# Estimated Age of the Universe in Fractional Cosmology

Emanuel Wallison de Oliveira Costa  Raheleh Jalalzadeh  Pedro Felix da Silva
Júnior  Seyed Meraj Mousavi Rasouli  Shahram Jalalzadeh  Departamento de
Física, Universidade Federal de Pernambuco, Recife, PE 50670-901, Brazil
Department of Physics, University of Kurdistan, Pasdaran St., Sanandaj P.O.
Box 66177-15175, Iran  Departamento de Física, Centro de Matemática e
Aplicações (CMA-UBI), Universidade da Beira Interior, Rua Marquês d’Avila e
Bolama, 6200-001 Covilhã, Portugal.  Department of Physics, Qazvin Branch,
Islamic Azad University, Qazvin 341851416, Iran

###### Abstract

Our proposed cosmological framework, which is based on fractional quantum
cosmology, aims to address the issue of synchronicity in the age of the
universe. To achieve this, we have developed a new fractional $\Lambda$CDM
cosmological model. We obtained the necessary formalism by obtaining the
fractional Hamiltonian constraint in a general minisuperspace. This formalism
has allowed us to derive the fractional Friedmann and Raychaudhuri equations
for a homogeneous and isotropic cosmology. Unlike the traditional de Sitter
phase, our model exhibits a power-law accelerated expansion in the late-time
universe, when vacuum energy becomes dominant. By fitting the model’s
parameters to cosmological observations, we determined that the fractional
parameter of Lévy equals $\alpha=1.986$. Additionally, we have calculated the
age of the universe to be 13.8196 Gyr. Furthermore, we have found that the
ratio of the age to Hubble time from the present epoch to the distant future
is finite and confined within the interval $0.9858\leq Ht<95.238$.

###### keywords:

fractional cosmology , age of the universe , Hubble parameter , synchronicity
problem

##  1 Introduction

Estimates for the age of the universe have been made over time. Einstein and
de Sitter’s model [1] predicted an age of the universe that was roughly $9$
Gyr, which is quite different from the age of the ancient stars. The age of
ancient stars has been estimated to be larger than $12$ Gyr, as per the
research by Chaboyer [2]. This discrepancy led to the development of new
models that could account for the universe’s observed age and expansion rate.
Despite the universe experiencing deceleration for the first 9 billion years
and then transitioning to acceleration through cosmic jerk for the past 5
billion years [3], the present value of the dimensionless age, $H_{0}t_{0}$,
where $H_{0}$ and $t_{0}$ are the current value of the Hubble parameter and
the universe’s age, respectively, is limited by supernova distances to be very
close to 1. For example, Tonry et al. [4] estimated this to be
$H_{0}t_{0}=0.96\pm 0.04$. We seem to be experiencing a period of privilege at
the moment, wherein the age of the universe in terms of Hubble time,
especially in the case of a $\Lambda$CDM Universe, is extremely close to
unity. This coincidence is known as the _synchronicity problem_ of the
universe’s age [3, 5].

Various other methods have verified this unexpected closeness to unity, which
has been the subject of intense speculation and debate. The synchronicity
problem has been investigated by cosmological models with the inclusion of
scalar fields [6, 7], effective dependency between dark energy and dark matter
[8, 9, 10, 11, 12], interacting holographic dark energy [13, 14, 15], and
other interpretations at the level of standard cosmology [16, 17]. From
another perspective, quantum cosmology in the paradigm of the Wheeler–DeWitt
(WDW) equation in the minisuperspace of a homogeneous and isotropic universe
was used to propose a solution to the question in a unified way to other gaps
in the $\Lambda$CDM model [18]. Proposals that modify the
Friedmann–Lamaître–Robertson–Walker (FLRW) cosmology were adopted to solve
cosmological problems. They have a linear or quasi-linear evolution of the
universe in common and present synchronicity $H_{0}t_{0}=1$.

Additionally, some proposals in cosmology, including the widely accepted Kolb
model ($1989$) [19], explain the presence of K-matter, a unique form of exotic
matter that dominates at low redshifts. K-matter has an equation of state with
a value of $\omega=-1/3$, and its density decreases as the universe expands,
following an inverse square relation with the scale factor. Another proposal
worth mentioning is the Allen model [20], which incorporates a $SU(2)$
cosmological instanton in addition to the Einstein field equations. The
instanton dominates during the later stages of the Universe’s evolution,
resulting in an approximately proportional relationship between the scale
factor and cosmic time $t$. Moving on, Pimentel and Diaz–Rivera [21] presented
a solution to the synchronicity problem by considering the effect of a time-
dependent cosmological constant $\Lambda$ in a family of scalar–tensor
theories. Interestingly, the cosmological constant in these models exhibits an
inverse square relationship with time, reaching its minimum value in the
present era. However, a point of concern arises when considering the
gravitational constant $G$, which decreases inversely with time and
contradicts the experimental constraints on time variation in $G$. Moreover,
Melia and Shevchuk [22] raised an intriguing question regarding the
coincidence between the cosmic time $t_{0}=H_{0}^{-1}$, describing the
distance that light has traveled since the Big Bang and the gravitational
horizon $R_{h}(t_{0})$. The query was whether these two quantities were indeed
equal or not. However, Lewis [23] criticized this model by highlighting that
if the dark energy component of the universe possesses an equation of state
with $\omega=-1/3$, the addition of any amount of matter will inevitably lead
to a deviation from the strictly linear evolution of the $R_{h}=ct$ universe.
This deviation would manifest the universe transitioning to a matter-dominated
phase during its earliest epochs.

The primary aim of this study is to establish an appropriate fractional
cosmological framework to examine the synchronicity issue, among other enigmas
linked to the $\Lambda$CDM model. Our investigation endeavors to tackle the
synchronicity problem through the fractional quantum cosmology (FQC) proposal
[24, 25, 26, 27, 28, 18, 29]. The basic and preliminary methodology of the FQC
is to combine fractional quantum mechanics (FQM) [30, 31, 32] and quantum
cosmology (QC) [33] to achieve the corresponding cosmological consequences
(see [29] for a detailed discussion). However, it should be mentioned that in
this study, we only consider the (semi-)classical limit of this theory; to see
other semiclassical models established applying various gravitational theories
as underlying frameworks, see, for instance, [34, 35, 36, 37, 38, 39, 40, 41,
42]. Several studies have put forth the idea of fractional classical cosmology
(FCC) [43, 44, 45], since the application of fractional calculus in cosmology
is now producing encouraging and promising results [46].

There have been numerous instances in the fields of gravity and cosmology
where fractional calculus has been employed, showcasing the broad and
significant applications of this mathematical tool. The current research in
this domain is thriving, with scientists actively exploring the potential of
fractional calculus. Its effectiveness as a valuable instrument becomes
apparent when confronting a wide range of problems associated with
gravitational forces and cosmological models, as evidenced by the wealth of
studies conducted [46, 45, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57].
Moreover, fractional calculus finds application in various other areas, such
as the study of the stochastic gravitational-wave background in quantum
gravity [58], the investigation of gravitational-wave luminosity distance
[59], the exploration of inflation and CMB spectrum [28, 60, 61], the
development of fractional action cosmology [62, 63, 64], the analysis of the
fractional geodesic equation and discrete gravity [65], and the examination of
nonminimal coupling and chaotic inflation [66]. Additionally, fractional
calculus has been utilized in investigating phantom cosmology with conformal
coupling [67], Ornstein–Uhlenbeck-like fractional differential equations in
cosmology [68], fractional action cosmology with a variable-order parameter
[69], and wormholes in fractional action cosmology [70]. Notably, new metrics
have been considered [71], while the application of fractional calculus has
also extended to the study of dark energy models in emergent, logamediate, and
intermediate scenarios of the universe [72, 73]. As an example, the work of
[50, 52] yielded a value of $\alpha=0.926$ for the order of the
Riemann–Liouville fractional integral, providing further insights into the
intricacies of the field.

In [43, 47, 48], several exact solutions were obtained for cosmological
models, which, due to the fractal nature of space–time, exhibit a significant
deviation from the standard model [74, 75]. The investigation carried out in
ref. [27] delves into the interval $1\leq\alpha<2$ and employs Riesz’s
fractional derivative to derive the nonboundary and tunneling wave functions
for a closed de Sitter geometry. Furthermore, ref. [28] explores the
preinflation epoch within the framework of fractional quantum cosmology. With
regard to the thermodynamics of fractional BHs, this subject has been
thoroughly examined in ref. [27], and the findings from this research have
been effectively utilized to obtain the Friedmann equations in the context of
emergent gravity in ref. [76]. It is worth mentioning that fractional calculus
has also been employed to modify the Friedmann and Raychaudhuri equations,
thereby enabling an investigation into the dynamics of the universe without
the need for cold dark matter (CDM) or dark energy [77]. In addition, an
alternative approach involves the utilization of fractional calculus to
determine the value of the cosmological constant, which necessitates a
restructuring due to the well-recognized ultraviolet divergence in
conventional quantum field theory [78, 79, 80]. To expand the scope of this
fractional approach, [81, 82] explore modified Newtonian dynamics (MOND) and
quantum cosmology. Lastly, it is crucial to note that there exists a multitude
of definitions for fractional derivatives and fractional integrals, including
those proposed by Riemann–Liouville, Caputo, Riesz, Hadamard, Marchand, and
Griinwald–Letnikov, as well as more recent formulations (as comprehensively
documented in [83] and [84], along with their respective references). Despite
the extensive research conducted on these operators, they do not always
exhibit the conventional properties associated with function differentiation,
such as Leibniz’s rule, the chain rule, and the semigroup property [84, 83].

This paper is organized as follows: We begin in Section 2 by introducing
fractional quantum cosmology, which allows us to define the fractional
extension of the ADM Hamiltonian and Hamiltonian constraint within the
minisuperspace approximation of gravity. Next, in Section 3, we apply the
general formalism obtained in Section 3 to the FLRW cosmology. This leads to
deriving the fractional extensions of the Friedmann and Raychaudhuri
equations. We then proceed with a thorough discussion of various fractional
cosmological parameters, including the Hubble and deceleration parameters. In
Section 4, we briefly review the available cosmological datasets. Afterwards,
we consider the observational datasets, such as SNIa, CMB, BAO, BBN, and OHD,
to estimate both the universe’s age and the model’s free parameters. Section 5
focuses on the perspective of fractional cosmology regarding the synchronicity
problem. Finally, in Section 6, we summarize the results and offer additional
discussions.

##  2 Fractional Quantum Cosmology

This section briefly explains the FQC framework. To develop a fractional WDW
equation, we first briefly describe how to construct a fractional Schrödinger
equation (SE).

In two seminal papers, Laskin [30, 85] introduced a groundbreaking concept
known as space fractional quantum mechanics (SFQM) through the application of
a purely physics-based approach. Laskin’s revolutionary work was built upon
the solid foundation of nonrelativistic quantum mechanics as formulated by
Feynman and Hibbs, which extensively utilized a path integral representation
employing Brownian paths. However, Laskin’s innovative contribution to the
field was to replace these conventional Brownian paths with paths
characterized by Lévy flights, thus giving rise to the fascinating realm of
SFQM and expanding our understanding of quantum phenomena in space.

