\frac{1}{H}\sqrt{\frac{\rho_{\mathrm{r}}}{3F}}$ (3.35bbbm)

and density parameters for the scalar field, non-relativistic matter, and
radiation

$\Omega_{\phi}\equiv(1-6\alpha^{2})x_{\mathrm{K}}^{2}+y_{\mathrm{V}}^{2}+2\sqrt{6}\alpha
x_{\mathrm{K}},\quad\Omega_{\mathrm{d}}\equiv\frac{\rho_{\mathrm{d}}}{3FH^{2}},\quad\Omega_{\mathrm{r}}\equiv
y_{\mathrm{r}}^{2}.$ (3.35bbbn)

A similar relation

$\Omega_{\phi}+\Omega_{\mathrm{d}}+\Omega_{\mathrm{r}}=1$ (3.35bbbo)

comes from Equation (3.35bbbj). The deceleration parameter $q$ and the
effective EOS parameter of the Universe are

$q=-1-\frac{\dot{H}}{H^{2}},\quad
w_{\mathrm{eff}}=-1-\frac{2}{3}\frac{\dot{H}}{H^{2}},$ (3.35bbbp)

where

$\frac{\dot{H}}{H^{2}}=-\frac{1-6\alpha^{2}}{2}(3+3x_{\mathrm{K}}^{2}-3y_{\mathrm{V}}^{2}+y_{\mathrm{r}}^{2}-6\alpha^{2}x_{\mathrm{K}}^{2}+2\sqrt{6}\alpha
x_{\mathrm{K}})+3\alpha(\lambda y_{\mathrm{V}}^{2}-4\alpha).$ (3.35bbbq)

Using Equations (3.35bbbj) to (3.35bbbl), one obtains the differential
equations for $x_{\mathrm{K}}$, $y_{\mathrm{V}}$ and $y_{\mathrm{r}}$ [187],

$\displaystyle\frac{\rmd x_{\mathrm{K}}}{\rmd
N}=\frac{\sqrt{6}}{2}\left(\lambda
y_{\mathrm{V}}^{2}-\sqrt{6}x_{\mathrm{K}}\right)$
$\displaystyle+\frac{\sqrt{6}\alpha}{2}\left[\left(5-6\alpha^{2}\right)x_{\mathrm{K}}^{2}+2\sqrt{6}\alpha
x_{\mathrm{K}}-3y_{\mathrm{V}}^{2}+y_{\mathrm{r}}^{2}-1\right]-x_{\mathrm{K}}\frac{\dot{H}}{H^{2}},$
(3.35bbbra) $\displaystyle\frac{\rmd y_{\mathrm{V}}}{\rmd
N}=\frac{\sqrt{6}}{2}(2\alpha-\lambda)x_{\mathrm{K}}y_{\mathrm{V}}-y_{\mathrm{V}}\frac{\dot{H}}{H^{2}},$
(3.35bbbrb) $\displaystyle\frac{\rmd y_{\mathrm{r}}}{\rmd N}=\sqrt{6}\alpha
x_{\mathrm{K}}y_{\mathrm{r}}-2y_{\mathrm{r}}-y_{\mathrm{r}}\frac{\dot{H}}{H^{2}},$
(3.35bbbrc)

where $N=\ln a$ and $\lambda=-\partial_{\phi}U/U$ as above.

Table 4: Properties of fixed points for the extended quintessence with a
constant $\alpha$ and potential (3.23) in the absence of radiation [187].