In the framework of SFQM, one can initiate the analysis by considering the
conventional time-dependent SE, which is expressed as follows:

$i\hbar\frac{\partial\psi(\textbf{r},t)}{\partial{t}}=-\frac{\hbar^{2}}{2m}\Delta\psi(\textbf{r},t)+V(\textbf{r},t)\psi(\textbf{r},t).$
(1)

This equation serves as a starting point for further exploration, wherein one
can delve into its fractional counterpart, as proposed by Laskin [31]. The
fractional SE is obtained by replacing the traditional Laplace operator
$\Delta$ with the Riesz fractional derivative $(-\hbar^{2}\Delta)^{\alpha/2}$,
which can be expressed as

$-\frac{\hbar^{2}}{2m}\Delta\,\rightarrow\,D_{\alpha}\left(-\hbar^{2}\Delta\right)^{\alpha/2}.$
(2)

This replacement paves the way for the exploration of the space-fractional SE:

$i\hbar\frac{\partial\psi(\mathbf{r},t)}{\partial
t}=\hat{H}_{\alpha}(\hat{\mathbf{p}},\hat{\mathbf{r}})\psi(\mathbf{r},t)=D_{\alpha}(-\hbar^{2}\Delta)^{\alpha/2}\psi(\mathbf{r},t)+V(\mathbf{r},t)\psi(\mathbf{r},t),~{}~{}~{}~{}1<\alpha\leq
2,$ (3)

where $\alpha\in(1,2]$ is the Lévy index, and $D_{\alpha}$ is, according to
Laskin, a generalized “fractional quantum diffusion coefficient” of dimension
$\text{J}^{1-\alpha}\text{m}^{\alpha}\text{s}^{-\alpha}$. As mentioned
earlier, $(-\hbar^{2}\Delta)^{\alpha/2}$ is a generalization of the fractional
Riesz derivative, known as the fractional Laplacian [86]:

$(-\hbar^{2}\Delta)^{\alpha/2}\psi(\mathbf{r},t)=\frac{1}{(2\pi\hbar)^{3}}\int
d^{3}pe^{i\frac{\mathbf{p}\cdot\mathbf{r}}{\hbar}}|\mathbf{p}|^{\alpha}\varphi(\mathbf{p},t),$
(4)

which, by means of Fourier transforms, relates $\psi(\mathbf{r},t)$ and
$\varphi(\mathbf{p},t)$. For the special case when $\alpha=2$, $D_{\alpha}$
reduces to $D_{2}=1/2m$, where $m$ is the particle’s mass.

The Riesz fractional derivative (the fractional Laplacian), in general,
characterizes a unique diffusion process that arises from the random
displacements of individuals who can move to nearby or neighboring locations
and even venture to distant sites by means of Lévy flights. Both literal and
conceptual flights have been observed or claimed to occur in various
scenarios, such as turbulent fluid motion and the transportation of materials
in fractured media. In the field of mechanics, the fractional Laplacian
describes the motion of a chain or group of particles interconnected by
elastic springs that connect not only to their immediate neighbors but also to
all other particles. The strength of the springs decreases as the particles
move apart, while the particle arrangement can take on either a regular or
fractal pattern. In a more abstract sense, the fractional Laplacian represents
the influence of a nonlocal process on a conservation law, which is not only
influenced by local conditions but also by the overall state of a field of
interest at a specific moment in time. We suggest that the reader consults
Pozrikidis’s exceptionally well-written and comprehensive book [86] in order
to obtain detailed explanations and develop a deeper understanding of
fractional Laplacian.

On the other hand, as demonstrated by Laskin in his work [87], the evolution
of the fractional wave function (3) can be effectively described by a kernel.
This kernel is expressed as an integral equation, which can be written as
follows:

$\psi(\mathbf{r}_{b},t_{b})=\int
d^{3}r_{a}K_{L}(\mathbf{r}_{b},t_{b}|\mathbf{r}_{a},t_{a})\psi(\mathbf{r}_{a},t_{a}),$
(5)

In this equation, $\psi(\mathbf{r}_{a},t_{a})$ represents the wave function of
the initial state, while $\psi(\mathbf{r}_{b},t_{b})$ represents the wave
function of the final state. The kernel itself, denoted as
$K_{L}(\mathbf{r}_{b},t_{b}|\mathbf{r}_{a},t_{a})$, is given by the following
expression:

$K_{L}(\mathbf{r}_{b},t_{b}|\mathbf{r}_{a},t_{a})=\int_{\mathbf{r}_{a}}^{\mathbf{r}_{b}}D\mathbf{r}(\tau)\int
D\mathbf{p}(\tau)e^{\frac{i}{\hbar}\int_{t_{a}}^{t_{b}}(\mathbf{p}(\tau)\cdot\dot{\mathbf{r}}(\tau)-H_{\alpha})d\tau}.$
(6)

Here, it is important to note that $H_{\alpha}(\mathbf{r},\mathbf{p},t)$
serves as the fractional extension for the Hamiltonian of the system under
consideration. This Hamiltonian can be mathematically expressed as

$H_{\alpha}(\textbf{p},\textbf{r})={D_{\alpha}}|\textbf{p}|^{\alpha}+V(\textbf{r},t),\,\,\,\,\,\,\,1<\alpha\leq{2}.$
(7)

It is worth mentioning that in the particular scenario where $\alpha$ takes a
value of 2 and $D_{\alpha}$ is equal to $1/{2m}$, the equation (7) simplifies
to the standard Hamiltonian, which is defined as
$H(\textbf{p},\textbf{r},t)=\textbf{p}^{2}/2m+V(\textbf{r},t)$.

We can further expand the aforementioned procedure in order to acquire
fractional quantum cosmology. In cosmology, we assume that the space–time
metric and matter field(s) exhibit homogeneity and isotropy. Consequently, the
lapse function is considered to be homogeneous, denoted as $N=N(t)$, while the
shift is assigned a value of zero, $N^{i}=0$. As a result, the space–time line
element reduces to

$ds^{2}=-N(t)^{2}dt^{2}+h_{ij}(\mathbf{x},t)dx^{i}dx^{j}.$ (8)

Inserting the restricted form of the line element (8) into the ADM action
functional, one generally obtains the following reduced form of the action
[29]:

$S[q^{\mu}(t),N(t)]=\int\left\\{\frac{1}{2N(t)}f_{\mu\nu}(q)\dot{q}^{\mu}\dot{q}^{\nu}-N(t)U(q)\right\\}dt,~{}~{}~{}~{}~{}\mu,\nu=1,2,...,n.$
(9)

In this equation, $f_{\mu\nu}$ denotes the reduced DeWitt metric of
$n$-dimensional minisuperspace with signature $(-,+,...,+)$. Also, $U(q)$, the
potential, includes the Ricci scalar of the three-dimensional $t=const.$
hypersurfaces and the potential of matter fields. Generally, the
minisuperspace coordinates $q^{\mu}$ may include matter fields and three-
metric components. Variation in the action with respect to $q^{A}$ yields the
Euler–Lagrange equations

$\displaystyle\frac{1}{N}\frac{d}{dt}\Big{(}\frac{\dot{q}^{\mu}}{N}\Big{)}+\frac{1}{N^{2}}\Big{\\{}^{\mu}_{\alpha\beta}\Big{\\}}\dot{q}^{\alpha}\dot{q}^{\beta}=-f^{\mu\nu}\partial_{\nu}U(q),$
(10)

where $\Big{\\{}^{\mu}_{\alpha\beta}\Big{\\}}$ are the connection coefficients
(Christoffel symbols) determined from the metric of the minisuperspace
$f_{\mu\nu}$. Also, in a variation with respect to the lapse function, $N$,
one obtains the constraint

$\frac{1}{2N}f_{\mu\nu}\dot{q}^{\mu}\dot{q}^{\nu}+NU(q)=0.$ (11)

To ensure consistency, it is necessary for Equations (10) and (11) to be equal
to the $00$ and $ij$ components of the complete Einstein equations,
respectively.

To define the Hamiltonian, we first need to define canonical momenta in a
usual way, which is

$\Pi_{\mu}=\frac{\partial
L}{\partial\dot{q}^{\mu}}=\frac{1}{N}f_{\mu\nu}\dot{q}^{\nu}.$ (12)

This leads us to the canonical ADM Hamiltonian in the minisuperspace

$H_{\text{ADM}}=N\left(\frac{1}{2}f^{\mu\nu}(q)\Pi_{\mu}\Pi_{\nu}+U(q)\right)\equiv
N\mathcal{H},$ (13)

where $f^{\mu\nu}$ is the inverse metric of the minisuperspace. The lapse
function $N$ serves as a Lagrange multiplier with the purpose of ensuring the
satisfaction of the Hamiltonian constraint

$\mathcal{H}(q^{\mu},\Pi_{\mu})=\frac{1}{2}f^{\mu\nu}\Pi_{\mu}\Pi_{\nu}+U(q)=0.$
(14)

In the coordinate representation of canonical quantization, the process
involves the incorporation of a wave function, denoted as $\Psi(q)$, which is
independent of time and necessitates the condition that it is rendered null by
the operator associated with the aforementioned Hamiltonian constraint. As a
result, the WDW equation is obtained:

$\left\\{-\frac{1}{2}\Delta+\zeta\mathcal{R}+U(q)\right\\}\Psi(q)=0,$ (15)

where
$\Delta=\frac{1}{\sqrt{-f}}\partial_{\alpha}(\sqrt{-f}f^{\alpha\beta}\partial_{\beta})$
is the d’Alembertian, $\mathcal{R}$ is the curvature of the minisuperspace
metric $f_{\mu\nu}$, and $\zeta$ is an arbitrary constant [29]. It is
important to note that the minisuperspace metric’s dependence on the local
coordinates of the minisuperspace leads to a nontrivial operator ordering
issue in the resulting WDW equation. However, this issue can be somewhat
addressed by ensuring that the quantization process is covariant in
minisuperspace. This means that it should not be affected by general
coordinate transformations in the minisuperspace. The factor ordering
procedure is the origin of the minisuperspace curvature term, denoted by
$\mathcal{R}$.

To acquire the fractional equivalent of the WDW equation (15), it is necessary
to substitute the typical d’Alembertian operator with the fractional
Riesz–d’Alembertian operator given by [88, 89]

$(-\nabla)^{\frac{\alpha}{2}}\Psi(q^{\alpha})={\mathcal{F}}^{-1}\Big{(}|{\mathbf{\Pi}}|^{\alpha}{\mathcal{F}}\Psi({\mathbf{\Pi}})\Big{)},$
(16)

where $|{\mathbf{\Pi}}|=\sqrt{f^{\mu\nu}\Pi_{\mu}\Pi_{\nu}}$, and
${\mathcal{F}}$ represents a Fourier transformation. Therefore, the fractional
equivalent of the WDW equation (15) can be expressed as [24, 25, 26, 27, 29]

$\left\\{\frac{D_{\alpha}(m_{\text{P}})}{2}(-\nabla)^{\frac{\alpha}{2}}+\zeta\mathcal{R}+U(q)\right\\}\Psi(q^{\nu})=0,$
(17)

where $D_{\alpha}(m_{\text{P}})$ is the coefficient, depending on the Planck
mass $m_{\text{P}}=1/\sqrt{G}$, which reduces to unity at $\alpha=2$ limit,
and the value of Lévy’s fractional parameter, denoted as $\alpha$, is assigned
the range of values $1<\alpha\leq 2$.

By following the methodology described by Laskin [87], it becomes feasible to
illustrate the comprehensive procedure for tje fractional path integral
quantization of constrained systems, as expounded upon by Balatin, Fradkin,
and Vilkovisky (for further elaboration, please refer to [90]). This approach
culminates in the derivation of the resulting propagation amplitude which
arises when this technique is applied to the minisuperspace model that is
precisely defined by the Hamiltonian (14)

$\begin{split}G_{\alpha}(\mathbf{q}_{2},\mathbf{q}_{1})&=\int
dN(t_{2}-t_{1})\int
Dq^{\mu}D\Pi_{\mu}\exp\left\\{i\int_{t_{1}}^{t_{2}}(\dot{q}^{\nu}\Pi_{\nu}-N{\mathcal{H}}_{\alpha})dt\right\\}\\\
&=\int dT\int
Dq^{\mu}D\Pi_{\mu}\exp\left\\{i\int_{0}^{T}(\dot{q}^{\nu}\Pi_{\nu}-N{\mathcal{H}}_{\alpha})dt\right\\}\\\
&=\int K_{L}(\mathbf{q}_{2},T|\mathbf{q}_{1},0)dT,\end{split}$ (18)

where $\mathcal{H}_{\alpha}(q^{\mu},\Pi_{\mu})$ is the fractional Hamiltonian
constraint, counterpart to (14), defined by

$\mathcal{H}_{\alpha}(q^{\mu},\Pi_{\mu})=\frac{D_{\alpha}(m_{\text{P}})}{2}|f^{\mu\nu}\Pi_{\mu}\Pi_{\nu}|^{\frac{\alpha}{2}}+U(q)=0.$
(19)

This equation gives us the fractional extension of Equation (11). Also, one
can easily obtain the fractional Euler–Lagrange counterpart of (10) by
Hamilton’s equations

$\dot{q}^{\mu}=\frac{\partial
H^{(\alpha)}_{\text{ADM}}}{\partial\Pi_{\mu}},~{}~{}~{}~{}~{}\dot{\Pi}_{\mu}=-\frac{\partial
H^{(\alpha)}_{\text{ADM}}}{\partial q^{\mu}},$ (20)

where $H^{(\alpha)}_{\text{ADM}}$ is the fractional ADM Hamiltonian defined by

$H^{(\alpha)}_{\text{ADM}}=N\left(\frac{D_{\alpha}(m_{\text{P}})}{2}\left|f^{\mu\nu}\Pi_{\mu}\Pi_{\nu}\right|^{\frac{\alpha}{2}}+U(q)\right).$
(21)

In the following section, we derive the fractional $\Lambda$CDM cosmology
using Equations (19) and (20).