  * Fixed point | $(x_{K},y_{V})$ | $\Omega_{d}$ | $w_{\mathrm{eff}}$
---|---|---|---
$\mathrm{D}^{\prime\prime}$ | $\Big{(}\frac{\sqrt{6}\alpha}{3(2\alpha^{2}-1)},0\Big{)}$ | $\frac{3-2\alpha^{2}}{3(1-2\alpha^{2})^{2}}$ | $\frac{4\alpha^{2}}{3(1-2\alpha^{2})}$
$\mathrm{D_{sc}^{\prime\prime}}$ | $\Big{(}\frac{\sqrt{6}}{2\lambda},\sqrt{\frac{3+2\alpha\lambda-6\alpha^{2}}{2\lambda^{2}}}\Big{)}$ | $1-\frac{3-12\alpha^{2}+7\alpha\lambda}{\lambda^{2}}$ | $-\frac{2\alpha}{\lambda}$
$\mathrm{K}_{\pm}^{\prime}$ | $\Big{(}\frac{1}{\sqrt{6}(\alpha+1)},0\Big{)}$ | $0$ | $\frac{3\mp\sqrt{6}\alpha}{3(1\pm\sqrt{6}\alpha)}$
$\mathrm{Q}^{\prime}$ | $\Big{(}\frac{\sqrt{6}(4\alpha-\lambda)}{6(4\alpha^{2}-\alpha\lambda-1)},\sqrt{\frac{6-\lambda^{2}+8\alpha\lambda-16\alpha^{2}}{6(4\alpha^{2}-\alpha\lambda-1)^{2}}}\Big{)}$ | $0$ | $-\frac{20\alpha^{2}-9\alpha\lambda-3+\lambda^{2}}{3(4\alpha^{2}-\alpha\lambda-1)}$
$\mathrm{Q_{dS}}$ | $(0,1)$ | $0$ | $-1$

In the absence of radiation ($y_{\mathrm{r}}=0$), the fixed points of the
system for a constant $\lambda$ are listed in Table 4. The dust-dominated
epoch can be realized either by the point $\mathrm{D}^{\prime\prime}$ or by
the point $\mathrm{D_{sc}^{\prime\prime}}$. If the point
$\mathrm{D}^{\prime\prime}$ is responsible for the matter domination, the
condition $\alpha^{2}\ll 1$ is required, leading to $\Omega_{\mathrm{m}}\simeq
1+10\alpha^{2}/3>0$ and $w_{\mathrm{eff}}\simeq{4\alpha^{2}}/{3}$. When
$\alpha^{2}\ll 1$, the scalar-field dominated point $\mathrm{Q}^{\prime}$
yields an accelerated expansion of the Universe provided that
$-\sqrt{2}+4\alpha<\lambda<\sqrt{2}+4\alpha$ [187]. The scaling solution
$\mathrm{D_{sc}^{\prime\prime}}$ can give rise to the EOS,
$w_{\mathrm{eff}}\simeq 0$ for $|\alpha|\ll|\lambda|$. In this case, however,
the condition $w_{\mathrm{eff}}<-1/3$ for the point $\mathrm{Q}^{\prime}$
gives $\lambda^{2}<2$. Then the energy fraction of the matter for the point
$\mathrm{D_{sc}^{\prime\prime}}$ does not satisfy the condition
$\Omega_{\mathrm{m}}\simeq 1$. As a result, the only way to describe the
evolution from matter domination to scalar-field domination in the phase
diagram is the trace from point $\mathrm{D}^{\prime\prime}$ to point
$\mathrm{Q}^{\prime}$. Note that fixed points in Table 4 tend to become the
corresponding fixed points in Table 1 as $\alpha\to 0$ in the limit of general
relativity.

Not surprisingly, the GR-de Sitter phase arises in the extended quintessence
model. If $\lambda=4\alpha$, the fixed point $\mathrm{Q}^{\prime}$ turns into
GR-de Sitter point $\mathrm{Q_{dS}}$, with which corresponds
$\Omega_{\mathrm{d}}=0$ and $w_{\mathrm{eff}}=-1$. Similar to the mentioned
quintessence model in Section 3.1.3, we have to consider the varying-$\lambda$
case in order to make it possible for the Universe to evolve from the scaling
solution $\mathrm{D_{sc}^{\prime\prime}}$ to the acceleration phase
$\mathrm{Q}^{\prime}$ or $\mathrm{Q_{dS}}$. One of feasible potentials has
been found by Tsujikawa et al. [187].

The dynamics of the extended quintessence system is much richer than in the
minimally coupled case [188, 189, 190, 175, 191]. One can have spontaneous
bouncing, spontaneous entry into and exit from inflation, and super-
acceleration ($\dot{H}>0$) in extended models [190]. The phantom EOS and the
boundary crossing of cosmological constants naturally arise in scalar-tensor
theories with large couplings [187]. Even in the limit $|\alpha|\ll 1$, the
phantom EOS can be realized without introducing a ghost field [192, 193, 194,
195].

####  4.3.4 Cosmological Attractor to General Relativity.

In Section 4.3.2, we demonstrate that the $\Lambda$CDM model serves as an
attractor in BD theory with certain potentials. Additionally, we highlight in
this section that even within more intricate scalar-tensor theories, general
relativity itself can emerge as an attractor.

Different conformal frames are not equivalent to each other in a cosmological
perspective. However, once the physical conformal frame is determined,
different conformal frames are equivalent mathematically. Usually, people
regard the Jordan frame as the physical one, as baryons follow the geodesic
and we can directly compare the results with observations. In this case, the
Einstein frame is a mathematical treatment that can sometimes be used to avoid
complicated calculations in the original Jordan frame, e.g. Damour’s famous
work [196, 197].

According to the transformation (3.35c), it is easy to notice that

$T^{*\mathrm{m}}_{\mu\nu}=A^{2}T^{\mathrm{m}}_{\mu\nu},\quad
T^{\mu}_{*\mathrm{m}\nu}=A^{4}T^{\mu}_{\mathrm{m}\nu},\quad
T^{\mu\nu}_{*\mathrm{m}}=A^{6}T_{\mathrm{m}}^{\mu\nu},$ (3.35bbbrbs)

and the transformation of the trace is

$T_{\mathrm{m}}^{*}=A^{4}T_{\mathrm{m}}.$ (3.35bbbrbt)

One also defines $t_{*}$ and $a_{*}$ as

$\displaystyle t_{*}\equiv\int A^{-1}(t)\rmd t,$ (3.35bbbrbu) $\displaystyle
a_{*}\equiv A^{-1}a,$ (3.35bbbrbv)

so that the line element is still in the form of the FLRW metric

$\rmd s_{*}^{2}=-\rmd t^{2}_{*}+a_{*}^{2}(t_{*})\left(\frac{\rmd
r^{2}}{1-Kr^{2}}+r^{2}\rmd\Omega^{2}_{2}\right).$ (3.35bbbrbw)

Substituting the metric (3.35bbbrbw) into equations of motion (3.35i) and
(3.35j), we obtain

$\displaystyle 3H_{*}^{2}=\kappa_{*}(\rho_{\mathrm{m}}^{*}+\rho_{\phi}),$
(3.35bbbrbx)
$\displaystyle-2\dot{H}_{*}=\kappa_{*}(\rho_{\mathrm{m}}^{*}+p_{\mathrm{m}}^{*}+\rho_{\phi}+p_{\phi}),$
(3.35bbbrby) $\displaystyle\ddot{\phi}+3H_{*}\dot{\phi}+\frac{\rmd
V}{\rmd\phi}=-\alpha(\rho_{\mathrm{m}}^{*}-3p_{\mathrm{m}}^{*}),$ (3.35bbbrbz)

where the overdot represents the derivative with respect to the cosmic time
$t_{*}$, and $H_{*}\equiv\dot{a}_{*}/a_{*}$ is the Hubble parameter in the
Einstein frame. The Bianchi identity

$\rmd(\rho_{\mathrm{m}}^{*}a_{*}^{3})+p_{\mathrm{m}}^{*}\rmd
a_{*}^{3}=(\rho^{*}_{\mathrm{m}}-3p^{*}_{\mathrm{m}})a_{*}^{3}\rmd\ln A(\phi)$
(3.35bbbrca)

tells us that the Einstein frame scaling laws are $\rho_{\mathrm{r}}\propto
a_{*}^{-4}$ for the radiation and $\rho_{\mathrm{d}}\propto A(\phi)a_{*}^{-3}$
for the dust.