##  3 Fractional FLRW Cosmology

Let us commence our discussion by considering the FLRW metric, which serves as
the underlying geometric structure

$ds^{2}=-N^{2}(t){\rm{d}}t^{2}+a^{2}(t)\left\\{\frac{{{{\rm\,d}r^{2}}}}{{1-kr^{2}}}+r^{2}{\rm{d}}\Omega_{2}^{2}\right\\}.$
(22)

In this metric, we can observe various components contributing to the overall
structure. Firstly, the term $N(t)$ represents the lapse function, which
characterizes the rate at which time elapses within the cosmic framework.
Secondly, the variable $t$ represents the cosmic time, enabling us to measure
and understand the progression of events within the universe. Furthermore, the
term ${\rm{d}}\Omega_{2}^{2}$ stands for the standard line element for
$\mathbb{S}^{2}$, encompassing both the angular coordinates $\theta$ and
$\phi$. Lastly, the parameter $k$ assumes values of $-1,0,$ and $1$,
signifying the sign of spatial curvature. These values correspond to an open,
flat, and closed universe, respectively.

The functional, which encompasses both gravitational and matter components,
with the matter field being treated as a perfect fluid [91, 92, 93, 94, 95,
36, 96], can be expressed as follows:

$S=\frac{1}{16\pi\,G}\left[\int_{\mathcal{M}}{\sqrt{-g}R\,\rm{d}^{4}x}+\int_{\partial\mathcal{M}}{2K\sqrt{h}}{\rm{d}}^{3}{x}\right]-\int_{\mathcal{M}}\sqrt{-g}\rho{\rm{d}}^{4}x,$
(23)

where $K$ denotes the trace of the extrinsic curvature of the spacelike
submanifolds, $(\Sigma,h)$, of the space–time, $(\mathcal{M},g)$, in which
$\mathcal{M}=\mathbb{R}\times\Sigma$ and $\mathcal{M}$ represents the manifold
of the space–time with boundary $\partial M$. In addition, $\rho$ is the total
energy density of the cosmic fluid with components

$\rho=\sum_{i}\rho^{(i)}=\rho^{(r)}+\rho^{(m)}+\rho^{(\Lambda)},$ (24)

where $\rho^{(r)}$, $\rho^{(m)}$, and $\rho^{(\Lambda)}=\frac{\Lambda}{8\pi
G}$ are the energy density of the radiation, matter density of the cosmic dust
(constitute of baryonic and cold matter), and the energy density of
cosmological constant, respectively.

The act of introducing the homogeneous and isotropic metric (22) into the
action functional (23) results in a simplified Arnowitt–Deser–Misner (ADM)
form as follows [29]:

$S_{ADM}=\int_{t_{i}}^{t_{f}}{\rm\,d}t\left[{\frac{3\mathcal{V}_{k}}{8\pi\,G}\left(kNa-\frac{a\,\dot{a}^{2}}{N}\right)-{\mathcal{V}_{k}Na^{3}\rho(t)}}\right],$
(25)

where

$\mathcal{V}_{k}=\int_{\Sigma}{{\rm{d}}r{\rm{d}}\theta{\rm{d}}\phi\frac{r^{2}\sin{\theta}}{\sqrt{1-kr^{2}}}},$
(26)

is the spatial volume of the spacelike hypersurfaces $(\Sigma,h)$.

To ensure that the spacelike sections, represented by $\Sigma$, have a finite
volume, it is assumed that they are both compact and without boundary. In the
scientific community, it is widely accepted that any compact Riemannian three-
manifold with constant curvature is homeomorphic to $\tilde{\Sigma}/\Gamma$,
where $\Gamma$ represents the group of covering transformations, and
$\tilde{\Sigma}$ represents the universal covering space. The shape of
$\tilde{\Sigma}$ can take on different forms depending on the sign of the
spatial curvature $k$ (where $k$ can be 0, 1, or -1), such as $\mathbb{R}^{3}$
(representing three-dimensional Euclidean space), $\mathbb{S}^{3}$
(representing a 3-sphere), or $\mathbb{H}^{3}$ (representing three-dimensional
hyperbolic space) [29].

Using this particular approach, the action mentioned above is clearly defined
and has a finite 3-volume, which is an important characteristic. The formula
for accurately determining the volume of spherical 3-spaces is given by
$\mathcal{V}_{k=1}=2\pi^{2}/|\Gamma|$, where $|\Gamma|$ represents the order
of the group $\Gamma$. It is worth noting that when the 3-manifold is
topologically complex, the order of the group $\Gamma$ tends to be large,
resulting in a relatively small corresponding volume. Additionally, it is
important to mention that there is no lower bound on the volume since the
group $\Gamma$ can have an arbitrarily large number of elements according to
[29].

Ongoing research and exploration within the scientific community are still
being carried out to determine the allowed topologies of
$\mathbb{H}^{3}/\Gamma$. However, Thurston and Jørgensen have made significant
contributions in this field by demonstrating the existence of the smallest
volume for compact orientable hyperbolic 3-manifolds, as discussed in [97].
This achievement can be attained through a finite number of manifolds, with
the prime candidate being the Weeks–Matveev–Fomenko manifold. Its volume is
approximately 0.942707. The research conducted by Thurston and Jørgensen has
greatly enhanced our understanding of the intricate and fascinating properties
of hyperbolic 3-manifolds and their volumes.

The Lagrangian that corresponds to the aforementioned action (25) is

$L_{ADM}=\frac{3\mathcal{V}_{k}}{8\pi\,G}\left(kNa-\frac{a\,\dot{a}^{2}}{N}\right)-{\mathcal{V}_{k}Na^{3}\rho(t)}.$
(27)

As a result, the corresponding ADM Hamiltonian is given by

$H_{ADM}=N\left[\frac{2\pi\,G\,\Pi^{2}}{3\mathcal{V}_{k}\,a}+\frac{3\mathcal{V}_{k}\,k\,a}{8\pi\,G}-\mathcal{V}_{k}\,a^{3}\,\rho\right],$
(28)

where

$\Pi=-\frac{3\mathcal{V}_{k}a\dot{a}}{4\pi GN},$ (29)

is the conjugate momenta of the scale factor. The ability to select the lapse
function provides us with gauge freedom, resulting in the super-Hamiltonian
constraint

$\mathcal{H}=\frac{2\pi
G\Pi^{2}}{3\mathcal{V}_{k}\,a}+\frac{3\mathcal{V}_{k}\,k\,a}{8\pi\,G}-\mathcal{V}_{k}\,a^{3}\,\rho=0.$
(30)

The minisuperspace of the FLRW cosmology, corresponding to the Hamiltonian
constraint (30), is unidimensional with
$f^{\mu\nu}\Pi_{\mu}\Pi_{\nu}=\Pi^{2}/a$. Regarding dimensions of
$[\Pi^{2}/a]=1/(\text{length})^{3}$ and $[\mathcal{H}]=1/(\text{length})$, one
can easily verify that $D_{\alpha}(m_{\text{P}})=m_{\text{P}}^{3(2-\alpha)}$.
Therefore, the fractional extension of the ADM Hamiltonian (28) and
Hamiltonian constraints (30) are given by

$H^{(\alpha)}_{\text{ADM}}=N\left[\frac{2\pi
l_{\text{P}}^{3\alpha-4}}{3\mathcal{V}_{k}}\left|\frac{\Pi^{2}}{a}\right|^{\frac{\alpha}{2}}+\frac{3\mathcal{V}_{k}\,k\,a}{8\pi\,G}-\mathcal{V}_{k}\,a^{3}\,\rho\right],$
(31) $\frac{2\pi
l_{\text{P}}^{3\alpha-4}}{3\mathcal{V}_{k}}\left|\frac{\Pi^{2}}{a}\right|^{\frac{\alpha}{2}}+\frac{3\mathcal{V}_{k}\,k\,a}{8\pi\,G}-\mathcal{V}_{k}\,a^{3}\,\rho=0,$
(32)

where $l_{\text{P}}=\sqrt{G}$ is the Planck length.

In the comoving gauge $N=1$, using the Hamilton’s equation, $\dot{a}=\partial
H^{(\alpha)}_{\text{ADM}}/\partial{\Pi}$, and rescaling the cosmic time by

$\displaystyle
t\rightarrow\frac{\alpha}{4^{\frac{\alpha-1}{\alpha}}}\left(\frac{3\mathcal{V}_{k}}{2\pi}\right)^{\frac{\alpha-2}{\alpha}}t,$
(33)

lead us to the following fractional extension of the Friedmann equation

$H^{\frac{2}{3-D}}=\frac{1}{\left(l_{\text{P}}^{2}a^{3}\right)^{\frac{D-2}{3-D}}}\left[\frac{8\pi
G}{3}\rho-\frac{k}{a^{2}}\right],$ (34)

where $\rho$ is given by relation (24), and the parameter $D$ is related to
the Lévy fractional parameter $\alpha$ by

$D=\frac{2}{\alpha}+1,~{}~{}~{}~{}~{}~{}2\leq D<3.$ (35)

Substituting $\alpha=2$ (or equivalently $D=2$) into Equation (34), we obtain
the standard counterpart, i.e., (48). We should note that there are several
variants of the fractional Friedmann Equation in the literature; see, for
instance, [43, 44].

Similar to the standard model of cosmology, one can define the fractional
extension of the density parameters $\\{\Omega^{(i)}=\frac{8\pi
G\rho^{(i)}}{3H^{2}},\Omega^{(k)}=-\frac{k}{a^{2}H^{2}}\\}$ by

$\tilde{\Omega}^{\text{(i)}}=\frac{1}{\left(l_{\text{P}}^{2}a^{3}\right)^{\frac{D-2}{3-D}}}\frac{8\pi
G\rho^{(i)}}{3H^{\frac{2}{3-D}}},~{}~{}~{}~{}~{}\tilde{\Omega}^{(k)}=-\frac{1}{\left(l_{\text{P}}^{2}a^{3}\right)^{\frac{D-2}{3-D}}}\frac{k}{a^{2}H^{\frac{2}{3-D}}}.$
(36)

It should be noted that the density parameters mentioned above follow the
standard definitions in $\Lambda$CDM when $D=2$. As a result, the fractional
Friedmann (34) takes the following standard form

$\tilde{\Omega}^{(m)}+\tilde{\Omega}^{(rad)}+\tilde{\Omega}^{(\Lambda)}+\tilde{\Omega}^{(k)}=1,$
(37)

which is also known as the closure relation.

The Raychaudhuri equation, also known as the second Friedmann equation, can be
derived by taking the time derivative of the Friedmann Equation (34) and
combining it with the continuity equation. In fractional cosmology, the
energy–momentum tensor of a perfect fluid is subject to the same covariant
conservation equation as in standard cosmology. This equation can be expressed
as follows:

$\dot{\rho}^{(i)}+3H\Big{(}\rho^{(i)}+p^{(i)}\Big{)}=0,$ (38)

where $p^{(i)}=\omega_{i}\rho^{(i)}$.