Defining the $e$-folding number $N_{*}\equiv\ln a_{*}$ in the Einstein frame
and introducing parameters [198]

$w(N_{*},\phi)\equiv\frac{p_{\mathrm{m}}^{*}}{\rho_{\mathrm{m}}^{*}},\quad
v(N_{*},\phi)\equiv\frac{V}{\rho_{\mathrm{m}}^{*}},$ (3.35bbbrcb)

the equation of motion of the scalar field becomes

$\frac{4(1+v)}{6-\kappa_{*}\phi^{\prime
2}}\phi^{\prime\prime}+(1-w+2v)\phi^{\prime}+\frac{2}{\kappa_{*}}\left[(1-3w)\alpha+v\frac{\rmd\ln
V}{\rmd\phi}\right]=0,$ (3.35bbbrcc)

where the prime denotes the derivative with respect to $N_{*}$.

Each scalar-tensor theory is specified by a particular choice for
$\alpha(\phi)$ and $V(\phi)$. For example, a constant
$\alpha(\phi)=\alpha_{\mathrm{BD}}$ and $V(\phi)=0$ select the traditional BD
theory (3.35a). The mechanism of attraction towards general relativity can be
illustrated by the simplest case where
$\alpha(\phi)=\beta_{\mathrm{DEF}}\phi$, known as the Damour-Esposito-Farèse
(DEF) theory [199], where in cosmology one usually consider the case
$\beta_{\mathrm{DEF}}>0$. Choosing $\kappa_{*}=2$ and $V=0$, Equation
(3.35bbbrcc) takes the form of the equation of motion of a particle with a
velocity-dependent mass $m(\phi^{\prime})=2/(3-\phi^{\prime 2})$ in a
parabolic potential $V_{*}(\phi)=(1-3w)\beta_{\mathrm{DEF}}\phi^{2}/2$, and
subjecting to a damping force proportional to $1-w$ [196]. Notice that the
potential $V_{*}$ vanishes in the radiation-dominated epoch no matter what
value $\alpha$ is. The exact solution to the initial condition
$(\phi,\phi^{\prime})|_{N_{*}=0}=(\phi_{\rmi},\phi_{\rmi}^{\prime})$ for the
radiation epoch is [196]

$\phi(N_{*})=\phi_{\rmi}-\sqrt{3}\ln\left(\frac{K_{\rmi}\rme^{-N_{*}}+\sqrt{1+K_{\rmi}^{2}\rme^{-2N_{*}}}}{K_{\rmi}+\sqrt{1+K_{\rmi}^{2}}}\right),\quad
K_{\rmi}=\frac{\phi_{\rmi}^{\prime}}{\sqrt{3-\phi_{\rmi}^{\prime 2}}}.$
(3.35bbbrcd)

The duration of the radiation-dominated epoch $\Delta N_{*}^{\mathrm{rad}}\sim
20$ is long enough to slow down the particle even if it is ultra-relativistic
as long as $\phi^{\prime}_{i}\lesssim\sqrt{3}$ [200, 201]. Then the particle
is stationary at the beginning of the matter-dominated epoch. Hence, it is
easy to realize that at late times the field $\phi$ will settle down at the
minimum of the potential $V_{*}$ with $\phi=0$, where
$\alpha=\beta_{\mathrm{DEF}}\phi=0$ and the theory flows towards general
relativity as the Universe evolves. Figure 7 illustrates the evolution of
$\alpha$ as the Universe evolves, revealing that the coupling between matter
and scalar fields is currently negligibly small.

Figure 7: The evolution of the coupling $\alpha$ between matter and scalar
field with different $\beta_{\mathrm{DEF}}$ in DEF theory. The initial
conditions are $\phi^{\prime}=0$ and $\phi$ saturating the constraint of
speed-up factor during nucleosynthesis [202]. The present coupling
$\alpha_{0}=\alpha|_{N_{*}=0}$ is negligibly small.

Another interesting feature of the DEF theory is that when the gravitational
field gets stronger, the compact star may scalarize spontaneously and become a
scalarized star if the parameter $\beta_{\mathrm{DEF}}\lesssim-4$ [199, 34].
Spontaneous scalarization and general relativity attractor share different
parameter spaces in the DEF theory and some generalized theories [198, 200,
203]. Adding another extra coupling between the scalar field and the Gauss-
Bonnet term is a feasible solution to exhibit spontaneous scalarization in the
strong-field regime and the spacetime behaves like general relativity at
infinity as a cosmological attractor [204, 205].

In order to explain the accelerated expansion, the potential $V(\phi)$ cannot
be zero. If one is interested in a scalar field with energy density of the
same order as that of matter today, then the term $\rmd\ln V/\rmd\phi$ in
Equation (3.35bbbrcc) turns out to be subdominant with respect to
$(1-3w)\alpha$ during the radiation-domination epoch and most of the matter-
domination epoch. At late times, the scalar potential term starts to dominate
and the gravity is approximately described by general relativity as $\alpha\to
0$. By choosing the coupling function as

$\alpha(\phi)=-Be^{-\beta\phi}$ (3.35bbbrce)

and the Ratra-Peebles potential (3.28) for the scalar field, one obtains an
approximate solution during the radiation and matter domination [206]

$\phi(a_{*})\simeq\frac{1}{\beta}\ln\left[\beta
B\ln\left(\frac{2}{3}+\frac{a_{*}}{a_{*}^{\mathrm{eq}}}\right)+C\right],$
(3.35bbbrcf)

where $a_{*}^{\mathrm{eq}}$ is the value of the scale factor at the
equivalence epoch (the cosmic time when $\rho_{\mathrm{r}}=\rho_{\mathrm{d}}$)
and $C$ is a constant. This solution is an attractor in phase space and
behaves as a subdominant contribution to the energy density that starts
increasing during the matter-dominated era and eventually comes to dominate in
the present era. The Doppler peaks are shifted toward higher multipoles and
their height is changed with respect to minimally coupled models, providing a
signature to look for in future CMB experiments [206].

###  4.4 Recapitulation and Possible Extension

A general action describing various couplings to the scalar field for
different components of the Universe in the Einstein frame is given by

$S[g^{*}_{\mu\nu},\phi,\Psi_{\mathrm{m}}^{(i)}]=\int\rmd^{4}x\sqrt{-g_{*}}\left[\frac{R_{*}}{2\kappa_{*}}-\frac{1}{2}g_{*}^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-V(\phi)\right]+S_{\mathrm{m}}[g_{\mu\nu}^{(i)},\Psi_{\mathrm{m}}^{(i)}]$
(3.35bbbrcg)

where $\Psi_{m}^{(i)}$ is the matter field coupled to the Jordan frame metric
$g_{\mu\nu}^{(i)}$, related to the Einstein frame metric $g^{*}_{\mu\nu}$ via

$g_{\mu\nu}^{(i)}=A^{2}_{i}(\phi)g^{*}_{\mu\nu}.$ (3.35bbbrch)

The action (3.35bbbrcg) reduces to the general relativity if $A_{i}=1$, which
means that each component of the Universe does not couple to the metric in the
Einstein frame. In that case, metrics in the Jordan frame and the Einstein
frame are the same. The quintessence model, in the framework of general
relativity, relies on a self-interacting potential of the scalar field and a
violation of the strong energy condition to provide necessary cosmic
acceleration. During a slow-roll phase of the massive scalar field when its
kinetic energy is much less than its self-interacting one, negative pressures
are achieved, producing the desired cosmic acceleration. The potential, as the
only degree of freedom in the quintessence model, can be quite sophisticated
and has to be well shaped to reproduce the cosmic evolution or to exhibit
interesting tracking properties. The coincidence problem is somehow moved to
the specific choice of both an appropriate shape and energy scale for the
self-interaction potential. Most of the potentials used in quintessence models
come out of an effective theory of high-energy physics, leaving a glimpse into
new physics. However, deriving such quintessence models (and also the
k-essence extension models) from high-energy physics in a self-consistent
framework is a challenging task, as combining all the cosmological,
gravitational, and particle physics aspects raises many technical
difficulties.