These two lead us to the fractional Raychaudhuri equation

$\frac{\ddot{a}}{a}=-\frac{4\pi(3-D)G}{3(H^{2}l_{\text{P}}^{2}a^{3})^{\left(\frac{D-2}{3-D}\right)}}(\rho+3p)-\frac{D-2}{2}H^{2}.$
(39)

In addition, the fractional Friedmann Equation (34), utilizing the continuity
Equation (38), can be rewritten as

$\left(\frac{H}{H_{0}}\right)^{\frac{2}{3-D}}=\frac{1}{a^{3\left(\frac{D-2}{3-D}\right)}}\left\\{\sum_{i}\frac{\tilde{\Omega}^{(i)}_{0}}{a^{3(1+\omega_{i})}}+\frac{\tilde{\Omega}^{(k)}_{0}}{a^{2}}\right\\},$
(40)

where $\tilde{\Omega}^{(i)}_{0}$ and $\tilde{\Omega}^{(k)}_{0}$ denote the
density parameters at the present epoch, $(a=1)$.

In the same method, one can obtain the deceleration parameter $q$. By taking a
time derivative of the Friedmann equation (34) and employing the continuity
equation and fractional density parameters (36), we find

$\begin{split}q&=-\frac{\ddot{a}}{aH^{2}}=\frac{3-D}{2}\sum_{i}(1+3\omega_{i})\tilde{\Omega}^{(i)}+\frac{D-2}{2}\\\
&=\frac{3-D}{2\left(\frac{H}{H_{0}}\right)^{\frac{2}{3-D}}a^{3\left(\frac{D-2}{3-D}\right)}}\sum_{i}(1+3\omega_{i})(1+z)^{3(1+\omega_{i})}\tilde{\Omega}^{(i)}_{0}+\frac{D-2}{2},\end{split}$
(41)

where, in the second equality, we used the definition of the redshift
$1+z=1/a$.

To observe the overall impact of the Lévy fractional parameter on cosmic
evolution, it is instructive to examine a simple flat ($k=0$) model of the
universe, consisting of cosmic dust and the cosmological constant. The
behavior of the $q$ in relation to the redshift $z$ is depicted in Figures 1
and 2. In Figure 1, we maintain a constant value of $D=2.2$ and examine the
impact of $\tilde{\Omega}^{(m)}_{0}$ on the variable $q$ in terms of $z$. Our
observation reveals that $q$ does not respond significantly to the current
values of $\tilde{\Omega}^{(m)}_{0}$. However, in Figure 2, it becomes
apparent that $q$ is highly sensitive to the Lévy fractional parameter. By
analyzing Figure 2, we can deduce that as $\alpha$ increases, the transition
from a deceleration phase to an acceleration phase occurs at lower redshifts.

Figure 1: The evolution of the deceleration parameter, $q$, as a function of
redshift, $z$, for $D=2.2$ and $\tilde{\Omega}^{(m)}_{0}=0.2$ (solid line),
$\tilde{\Omega}^{(m)}_{0}=0.26$ (dashed), $\tilde{\Omega}^{(m)}_{0}=0.3$ (dot-
dashed), and $\tilde{\Omega}^{(m)}_{0}=0.4$ (doted). Figure 2: The evolution
of the deceleration parameter, $q$, as a function of redshift, $z$, for
$\tilde{\Omega}^{(m)}_{0}=0.264$ and $D=2$ (solid line), $D=2.1$ (dashed),
$D=2.4$ (dot-dashed), and $D=2.8$ (doted).

##  4 Fractional Cosmology Versus Data

It is important to mention that the analysis in this section is limited to the
background level and does not take any disturbances into account. In the
following section, we compare our theoretical findings with observations. We
examine four significant datasets, namely SNIa, CMB, BAO, BBN, and OHD. Some
articles suggest that OHD, which is obtained based on redshift, can be
compared with the SNIa dataset. If one is interested, one can refer to
reference [98] and other relevant sources. The largest combined sample of
SNIa, named the Pantheon Sample, consists of 1048 SNIa in the redshift range
of $[0.01,2.3]$, as presented by Scolnic et al. [99]. In this analysis, we use
the binned Pantheon SNIa dataset, which consists of 40 data points within the
$0.014<z<1.61$ range. We perform an analytical marginalization of the nuisance
parameter in the distance estimates $M$. In ref. [100], the introduction of
$\chi^{2}_{SN}$ as

$\chi_{SN}^{2}(p)=S_{2}(p)-\frac{S_{1}^{2}(p)}{S_{0}}+\ln\frac{S_{0}}{2\pi}+\ln|2\pi
S_{SN}|,$ (42)

in which

$\begin{split}S_{0}=V.S_{SN}^{-1}.V^{T},~{}~{}~{}S_{1}=W.S_{SN}^{-1}.V^{T},~{}~{}~{}S_{2}=W.S_{SN}^{-1}.W^{T},\end{split}$
(43)

the covariance matrix, denoted by $S$, is calculated using the binned Pantheon
sample, taking into account both statistical and systematic errors.
Additionally, $V$ is a row vector consisting of unitary elements, while
$W_{i}$ represents the difference between $\mu_{b,i}$ and $\mu(z_{i})$. The
distance modulus, $\mu$, is predicted theoretically using the luminosity
distance $d_{L}$, which is given by the following equation:

$\mu(z)=5\log_{10}\left[(1+z)\int^{z}_{0}\frac{dx}{E(x)}\right]+42.384-5\log_{10}h-19.37.$
(44)

Here, $E(x)$ is the dimensionless Hubble parameter, and $h$ is the Hubble
constant.

The normalization constants independent of the cosmological model can be
eliminated for this particular dataset. Afterwards, we incorporate the data
obtained by observing acoustic signatures within the large-scale clustering of
galaxies. By utilizing the BAO data, it is possible to minimize the
$\chi_{BAO}^{2}$ that is defined as [101]

$\chi_{BAO}^{2}=Y^{T}C_{BAO}^{-1}Y.$ (45)

Since the SNIa and BAO data contain valuable information regarding the
universe at lower redshifts, we incorporate the CMB shift information by
considering the compressed CMB likelihood. This likelihood is discussed in
Table 1 of ref. [102]) and focuses on the angular scale of the sound horizon
at the last scattering, denoted as $l_{a}$, as well as the baryon density
parameter. By including this information, we aim to explore the entire history
of expansion leading up to the last scattering surface. To achieve this, we
refer to Equations (22)–(33) in ref. [103]. Additionally, the Big Bang
Nucleosynthesis (BBN) offers a data point that helps constrain mainly
$\Omega^{(b)}_{0}$, as mentioned in the work by Serra et al. [104].

The $\chi_{BBN}^{2}$ is given by

$\chi_{BBN}^{2}=\frac{(\Omega^{(b)}_{0}h^{2}-0.022)^{2}}{0.002^{2}}.$ (46)

Finally, we additionally incorporate the data obtained from the observational
Hubble parameter. In this particular investigation, we utilize the 31 data
points derived from the recent and precise estimates of $H(z)$ within the
redshift range of $0.07\leq z\leq 1.965$. These particular data points are
independent of the Baryon Acoustic Oscillation (BAO) data points and were
presented initially in [105]. In this case, we can write

$\chi_{OHD}^{2}(p)=\Sigma_{i}\frac{(H(z_{i},p)-H_{i})^{2}}{\sigma_{i}^{2}}$
(47)

where $\sigma_{i}$ is the Gaussian error on the measured value of $H_{i}$.

The $\Lambda$CDM model, also known as the Concordance Model, is widely
accepted in cosmology as it accurately fits current observations. It has been
extensively tested using various cosmological measurements. Our statistical
analysis results are presented in Table 1. The fractional $\Lambda$CDM model,
on the other hand, introduces an additional independent parameter ($\alpha$ or
$D$) that represents the attribute of nonlocality due to the fractional
derivative. The statistical analysis of this model is presented in Table 2.
Figure 3 shows the evolution of the deceleration parameter $q(z)$ against
redshift for the $\Lambda$CDM and fractional $\Lambda$CDM models. As we see
from this figure, the deceleration parameter of the standard model of
cosmology is relatively greater than the fractional deceleration parameter in
all redshifts. Figure 4 illustrates the $1\sigma$ and $2\sigma$ confidence
regions that are obtained from fitting the fractional $\Lambda$CDM model to a
comprehensive set of datasets, including Baryon Acoustic Oscillations (BAO),
Supernovae (SN), Cosmic Microwave Background (CMB), Big Bang Nucleosynthesis
(BBN), and Observational Hubble Data (OHD). This fitting process determines
the best possible agreement between the model and the observed data. In order
to further evaluate the accuracy of the fractional $\Lambda$CDM model, Table 2
was prepared to showcase the consistency between the derived values of
$\tilde{\Omega}_{m}$ and $H_{0}$ obtained from the BAO+SN+CMB+BBN+OHD datasets
and the corresponding values reported by the Planck 2018 Collaboration. By
comparing the two, we can gain valuable insights into the reliability of our
fits. Additionally, Figure 5 was included to provide a visual representation
of the performance of our fits as compared with the well-established
$\Lambda$CDM model. It is worth mentioning that both the fractional
$\Lambda$CDM model and the $\Lambda$CDM model exhibit agreement with the most
up-to-date cosmological observations when it comes to their respective values
of the deceleration parameter $q_{0}$ and the associated transition redshift
$z_{t}=0.72$. This further solidifies the credibility of our findings.

Figure 3: The evolution of the deceleration parameter $q(z)$ is shown against redshift. The dotted line represents the $\Lambda$CDM model, while the dashed line represents the fractional $\Lambda$CDM model. Figure 4: The two-dimensional contours and one-dimensional marginalized distributions represent the fractional $\Lambda$CDM model parameters with $68\%$ CL and $95\%$ CL, respectively. Table 1: The best-fit parameters with $1\sigma$ and $2\sigma$ confidence levels (CLs) for the FLRW model. Parameter | 68% CL | 95% CL | Best-Fit Value
---|---|---|---
$\Omega_{0}^{(cdm)}$ | $0.226^{+0.020}_{-0.018}$ | $0.226^{+0.033}_{-0.036}$ | $0.226^{+0.044}_{-0.048}$
$\Omega_{0}^{(b)}$ | $0.0476^{+0.0015}_{-0.0018}$ | $0.0476^{+0.0034}_{-0.0032}$ | $0.0476^{+0.0045}_{-0.0037}$
$\Omega_{0}^{(m)}$ | $0.274^{+0.021}_{-0.018}$ | $0.274^{+0.034}_{-0.037}$ | $0.274^{+0.040}_{-0.050}$
$\Omega_{0}^{(k)}$ | $0.052\pm 0.045$ | $0.052^{+0.089}_{-0.089}$ | $0.05^{+0.11}_{-0.11}$
$H_{0}$ | $68.9\pm 1.2$ | $68.9^{+2.3}_{-2.3}$ | $68.9^{+2.7}_{-3.0}$
$\chi^{2}_{min}$ | $68.6^{+1.3}_{-3.3}$ | $68.6^{+5.6}_{-4.2}$ | $69^{+9}_{-5}$
Figure 5: The $68\%$ and $95\%$ constraint contours on the cold dark matter density parameter $\Omega_{0}^{(cdm)}$, baryon density parameter $\Omega_{0}^{(b)}$, matter density parameter $\Omega_{0}^{(m)}$, curvature density parameter $\Omega_{0}^{(k)}$, and Hubble parameter $H_{0}$ when using data from the BAO+SN+CMB+BBN+OHD. Table 2: The best-fit parameters with $1\sigma$ and $2\sigma$ confidence levels (CLs) for fractional FLRW model. Parameter | 68% CL | 95% CL | Best-Fit Value
---|---|---|---
$\tilde{\Omega}_{0}^{(cdm)}$ | $0.226\pm 0.018$ | $0.226^{+0.036}_{-0.038}$ | $0.226^{+0.046}_{-0.050}$
$\tilde{\Omega}_{0}^{(b)}$ | $0.0480\pm 0.0018$ | $0.0480^{+0.0036}_{-0.0036}$ | $0.0480^{+0.0052}_{-0.0045}$
$\tilde{\Omega}_{0}^{(m)}$ | $0.274\pm 0.018$ | $0.274^{+0.032}_{-0.036}$ | $0.274^{+0.045}_{-0.049}$
$\tilde{\Omega}_{0}^{(k)}$ | $0.049^{+0.050}_{-0.060}$ | $0.05^{+0.11}_{-0.10}$ | $0.05^{+0.14}_{-0.13}$
$H_{0}$ | $68.6^{+1.2}_{-1.4}$ | $68.6^{+2.6}_{-2.4}$ | $68.6^{+3.5}_{-3.4}$
$D$ | $2.0069^{+0.0036}_{-0.0061}$ | $2.007^{+0.011}_{-0.0087}$ | $2.007^{+0.015}_{-0.010}$
$\chi^{2}_{min}$ | $68.4^{+1.5}_{-3.3}$ | $68.4^{+5.8}_{-4.4}$ | $68^{+9}_{-5}$

Observations of Type SNIa provide crucial data for understanding the expansion
of the universe. These observations serve as primary evidence for the
universe’s accelerated expansion. To obtain the best possible results from teh
SNIa data, we compare the observed distance modulus of SNIa detection with the
theoretical value. In this regard, we utilized the Pantheon sample, which is
an updated dataset of SNIa with 1048 distance moduli ($\mu$) at various
redshifts within the range of $0.01<z<2.26$. In Figure 6, we illustrate the
Pantheon Survey as the standard Hubble diagram of SNIa with an absolute
magnitude of $M_{0}=-19.37$. This figure also compares our model with the
$\Lambda$CDM model, showing significant similarities.

Figure 6: The distance modulus of supernovae (data points) is compared with
the theoretically predicted distance modulus (red line) in a fractional
$\Lambda$CDM model.

The Hubble parameter plotted against the redshift $z$ is depicted in the graph
illustrated in Figure 7, utilizing the data extracted from Table 1. Upon
perusing the figure, it becomes evident that our model demonstrates a
remarkable degree of concurrence with the conventional model of cosmology,
thereby substantiating its validity and reliability.

Figure 7: The graph displays the evolution of the Hubble parameter, $H(z)$, in
units of $\text{km}\cdot\text{sec}^{-1}\cdot\text{Mpc}^{-1}$ versus redshift
$z$, with error bars. The red line represents the dynamics of the Hubble
parameter obtained from Equation (40). The values of the density parameters,
fractional density parameters, and Hubble constant at the present epoch are
the best-fit values from Tables 2 and 1, respectively.

##  5 The Synchronicity Problem

Let us review the critical property of the $\Lambda$CDM model related to the
synchronicity problem. The limiting case of $D=2$ in the Friedmann Equation
(40) gives us the Friedmann equation in the standard ($\Lambda$CDM) model of
cosmology

$\left(\frac{H}{H_{0}}\right)^{2}=\sum_{i}\frac{\Omega^{(i)}_{0}}{a^{3(1+\omega_{i})}}+\frac{\Omega^{(k)}_{0}}{a^{2}},$
(48)

where $\Omega^{(i)}_{0}=\frac{8\pi G\rho^{(i)}_{0}}{3H_{0}^{2}}$, and
$\Omega^{(k)}_{0}=-\frac{k}{H_{0}^{2}}$. The solutions of the aforementioned
equation for a flat universe, when radiation, cosmic dust, or the cosmological
constant dominate, are given by

$a(t)=\begin{cases}\left(\frac{\sqrt{H_{0}\Omega^{(r)}_{0}}}{2}\right)^{\frac{1}{2}}t^{\frac{1}{2}},~{}~{}~{}~{}\omega=\frac{1}{3},\\\
\left(\frac{2\sqrt{H_{0}\Omega^{(m)}_{0}}}{3}\right)^{\frac{2}{3}}t^{\frac{2}{3}},~{}~{}~{}~{}\omega=0,\\\
C\exp{\left(\sqrt{\frac{3}{\Lambda}}t\right)},~{}~{}~{}~{}\omega=-1,\end{cases}$
(49)

where, in the last equality, $C$ is a constant of integration. It is easy to
verify that we have $Ht=1/2$ and $Ht=2/3$ in the first two cases,
respectively. In the context of power-law expansion, the ratio of the
universe’s age to the Hubble time is a constant. This is a general property.
However, when the cosmological constant dominates in late-time cosmology, we
have the equation $tH=\sqrt{\Lambda/3}t$. This means that at the present
epoch, concerning cosmological observations, we have
$t_{0}\propto\sqrt{3/\Lambda}$. This equation, in fact, is an expression of
the synchronicity problem of the universe’s age.

Let us analyze the effect of the Lévy fractional parameter on the
synchronicity problem of the age of the universe. We obtain explicit solutions
for the scale factor in a flat universe with a single-component perfect fluid.
Thus, let us consider the $i$-th component is dominated in the Friedmann
equation (40). Regarding Equations (40) and (41), the scale factor and the
deceleration parameter for a perfect fluid with the equation of state
$p^{(i)}=\omega_{i}\rho^{(i)}$ are

$a(t)=\left(\frac{3H_{0}[1+\omega_{i}(3-D)]\tilde{\Omega}^{(i)}_{0}}{2}\right)^{\frac{2}{3(1+\omega_{i}(3-D))}}t^{\frac{2}{3(1+\omega_{i}(3-D))}},$
(50) $q_{i}=\frac{1+3\omega_{i}(3-D)}{2}.$ (51)

This demonstrates that within the radiation-dominated universe ($\omega=1/3$),
the expansion rate is $a(t)\propto t^{\frac{3}{6-D}}$. Furthermore, it is
noteworthy that the nature of the scale factor in a matter-dominated universe
($\omega=0$) remains unaffected by the Lévy fractional parameter and is given
by $a\propto t^{\frac{2}{3}}$. On the other hand, for a de Sitter universe
($\omega=-1$) and $D\neq 2$, the above form of scale factor reduces to
$a(t)\propto t^{\frac{2}{3(D-2)}}$. Furthermore, according to Equation (51),
it can be inferred that in the context of fractional cosmology, our universe
has experienced a period of deceleration during the phases dominated by
radiation and pressureless matter. To summarize, when considering the presence
of radiation and pressureless matter, fractional cosmology is unable to
elucidate the cosmic phase transition from deceleration to acceleration that
occurred throughout the history of the universe unless the inclusion of dark
energy (in our model, a cosmological constant) is taken into consideration.
Also, the deceleration parameter (51) shows that we have an accelerated
universe for $D<8/3$ restricted values of $D$.

Solution (50) shows that in the fractional $\Lambda$CDM model, the universe
follows a power-law accelerated expansion even in recent times, unlike the
standard model, where the late-time acceleration is explained by exponential
de Sitter expansion. In the fractional $\Lambda$CDM model, for the $i$-th
component of cosmic fluid, we have

$Ht=\frac{2}{3(1+\omega_{i}(3-D))}.$ (52)

Thus, for a de Sitter universe, we find

$Ht=\frac{2}{3(D-2)}.$ (53)

Unlike the $\Lambda$CDM model, the present epoch in the history of the
universe is not a particular moment; instead, the age is always proportional
to the Hubble time at that moment. Figure 8 displays the evolution of $Ht$ in
relation to the scale factor for both the standard model of cosmology and its
fractional extension. It is evident that at the current epoch, both models
yield nearly identical values for $H_{0}t_{0}$. However, in the future, their
behaviors will diverge. In the $\Lambda$CDM model, $Ht$ increases for $a>1$,
whereas in the case of fractional $\Lambda$CDM, it remains constant, as we
observe in the aforementioned single-component fluid.

Figure 8: The evolution of $Ht$ with the scale factor is depicted for $D=2$
(dashed) and $D=2.3$ (line). The best-fit values for the density parameters
and fractional density parameters are substituted from Tables 2 and 1,
respectively.

The content matter of the universe has been predominantly composed of cosmic
dust throughout most of its entire history. Regarding this fact and ignoring
the radiation and curvature terms (cold matter plus the cosmological
constant), one can easily show that

$Ht=f(a)\int_{0}^{a}\frac{dx}{xf(x)},~{}~{}~{}~{}~{}f(x)=x^{\frac{3(2-D)}{2}}\left(\frac{\tilde{\Omega}^{(m)}_{0}}{x^{3}}+1-\tilde{\Omega}_{0}^{(m)}\right)^{\frac{3-D}{2}}.$
(54)

Utilizing the above equation, we find that when $2\leq D<3$, the current age
of the universe, in units of Hubble time, is restricted to

$0.6667<H_{0}t_{0}\leq 0.9887.$ (55)

Figure 9 illustrates how the universe’s current age is restricted for
different fractional parameter values. Based on inequality (55), we can
observe that when the fractional parameter $D$ approaches 2, the value of
$H_{0}t_{0}$ becomes almost equal to one. However, in the standard model of
cosmology, where $D=2$, $Ht$ tends towards infinity in the distant future. For
other fractional parameter values, $Ht$ remains finite. Based on the
information presented in Table 2, the fractional parameter has a fitted value
of $D=2.007$. For this value of $D$, we obtain

$H_{0}t_{0}=0.9858,~{}~{}~{}~{}~{}\lim_{a\rightarrow\infty}Ht=95.238,~{}~{}~{}~{}~{}t_{0}=13.8196~{}\text{Gyr}.$
(56)

In addition, in the upper limit of the fractional parameter, $D\rightarrow 3$,
we have

$\lim_{a\rightarrow\infty}Ht=0.6667.$ (57)

Figure 9: The dependence of $H_{0}t_{0}$ to the fractional parameter, $D$. The
best-fit value of the fractional density parameter of dust is substituted from
Table 2.

The analysis conducted shows that in the fractional $\Lambda$CDM model, the
value of $Ht$ is always finite, and at the present epoch, its value is
consistently close to unity. Therefore, it can be confidently concluded that
the value of $Ht$ is finite in the fractional $\Lambda$CDM model.

It is worth mentioning that fractional cosmology may have a considerable
impact on an inflationary phase. Utilizing a scalar field with an appropriate
and dominant potential compared with the universe’s initial energy density is
the typical method employed to achieve inflation. This inflationary approach
triggers a rapid expansion of the scale factor of the universe. In the initial
theories of inflation, this expansion would manifest as exponential growth,
resembling de Sitter space, as the scalar field gradually moves towards its
global minimum. By considering that the potential of the scalar field
primarily affects inflation, we can deduce, based on the fractional
deceleration parameter, that during the early universe, acceleration was
experienced when the value of $D$ lay between 2 and 2.667. This specific range
of $D$ values establishes a power-law inflationary model, and within this
model, the fractional parameter $\alpha$ assumes a significant role. If we
assume that the inflationary epoch took place within a time interval of
$t_{i}=10^{-36}$ s, and $t_{f}=10^{-33}$ s, we can calculate the number of
e-foldings:

$N_{e}=\ln\left(\frac{a(t_{f})}{a(t_{i})}\right)=\frac{2}{3(D-2)}\ln\left(\frac{t_{f}}{t_{i}}\right)=\frac{2\ln(10)}{D-2}.$
(58)

We obtain $D\leq 2.072$ or $\alpha\geq 1.866$ if we assume a minimum of 64
e-foldings for addressing common Big Bang issues. This conclusion is similar
to what we discovered in 4.

##  6 Conclusions

The concept of synchronicity is perplexing because the dimensionless cosmic
time, represented as $H_{0}t_{0}$, can vary significantly within the standard
model of cosmology. This model explains the late-time accelerated expansion as
a de Sitter universe. In this model, the current coincidence of cosmic time
and Hubble time is only valid for the present moment. However, the age of the
universe will not be as near to the Hubble time in the past or future, and
$tH$ might even be infinite in the future.

In this paper, we present a comprehensive demonstration indicating that in the
fractional $\Lambda$CDM model, during the late times, when the cosmological
constant is dominated, the universe exhibits a remarkable and intriguing
phenomenon characterized by a power-law accelerated expansion. This stands in
sharp contrast to the standard model, in which an exponential de Sitter
expansion is obtained to explain the late-time acceleration. Therefore, the
age of the universe is continuously linked to the Hubble time at any given
moment, creating a proportional relationship between the two phenomena. This
contrasts the traditional cosmology framework, where the present epoch in the
vast expanse of the universe is perceived as a specific instant in time.