The quintessence scalar field minimally couples to gravitation and does not
couple directly to ordinary matter. Hence, quintessence only modifies the
background cosmic expansion and all the observations of gravity theory without
the quintessence component remain the same. If matter fields couple to both
scalar field and gravitation directly, the new “fifth” interaction yields a
violation of the strong equivalence principle [207]. The non-minimal coupling
changes physical coupling constants or inertial masses and as a consequence,
modifies the way energies have been weighted in the cosmic history. In the
case where the scalar field directly affects the couplings to the
gravitational field and therefore makes the gravitational constant vary, one
gets the scalar-tensor theories of gravity,

$S[g^{*}_{\mu\nu},\phi,\Psi_{\mathrm{m}}]=\int\rmd^{4}x\sqrt{-g_{*}}\left[\frac{R_{*}}{2\kappa_{*}}-\frac{1}{2}g_{*}^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-V(\phi)\right]+S_{\mathrm{m}}[A^{2}(\phi)g_{\mu\nu}^{*},\Psi_{\mathrm{m}}].$
(3.35bbbrci)

If we consider that about 70% of the missing energy is the whole contribution
of such a non-minimally coupled scalar field, it might be difficult to match
the present tests of gravity without assuming the non-minimal couplings to be
extremely weak. In fact, the coupled quintessence model corresponds to the
case $A_{\mathrm{b}}=1$ for baryons while $A_{\rm dm}$ for dark matter varies
with $\phi$, which reveals a non-universal coupling. This non-universal
coupling solves the extremely small coupling problem, but the weak equivalent
principle is severely violated. As for the chameleon cosmology, however, both
baryons and dark matter are non-minimally coupled to the quintessence field as
$A_{\mathrm{b}}=A_{\rm dm}=e^{\alpha\phi}$, with the mass of the scalar field
being environmentally dependent. This offers the possibility of accounting for
the present bounds of local tests of the strong and weak equivalent principles
on the Earth and in the Solar system by the larger mass of the scalar field in
a denser neighborhood than in the cosmological low-density limit. In order to
do so, a specific self-interaction potential and a consequent violation of the
strong energy condition have to be re-introduced. Another alternative model
proposes that both $A_{\mathrm{b}}$ and $A_{\rm dm}$ depends on the scalar
field, but not identical [208]. The self-interacting potential $V(\phi)$ is
unnecessary in this model if two couplings are both assumed to be of order
unity, and the cosmic acceleration is a generic prediction of the competition
between different non-minimal couplings. Moreover, the attractor to which
gravitation is driven depends on the ratio between the energy densities of
ordinary matter and dark matter.

The most general 4-dimensional scalar-tensor theories having second-order
equations of motion are the Horndeski theory [209, 210]

$S_{\mathrm{H}}[g_{\mu\nu},\phi,\Psi_{\mathrm{m}}]=\sum_{i=2}^{5}\int\rmd^{4}x\sqrt{-g}\mathcal{L}_{i}+S_{\mathrm{m}}[g_{\mu\nu},\Psi_{\mathrm{m}}],$
(3.35bbbrcj)

where the Lagrangian densities are

$\displaystyle\mathcal{L}_{2}=G_{2}(\phi,X),$ (3.35bbbrcka)
$\displaystyle\mathcal{L}_{3}=G_{3}(\phi,X)\square\phi,$ (3.35bbbrckb)
$\displaystyle\mathcal{L}_{4}=G_{4}(\phi,X)R+G_{4X}\big{[}(\square\phi)^{2}-\nabla_{\mu}\nabla_{\nu}\phi\nabla^{\mu}\nabla^{\nu}\phi\big{]},$
(3.35bbbrckc)
$\displaystyle\mathcal{L}_{5}=G_{5}(\phi,X)G_{\mu\nu}\nabla^{\mu}\nabla^{\nu}\phi$
$\displaystyle-\frac{G_{5X}}{6}\Big{[}(\square\phi)^{3}-3\square\phi\nabla_{\mu}\nabla_{\nu}\phi\nabla^{\mu}\nabla^{\nu}\phi+2\nabla^{\mu}\nabla_{\alpha}\phi\nabla^{\alpha}\nabla_{\beta}\phi\nabla^{\beta}\nabla_{\mu}\phi\Big{]}.$
(3.35bbbrckd)