Our statistical analysis revealed that the fractional parameter proposed by
Lévy, which characterizes the degree of nonlocality in a system, is determined
to be precisely $\alpha=1.986$. We have also determined the age of the
universe through the meticulous study of various cosmological phenomena and
the use of sophisticated techniques. Our calculations show that the age of the
universe is $t_{0}=13.8196$ billion years. Furthermore, by calculating the
dimensionless age of the universe, we find that $H_{0}t_{0}=0.9858$, and
$Ht=95.238$ for the distant future. These findings indicate that our model is
a plausible and valid cosmological model. It successfully encompasses a
crucial early stage of matter domination, essential for forming and emerging
intricate structures within the universe. Furthermore, our model also
incorporates a more recent acceleration phase that corresponds harmoniously
with the observations made in the field of cosmology.

In conclusion, it is essential to emphasize that our fractional model, which
serves as a modified gravity theory, is currently in its nascent stage and
necessitates further exploration and development. By delving into numerous
issues within modern cosmology, such as the Hubble tension, gravitational
lensing, gravitational-wave propagation, and density perturbation generation,
we not only facilitate the advancement of fractional models but also
potentially uncover fascinating findings at both the quantum and classical
levels. However, it is prudent to defer the aforementioned evaluations to
future endeavors.

##  7 ACKNOWLEDGMENTS

We gratefully acknowledge the anonymous reviewers for their insightful
comments. S.J. acknowledges financial support from the National Council for
Scientific and Technological Development, Brazil–CNPq, Grant no.
308131/2022-3. S. M. M. R. acknowledges the FCT grants UID-B-MAT/00212/2020
and UID-P-MAT/00212/2020 at CMA-UBI plus the COST Action CA18108 (Quantum
gravity phenomenology in the multimessenger approach).

## References

  * Einstein and de Sitter [1932] Einstein, A.; de Sitter, W.  On the Relation between the Expansion and the Mean Density of the Universe.  Proc. Nat. Acad. Sci. USA 1932, 18, 213–214.  https://doi.org/10.1073/pnas.18.3.213.
  * Chaboyer et al. [1996] Chaboyer, B.; Kernan, P.J.; Krauss, L.M.; Demarque, P.  A Lower limit on the age of the universe.  Science 1996, 271, 957–961.  https://doi.org/10.1126/science.271.5251.957.
  * Avelino and Kirshner [2016] Avelino, A.; Kirshner, R.P.  The dimensionless age of the Universe: A riddle for our time.  Astrophys. J. 2016, 828, 35.  https://doi.org/10.3847/0004-637X/828/1/35.
  * Tonry et al. [2003] Tonry, J.L.; et al.  Cosmological results from high-z supernovae. Astrophys. J. 2003, 594, 1–24.  https://doi.org/10.1086/376865.
  * Casado [2020] Casado, J.  Linear expansion models vs. standard cosmologies: A critical and historical overview.  Astrophys. Space Sci 2020, 365, 16.  https://doi.org/10.1007/s10509-019-3720-z.
  * Tian [2022] Tian, S.X.  Cosmological consequences of a scalar field with oscillating equation of state. IV. Primordial nucleosynthesis and the deuterium problem.  Phys. Rev. D 2022, 106, 043524.  https://doi.org/10.1103/PhysRevD.106.043524.
  * Albuquerque et al. [2022] Albuquerque, I.S.; Frusciante, N.; Martinelli, M.  Constraining cosmological scaling solutions of a Galileon field.  Phys. Rev. D 2022, 105, 044056.  https://doi.org/10.1103/PhysRevD.105.044056.
  * Kleidis and Spyrou [2016] Kleidis, K.; Spyrou, N.K.  Dark Energy: The Shadowy Reflection of Dark Matter? Entropy 2016, 18, 94.  https://doi.org/10.3390/e18030094.
  * Kleidis and Spyrou [2017] Kleidis, K.; Spyrou, N.K.  Cosmological perturbations in the $\Lambda$CDM-like limit of a polytropic dark matter model.  Astron. Astrophys. 2017, 606, A116.  https://doi.org/10.1051/0004-6361/201630364.
  * Joseph and Saha [2022] Joseph, A.; Saha, R.  Forecast Analysis on Interacting Dark Energy Models from Future Generation PICO and DESI Missions.  Mon. Not. Roy. Astron. Soc. 2022, 519, 1809–1822.  https://doi.org/10.1093/mnras/stac3586.
  * Teixeira et al. [2022] Teixeira, E.M.; Barros, B.J.; Ferreira, V.M.C.; Frusciante, N.  Dissecting kinetically coupled quintessence: Phenomenology and observational tests.  JCAP 2022, 11, 059.  https://doi.org/10.1088/1475-7516/2022/11/059.
  * Liu et al. [2022] Liu, Y.; Liao, S.; Liu, X.; Zhang, J.; An, R.; Fan, Z.  Dark matter haloes in interacting dark energy models: Formation history, density profile, spin, and shape.  Mon. Not. Roy. Astron. Soc. 2022, 511, 3076–3088.  https://doi.org/10.1093/mnras/stac229.
  * Nayak [2020] Nayak, B.  Interacting Holographic Dark Energy, the Present Accelerated Expansion and Black Holes.  Grav. Cosmol. 2020, 26, 273–280.  https://doi.org/10.1134/S020228932003010X.
  * Escamilla-Rivera and Gamboa [2022] Escamilla-Rivera, C.; Gamboa, A.  A new parameterized interacting holographic dark energy.  Eur. Phys. J. Plus 2022, 137, 254.  https://doi.org/10.1140/epjp/s13360-022-02490-4.
  * Landim [2022] Landim, R.G.  Note on interacting holographic dark energy with a Hubble-scale cutoff.  Phys. Rev. D 2022, 106, 043527.  https://doi.org/10.1103/PhysRevD.106.043527.
  * Shimon [2024] Shimon, M.  Elucidation of ’Cosmic Coincidence’ _New Astron._ 2024.  106, 102126.  https://doi.org/10.1016/j.newast.2023.102126.
  * Anari and Sadjadi [2022] Anari, V.; Sadjadi, H.M.  Constant-roll, cosmic acceleration, and massive neutrinos.  JCAP 2022, 7, 043.  https://doi.org/10.1088/1475-7516/2022/07/043.
  * Jalalzadeh et al. [2023] Jalalzadeh, S.; Mohammadi, A.; Demir, D.  A quantum cosmology approach to cosmic coincidence and inflation.  Phys. Dark Univ. 2023, 40, 101227.  https://doi.org/10.1016/j.dark.2023.101227.
  * Kolb [1989] Kolb, E.W.  A coasting cosmology.  Astrophys. J. 1989, 344, 543–550.  https://doi.org/10.1086/167825.
  * Allen [1999] Allen, R.E.  Four testable predictions of instanton cosmology.  AIP Conf. Proc. 1999, 478, 204–207.  https://doi.org/10.1063/1.59392.
  * Pimentel and Díaz-Rivera [1998] Pimentel, L.O.; Díaz-Rivera, L.M.  Coasting Cosmologies with Time Dependent Cosmological Constant.  Int. J. Mod. Phys. A 1998, 14, 1523–1529.  https://doi.org/10.1142/S0217751X99000762.
  * Melia and Shevchuk [2012] Melia, F.; Shevchuk, A.  The $R_{h}=ct$ universe.  Mon. Not. Roy. Astron. Soc. 2012, 419, 2579–2586.  https://doi.org/10.1111/j.1365-2966.2011.19906.x.
  * Lewis [2013] Lewis, G.F.  Matter matters: Unphysical properties of the $R_{h}=ct$ universe.  Mon. Not. Roy. Astron. Soc. 2013, 432, 2324–2330.  https://doi.org/10.1093/mnras/stt592.
  * Moniz and Jalalzadeh [2020] Moniz, P.V.; Jalalzadeh, S.  From Fractional Quantum Mechanics to Quantum Cosmology: An Overture.  Mathematics 2020, 8, 313.  https://doi.org/10.3390/math8030313.
  * Rasouli et al. [2021] Rasouli, S.; Jalalzadeh, S.; Moniz, P.  Broadening quantum cosmology with a fractional whirl.  Mod. Phys. Lett. A 2021, 36, 2140005.  https://doi.org/10.1142/S0217732321400058.
  * Jalalzadeh et al. [2021] Jalalzadeh, S.; da Silva, F.R.; Moniz, P.  Prospecting black hole thermodynamics with fractional quantum mechanics.  Eur. Phys. J. C 2021, 81, 632.  https://doi.org/10.1140/epjc/s10052-021-09438-5.
  * Jalalzadeh et al. [2022] Jalalzadeh, S.; Costa, E.O.; Moniz, P.  de Sitter fractional quantum cosmology.  Phys. Rev. D 2022, 105, L121901.  https://doi.org/10.1103/PhysRevD.105.L121901.
  * Rasouli et al. [2022] Rasouli, S.M.M.; Costa, E.W.O.; Moniz, P.V.; Jalalzadeh, S.  Inflation and fractional quantum cosmology.  Fractal Fract. 2022, 6, 655.  https://doi.org/10.3390/fractalfract6110655.
  * Jalalzadeh and Vargas Moniz [2022] Jalalzadeh, S.; Vargas Moniz, P.  Challenging Routes in Quantum Cosmology; World Scientific: Singapore, 2022\.  https://doi.org/10.1142/8540.
  * Laskin [2000] Laskin, N.  Fractional Quantum Mechanics.  Phys. Rev. E 2000, 62, 3135–3145.  https://doi.org/10.1103/PhysRevE.62.3135.
  * Laskin [2002] Laskin, N.  Fractional Schrodinger equation.  Phys. Rev. E 2002, 66, 056108.  https://doi.org/10.1103/PhysRevE.66.056108.
  * Laskin [2010] Laskin, N.  Principles of Fractional Quantum Mechanics. _arXiv_ 2010, arXiv:1009.5533.
  * Calcagni [2017] Calcagni, G.  Classical and Quantum Cosmology; Graduate Texts in Physics; Springer: Berlin/Heidelberg, Germany, 2017.  https://doi.org/10.1007/978-3-319-41127-9.
  * Rasouli et al. [2011] Rasouli, S.M.M.; Farhoudi, M.; Khosravi, N.  Horizon Problem Remediation via Deformed Phase Space.  Gen. Rel. Grav. 2011, 43, 2895–2910.  https://doi.org/10.1007/s10714-011-1208-4.
  * Rasouli et al. [2014] Rasouli, S.M.M.; Ziaie, A.H.; Marto, J.; Moniz, P.V.  Gravitational Collapse of a Homogeneous Scalar Field in Deformed Phase Space.  Phys. Rev. D 2014, 89, 044028.  https://doi.org/10.1103/PhysRevD.89.044028.
  * Jalalzadeh et al. [2014] Jalalzadeh, S.; Rasouli, S.M.M.; Moniz, P.V.  Quantum cosmology, minimal length and holography.  Phys. Rev. D 2014, 90, 023541.  https://doi.org/10.1103/PhysRevD.90.023541.
  * Rasouli and Vargas Moniz [2014] Rasouli, S.M.M.; Vargas Moniz, P.  Noncommutative minisuperspace, gravity-driven acceleration, and kinetic inflation.  Phys. Rev. D 2014, 90, 083533.  https://doi.org/10.1103/PhysRevD.90.083533.
  * Rasouli et al. [2016] Rasouli, S.M.M.; Ziaie, A.H.; Jalalzadeh, S.; Moniz, P.V.  Non-singular Brans–Dicke collapse in deformed phase space.  Annals Phys. 2016, 375, 154–178.  https://doi.org/10.1016/j.aop.2016.09.007.
  * Rasouli and Vargas Moniz [2016] Rasouli, S.M.M.; Vargas Moniz, P.  Gravity-Driven Acceleration and Kinetic Inflation in Noncommutative Brans-Dicke Setting.  Odessa Astron. Pub. 2016, 29, 19.  https://doi.org/10.18524/1810-4215.2016.29.84956.
  * Rasouli et al. [2018] Rasouli, S.M.M.; Saba, N.; Farhoudi, M.; Marto, J.a.; Moniz, P.V.  Inflationary Universe in Deformed Phase Space Scenario.  Annals Phys. 2018, 393, 288–307.  https://doi.org/10.1016/j.aop.2018.04.014.
  * Rasouli et al. [2019] Rasouli, S.M.M.; Marto, J.a.; Vargas Moniz, P.  Kinetic inflation in deformed phase space Brans–Dicke cosmology.  Phys. Dark Univ. 2019, 24, 100269.  https://doi.org/10.1016/j.dark.2019.100269.
  * Rasouli [2022] Rasouli, S.M.M.  Noncommutativity, Sáez–Ballester Theory and Kinetic Inflation.  Universe 2022, 8, 165.  https://doi.org/10.3390/universe8030165.
  * Shchigolev [2011] Shchigolev, V.  Cosmological models with fractional derivatives and fractional action functional.  Commun. Theor. Phys. 2011, 56, 389.  https://doi.org/10.1088/0253-6102/56/2/34.
  * Varieschi [2021] Varieschi, G.U.  Relativistic Fractional-Dimension Gravity.  Universe 2021, 7, 387.  https://doi.org/10.3390/universe7100387.
  * García-Aspeitia et al. [2022] García-Aspeitia, M.A.; Fernandez-Anaya, G.; Hernández-Almada, A.; Leon, G.; Magaña, J.  Cosmology under the fractional calculus approach.  Mon. Not. Roy. Astron. Soc. 2022, 517, 4813–4826.  https://doi.org/10.1093/mnras/stac3006.
  * Leon et al. [2023] Leon, G.; García-Aspeitia, M.A.; Fernandez-Anaya, G.; Hernández-Almada, A.; Magaña, J.; González, E.  Cosmology under the fractional calculus approach: A possible $H_{0}$ tension resolution? _arXiv_ 2023, arXiv:2304.14465.
  * Shchigolev [2013a] Shchigolev, V.K.  Cosmic Evolution in Fractional Action Cosmology.  Discontinuitynlinearity Complex. 2013, 2, 115–123.  https://doi.org/10.5890/DNC.2013.04.002.
  * Shchigolev [2013b] Shchigolev, V.K.  Fractional Einstein-Hilbert Action Cosmology.  Mod. Phys. Lett. A 2013, 28, 1350056.  https://doi.org/10.1142/S0217732313500569.
  * Calcagni [2013] Calcagni, G.  Multi-scale gravity and cosmology.  JCAP 2013, 12, 041.  https://doi.org/10.1088/1475-7516/2013/12/041.
  * Shchigolev [2016] Shchigolev, V.K.  Testing Fractional Action Cosmology.  Eur. Phys. J. Plus 2016, 131, 256.  https://doi.org/10.1140/epjp/i2016-16256-6.
  * Calcagni [2017] Calcagni, G.  Multifractional theories: An unconventional review.  JHEP 2017, 3, 138. Erratum in 2017, _6_ , 20. https://doi.org/10.1007/JHEP03(2017)138.
  * Shchigolev [2021] Shchigolev, V.K.  Fractional-order derivatives in cosmological models of accelerated expansion.  Mod. Phys. Lett. A 2021, 36, 2130014.  https://doi.org/10.1142/S0217732321300147.
  * Calcagni and De Felice [2020] Calcagni, G.; De Felice, A.  Dark energy in multifractional spacetimes.  Phys. Rev. D 2020, 102, 103529.  https://doi.org/10.1103/PhysRevD.102.103529.
  * Calcagni [2021a] Calcagni, G.  Multifractional theories: An updated review.  Mod. Phys. Lett. A 2021, 36, 2140006.  https://doi.org/10.1142/S021773232140006X.
  * Calcagni [2021b] Calcagni, G.  Classical and quantum gravity with fractional operators.  Class. Quant. Grav. 2021, 38, 165005. Erratum in 2021, _38_ , 169601. https://doi.org/10.1088/1361-6382/ac1bea.
  * González et al. [2023] González, E.; Leon, G.; Fernandez-Anaya, G.  Exact solutions and cosmological constraints in fractional cosmology.  Fractal Fract. 2023, 7, 368.  https://doi.org/10.3390/fractalfract7050368.
  * Socorro and Rosales [2023] Socorro, J.; Rosales, J.J.  Quantum Fractionary Cosmology: K-Essence Theory.  Universe 2023, 9, 185.  https://doi.org/10.3390/universe9040185.
  * Calcagni and Kuroyanagi [2021] Calcagni, G.; Kuroyanagi, S.  Stochastic gravitational-wave background in quantum gravity. JCAP 2021, 3, 019.  https://doi.org/10.1088/1475-7516/2021/03/019.
  * Calcagni et al. [2019] Calcagni, G.; Kuroyanagi, S.; Marsat, S.; Sakellariadou, M.; Tamanini, N.; Tasinato, G.  Quantum gravity and gravitational-wave astronomy.  JCAP 2019, 10, 012.  https://doi.org/10.1088/1475-7516/2019/10/012.
  * Calcagni [2017] Calcagni, G.  Complex dimensions and their observability.  Phys. Rev. D 2017, 96, 046001.  https://doi.org/10.1103/PhysRevD.96.046001.
  * Calcagni et al. [2016] Calcagni, G.; Kuroyanagi, S.; Tsujikawa, S.  Cosmic microwave background and inflation in multi-fractional spacetimes.  JCAP 2016, 8, 039.  https://doi.org/10.1088/1475-7516/2016/08/039.
  * El-Nabulsi [2012] El-Nabulsi, R.A.  Gravitons in fractional action cosmology.  Int. J. Theor. Phys. 2012, 51, 3978–3992.  https://doi.org/10.1007/s10773-012-1290-8.
  * El-Nabulsi [2016] El-Nabulsi, R.A.  A Cosmology Governed by a Fractional Differential Equation and the Generalized Kilbas-Saigo-Mittag-Leffler Function.  Int. J. Theor. Phys. 2016, 55, 625–635.  https://doi.org/10.1007/s10773-015-2700-5.
  * Jamil et al. [2012] Jamil, M.; Momeni, D.; Rashid, M.A.  Fractional Action Cosmology with Power Law Weight Function.  J. Phys. Conf. Ser. 2012, 354, 012008.  https://doi.org/10.1088/1742-6596/354/1/012008.
  * El-Nabulsi [2013a] El-Nabulsi, R.A.  Nonstandard fractional exponential Lagrangians, fractional geodesic equation, complex general relativity, and discrete gravity.  Can. J. Phys. 2013, 91, 618–622.  https://doi.org/10.1139/cjp-2013-0145.
  * El-Nabulsi [2013b] El-Nabulsi, A.R.  Non-minimal coupling in fractional action cosmology.  Indian J. Phys. 2013, 87, 835–840.  https://doi.org/10.1007/s12648-013-0295-3.
  * Rami [2015] Rami, E.N.A.  Fractional action oscillating phantom cosmology with conformal coupling.  Eur. Phys. J. Plus 2015, 130, 102.  https://doi.org/10.1140/epjp/i2015-15102-9.
  * El-Nabulsi [2016] El-Nabulsi, R.A.  Implications of the Ornstein-Uhlenbeck-like fractional differential equation in cosmology.  Rev. Mex. Fis. 2016, 62, 240.
  * El-Nabulsi [2017a] El-Nabulsi, R.A.  Fractional Action Cosmology with Variable Order Parameter.  Int. J. Theor. Phys. 2017, 56, 1159–1182.  https://doi.org/10.1007/s10773-016-3260-z.
  * El-Nabulsi [2017b] El-Nabulsi, R.A.  Wormholes in fractional action cosmology.  Can. J. Phys. 2017, 95, 605–609.  https://doi.org/10.1139/cjp-2017-0109.
  * El-Nabulsi [2017c] El-Nabulsi, R.A.  New Metrics from a Fractional Gravitational Field.  Commun. Theor. Phys. 2017, 68, 309.  https://doi.org/10.1088/0253-6102/68/3/309.
  * Debnath et al. [2012] Debnath, U.; Jamil, M.; Chattopadhyay, S.  Fractional Action Cosmology: Emergent, Logamediate, Intermediate, Power Law Scenarios of the Universe and Generalized Second Law of Thermodynamics.  Int. J. Theor. Phys 2012, 51, 812–837.  https://doi.org/10.1007/s10773-011-0961-1.
  * Debnath et al. [2013] Debnath, U.; Chattopadhyay, S.; Jamil, M.  Fractional action cosmology: Some dark energy models in emergent, logamediate, and intermediate scenarios of the universe.  Int. J. Theor. Phys 2013, 7, 25.  https://doi.org/10.1186/2251-7235-7-25.
  * Calcagni [2010a] Calcagni, G.  Quantum field theory, gravity and cosmology in a fractal universe.  JHEP 2010, 3, 120.  https://doi.org/10.1007/JHEP03(2010)120.
  * Calcagni [2010b] Calcagni, G.  Fractal universe and quantum gravity.  Phys. Rev. Lett. 2010, 104, 251301.  https://doi.org/10.1103/PhysRevLett.104.251301.
  * Junior et al. [2023] Junior, P.F.d.S.; Costa, E.W.d.O.; Jalalzadeh, S.  Emergence of fractal cosmic space from fractional quantum gravity.  Eur. Phys. J. Plus 2023, 138, 862.  https://doi.org/10.1140/epjp/s13360-023-04506-z.
  * Barrientos et al. [2021] Barrientos, E.; Mendoza, S.; Padilla, P.  Extending Friedmann equations using fractional derivatives using a Last Step Modification technique: The case of a matter dominated accelerated expanding Universe.  Symmetry 2021, 13, 174.  https://doi.org/10.3390/sym13020174.
  * Landim [2021] Landim, R.G.  Fractional dark energy.  Phys. Rev. D 2021, 103, 083511.  https://doi.org/10.1103/PhysRevD.103.083511.
  * Calcagni [2021] Calcagni, G.  Quantum scalar field theories with fractional operators.  Class. Quant. Grav. 2021, 38, 165006.  https://doi.org/10.1088/1361-6382/ac103c.
  * Landim [2021] Landim, R.G.  Fractional dark energy: Phantom behavior and negative absolute temperature.  Phys. Rev. D 2021, 104, 103508.  https://doi.org/10.1103/PhysRevD.104.103508.
  * Giusti [2020] Giusti, A.  MOND-like Fractional Laplacian Theory.  Phys. Rev. D 2020, 101, 124029.  https://doi.org/10.1103/PhysRevD.101.124029.
  * Torres et al. [2020] Torres, I.; Fabris, J.C.; Piattella, O.F.; Batista, A.B.  Quantum Cosmology of Fab Four John Theory with Conformable Fractional Derivative.  Universe 2020, 6, 50.  https://doi.org/10.3390/universe6040050.
  * Kilbas et al. [2006] Kilbas, A.; Srivastava, H.; Trujillo, J.  Theory and applications Of Fractional Differential Equations.  North-Holland Math. Stud. 2006, 204,7–10.  https://doi.org/10.1016/S0304-0208(06)80001-0.
  * Podlubny [1998] Podlubny, I.  Fractional Differential Equations; Elsevier: Amsterdam, The Netherlands, 1998; Volume 198.
  * Laskin [2000] Laskin, N.  Fractional quantum mechanics and Levy paths integrals.  Phys. Lett. A 2000, 268, 298–305.  https://doi.org/10.1016/S0375-9601(00)00201-2.
  * Pozrikidis [2018] Pozrikidis, C.  The Fractional Laplacian; CRC Press: Boca Raton, FL, USA, 2018.  https://doi.org/10.1201/9781315367675.
  * Laskin [2018] Laskin, N.  Fractional Quantum Mechanics; World Scientific: Singapore, 2018.  https://doi.org/10.1142/10541.
  * Riesz [1949] Riesz, M.  L’intégrale de Riemann-Liouville et le problème de Cauchy.  Acta Mathematica 1949, 81, 1 – 222.
  * Tarasov [2018] Tarasov, V.E.  Fractional Derivative Regularization in QFT.  Adv. High Energy Phys. 2018, 2018, 7612490.  https://doi.org/10.1155/2018/7612490.
  * Batalin and Vilkovisky [1977] Batalin, I.A.; Vilkovisky, G.A.  Relativistic S Matrix of Dynamical Systems with Boson and Fermion Constraints.  Phys. Lett. B 1977, 69, 309–312.  https://doi.org/10.1016/0370-2693(77)90553-6.
  * Hawking and Ellis [1973] Hawking, S.W.; Ellis, G.F.R.  The Large Scale Structure of Space-Time; Cambridge Monographs on Mathematical Physics; Cambridge University Press: Cambridge, UK, 1973.  https://doi.org/10.1017/CBO9780511524646.
  * Fathi and Jalalzadeh [2017] Fathi, M.; Jalalzadeh, S.  Quantum Hamilton-Jacobi cosmology and classical-quantum correlation.  Int. J. Theor. Phys. 2017, 56, 2167–2177.  https://doi.org/10.1007/s10773-017-3363-1.
  * Rashki and Jalalzadeh [2017] Rashki, M.; Jalalzadeh, S.  The Quantum State Of The Universe From Deformation Quantization and Classical-Quantum Correlation.  Gen. Rel. Grav. 2017, 49, 14.  https://doi.org/10.1007/s10714-016-2178-3.
  * Jalalzadeh and Moniz [2014] Jalalzadeh, S.; Moniz, P.V.  Dirac observables and boundary proposals in quantum cosmology.  Phys. Rev. D 2014, 89, 083504.  https://doi.org/10.1103/PhysRevD.89.083504.
  * Fathi et al. [2016] Fathi, M.; Jalalzadeh, S.; Moniz, P.V.  Classical Universe emerging from quantum cosmology without horizon and flatness problems.  Eur. Phys. J. C 2016, 76, 527.  https://doi.org/10.1140/epjc/s10052-016-4373-5.
  * Rashki and Jalalzadeh [2015] Rashki, M.; Jalalzadeh, S.  Holography from quantum cosmology.  Phys. Rev. D 2015, 91, 023501.  https://doi.org/10.1103/PhysRevD.91.023501.
  * Benedetti and Petronio [1992] Benedetti, R.; Petronio, C.  Lectures on Hyperbolic Geometry; Springer: Berlin/Heidelberg, Germany, 1992.  https://doi.org/10.1007/978-3-642-58158-8_1.
  * Farooq and Ratra [2013] Farooq, O.; Ratra, B.  Hubble parameter measurement constraints on the cosmological deceleration-acceleration transition redshift.  Astrophys. J. Lett. 2013, 766, L7.  https://doi.org/10.1088/2041-8205/766/1/L7.
  * Scolnic et al. [2018] Scolnic, D.M.; et al.  The Complete Light-curve Sample of Spectroscopically Confirmed SNe Ia from Pan-STARRS1 and Cosmological Constraints from the Combined Pantheon Sample.  Astrophys. J. 2018, 859, 101.  https://doi.org/10.3847/1538-4357/aab9bb.
  * Camarena and Marra [2018] Camarena, D.; Marra, V.  Impact of the cosmic variance on $H_{0}$ on cosmological analyses.  Phys. Rev. D 2018, 98, 023537.  https://doi.org/10.1103/PhysRevD.98.023537.
  * Hinshaw et al. [2013] Hinshaw, G.; et al.  Nine-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Cosmological Parameter Results.  Astrophys. J. Suppl. 2013, 208, 19.  https://doi.org/10.1088/0067-0049/208/2/19.
  * Chen et al. [2019] Chen, L.; Huang, Q.G.; Wang, K.  Distance Priors from Planck Final Release.  JCAP 2019, 2, 028.  https://doi.org/10.1088/1475-7516/2019/02/028.
  * Davari and Rahvar [2021] Davari, Z.; Rahvar, S.  MOG cosmology without dark matter and the cosmological constant.  Mon. Not. Roy. Astron. Soc. 2021, 507, 3387–3399.  https://doi.org/10.1093/mnras/stab2350.
  * Serra et al. [2009] Serra, P.; Cooray, A.; Holz, D.E.; Melchiorri, A.; Pandolfi, S.; Sarkar, D.  No Evidence for Dark Energy Dynamics from a Global Analysis of Cosmological Data.  Phys. Rev. D 2009, 80, 121302.  https://doi.org/10.1103/PhysRevD.80.121302.
  * Marra and Sapone [2018] Marra, V.; Sapone, D.  Null tests of the standard model using the linear model formalism.  Phys. Rev. D 2018, 97, 083510.  https://doi.org/10.1103/PhysRevD.97.083510.