The theory is termed as Horndeski theory because Kobayashi et al. [211] found
that the action (3.35bbbrcj) is equivalent to the one discovered by Horndeski
in 1974 [212]. Here, $G_{i}$ ($i=2,3,4,5$) are arbitrary functions of the
scalar field $\phi$ and its kinetic energy $X$, while $G_{i\phi}$ and $G_{iX}$
represent the derivatives of functions $G_{i}$ with respect to $\phi$ and $X$,
respectively. Due to the second-order property, there is no Ostrogradski
instability [213] associated with the Hamiltonian unbounded from below.

In a flat FLRW background, the EOS parameter of the scalar field $\phi$ is

$w_{\phi}=-1+\frac{2(q_{\mathrm{t}}-M_{\mathrm{Pl}}^{2})\dot{H}-D_{6}\ddot{\phi}+D_{7}\dot{\phi}}{\rho_{\phi}},$
(3.35bbbrckcl)

where

$q_{\mathrm{t}}=2G_{4}-2\dot{\phi}^{2}G_{4X}+\dot{\phi}^{2}G_{5\phi}-H\dot{\phi}^{3}G_{5X},$
(3.35bbbrckcm)

and the definition of $D_{6,7}$ and energy density $\rho_{\phi}$ can be found
in Ref. [214]. Since the evolution of $w_{\phi}$ is different depending on
dark energy models, it is possible to distinguish them from observations, like
SN Ia, CMB, BAO, LSSs, and weak lensing.

The propagation speed squared of the tensor perturbations in the Horndeski
theories is [211]

$c_{\mathrm{t}}^{2}=\frac{1}{q_{\mathrm{t}}}\Big{(}2G_{4}-\dot{\phi}^{2}G_{5\phi}-\dot{\phi}^{2}\ddot{\phi}G_{5X}\Big{)}.$
(3.35bbbrckcn)

The detection of gravitational waves by GW170817 [215] from a binary neutron
star merger together with the short gamma-ray burst GRB 170817A [216]
constrained the propagation speed $c_{\mathrm{t}}$ of gravitational waves to
be [217]

$-3\times 10^{-15}<c_{\mathrm{t}}-1<7\times 10^{-16}.$ (3.35bbbrckco)

If any fine tuning among functions is forbidden, one has
$G_{4X}=G_{5\phi}=G_{5X}=0$, then the Horndeski theory is restricted to be of
the form [218, 219, 220, 221]

$S[g_{\mu\nu},\phi,\Psi_{\mathrm{m}}]=\int\rmd^{4}x\sqrt{-g}\big{[}G_{2}(\phi,X)+G_{3}(\phi,X)\square\phi+G_{4}(\phi)R\big{]}+S_{\mathrm{m}}[g_{\mu\nu},\Psi_{\mathrm{m}}].$
(3.35bbbrckcp)

Each dark energy model or scalar-tensor gravity mentioned above corresponds to
this reduced action (3.35bbbrckcp) with specific functions $G_{i}$
($i=1,2,3$). The k-essence model (3.35ax) is given by the choice

$G_{2}=P(\phi,X),\quad G_{3}=0,\quad G_{4}=\frac{M^{2}_{\mathrm{Pl}}}{2},$
(3.35bbbrckcq)

and the extended quintessence model (3.35ac) is acquired if

$G_{2}=\zeta(\phi)X-U(\phi),\quad G_{3}=0,\quad
G_{4}=\frac{M^{2}_{\mathrm{Pl}}}{2}F(\phi).$ (3.35bbbrckcr)

There are other modified gravity theories dubbed Galileon theories [209, 222,
210] containing scalar derivative self-interactions $G_{3}\neq 0$. In the
original Galileon theory [209], the field equations of motion are invariant
under the Galilean shift $\partial_{\mu}\phi\to\partial_{\mu}\phi+b_{\mu}$ in
the Minkowski spacetime. In a curved spacetime, the Lagrangian of covariant
Galileon theories [210] is constructed to keep the equations of motion up to