* [4] Konikowska B, Penczek W.  Model checking of multivalued logic of knowledge and time.  In: Proceedings of AAMAS. ACM Press, 2006 pp. 169–176. doi:10.1145/1160633.1160661.
  * [5] Bauer A, Leucker M, Schallhart C.  Monitoring of Real-Time Properties.  In: Proceedings of FSTTCS, volume 4337 of _Lecture Notes in Computer Science_. Springer, 2006 pp. 260–272.  10.1007/11944836_25.
  * [6] Bauer A, Leucker M, Schallhart C.  The Good, the Bad, and the Ugly, But How Ugly Is Ugly?  In: Proceedings of RV, volume 4839 of _Lecture Notes in Computer Science_. Springer, 2007 pp. 126–138.  10.1007/978-3-540-77395-5_11.
  * [7] Biere A, Cimatti A, Clarke E, Zhu Y.  Symbolic Model Checking Without BDDs.  In: Proceedings of TACAS, volume 1579 of _Lecture Notes in Computer Science_. Springer, 1999 pp. 193–207. doi:10.1007/3-540-49059-0_14.
  * [8] Penczek W, Lomuscio A.  Verifying Epistemic Properties of Multi-Agent Systems via Bounded Model Checking.  In: Proceedings of AAMAS. ACM Press, 2003 pp. 209–216.
  * [9] Lomuscio A, Qu H, Raimondi F.  MCMAS: An Open-Source Model Checker for the Verification of Multi-Agent Systems.  _International Journal on Software Tools for Technology Transfer_ , 2015. Availabe online.  10.1007/s10009-015-0378-x.
  * [10] Jamroga W, Konikowska B, Penczek W.  Multi-Valued Verification of Strategic Ability.  In: Proceedings of AAMAS. ACM Press, 2016 pp. 1180–1189.
  * [11] Fitting M.  Many-Valued Modal Logics.  _Fundamenta Informaticae_ , 1991.  15(3-4):335–350.
  * [12] Fitting M.  Many-Valued Modal Logics II.  _Fundamenta Informaticae_ , 1992.  17:55–73.
  * [13] Vijzelaar S, Fokkink W.  Multi-valued Abstraction Using Lattice Operations.  In: Proceedings of ACSD. 2015 pp. 70–79.  10.1109/ACSD.2015.18.
  * [14] Gurfinkel A, Chechik M.  Multi-Valued Model Checking via Classical Model Checking.  In: Proceedings of CONCUR, volume 2761 of _LNCS_. Springer-Verlag, 2003 pp. 266–280. doi:10.1007/978-3-540-45187-7_18.
  * [15] Bruns G, Godefroid P.  Model Checking with Multi-valued Logics.  In: ICALP. 2004 pp. 281–293. doi:10.1007/978-3-540-27836-8_26.
  * [16] Shoham S, Grumberg O.  Multi-valued model checking games.  _J. Comput. Syst. Sci._ , 2012.  78(2):414–429. doi:10.1016/j.jcss.2011.05.003.
  * [17] Pan H, Li Y, Cao Y, Ma Z.  Model Checking Computation Tree Logic over Finite Lattices.  _Theoretical Computer Science_ , 2016.  612:45–62. doi:10.1016/j.tcs.2015.10.014.
  * [18] B Aminof AM O Kupferman.  Improved model checking of hierarchical systems.  _Information Computation_ , 2012.  210:68–86. doi:10.1016/j.ic.2011.10.008.
  * [19] Kwiatkowska M, Norman G, Parker D.  PRISM: Probabilistic Symbolic Model Checker.  In: Proceedings of TOOLS, volume 2324 of _Lecture Notes in Computer Science_. Springer, 2002 pp. 200–204.  10.1007/3-540-46029-2_13.
  * [20] Godefroid P, Jagadeesan R.  On the Expressiveness of 3-Valued Models.  In: Proceedings of VMCAI, volume 2575 of _LNCS_. Springer-Verlag, 2003 pp. 206–222. doi:10.1007/3-540-36384-X_18.
  * [21] Huth M, Jagadeesan R, Schmidt DA.  A Domain Equation for Refinement of Partial Systems.  _Mathematical Structures in Computer Science_ , 2004.  14:469–505. doi:10.1017/S0960129504004268.
  * [22] Huth M, Pradhan S.  Consistent Partial Model Checking.  In: Proc. of the Workshop on Domains VI, volume 73 of _ENTCS_. Elsevier, 2004 pp. 45–85. doi:10.1016/j.entcs.2004.08.003.
  * [23] Kouvaros P, Lomuscio A.  Parameterised Verification of Infinite State Multi-Agent Systems via Predicate Abstraction.  In: Proceedings of AAAI. 2017 pp. 3013–3020.
  * [24] Belardinelli F, Lomuscio A, Malvone V.  Approximating Perfect Recall When Model Checking Strategic Abilities.  In: Principles of Knowledge Representation and Reasoning: Proceedings of KR. 2018 pp. 435–444. ID:52969829.
  * [25] Belardinelli F, Lomuscio A, Malvone V.  An Abstraction-Based Method for Verifying Strategic Properties in Multi-Agent Systems with Imperfect Information.  In: The Thirty-Third AAAI Conference on Artificial Intelligence, AAAI 2019, The Thirty-First Innovative Applications of Artificial Intelligence Conference, IAAI 2019, The Ninth AAAI Symposium on Educational Advances in Artificial Intelligence, EAAI 2019. 2019 pp. 6030–6037. doi:10.1609/aaai.v33i01.33016030.
  * [26] Maluszynski J, Szalas A.  Logical foundations and complexity of 4QL, a query language with unrestricted negation.  _Journal of Applied Non-Classical Logics_ , 2011.  21(2):211–232.  10.3166/jancl.21.211-232.
  * [27] Maluszynski J, Szalas A.  Partiality and Inconsistency in Agents’ Belief Bases.  In: Proceedings of KES-AMSTA, volume 252 of _Frontiers in Artificial Intelligence and Applications_. IOS Press, 2013 pp. 3–17.  10.3233/978-1-61499-254-7-3.
  * [28] Ball T, Kupferman O.  An Abstraction-Refinement Framework for Multi-Agent Systems.  In: Proceedings of LICS. IEEE Computer Society, 2006 pp. 379–388.  10.1109/LICS.2006.10.
  * [29] Lomuscio A, Michaliszyn J.  Verifying Multi-Agent Systems by Model Checking Three-valued Abstractions.  In: Proceedings of AAMAS. 2015 pp. 189–198.
  * [30] Belardinelli F, Lomuscio A.  Agent-based Abstractions for Verifying Alternating-time Temporal Logic with Imperfect Information.  In: Proceedings of AAMAS. 2017 pp. 1259–1267.
  * [31] de Alfaro L, Faella M, Henzinger T, Majumdar R, Stoelinga M.  Model checking discounted temporal properties.  _Theoretical Computer Science_ , 2005.  345:139–170. doi:10.1016/j.tcs.2005.07.033.
  * [32] Lluch-Lafuente A, Montanari U.  Quantitative $\mu$-Calculus and CTL Based on Constraint Semirings.  _Electr. Notes Theor. Comput. Sci._ , 2005.  112:37–59. doi:10.1016/j.entcs.2004.02.063.
  * [33] Jamroga W.  A Temporal Logic for Markov Chains.  In: Proceedings of AAMAS. 2008 pp. 697–704.
  * [34] Jamroga W.  A Temporal Logic for Stochastic Multi-Agent Systems.  In: Proceedings of PRIMA, volume 5357 of _Lecture Notes in Computer Science_. 2008 pp. 239–250. doi:10.1007/978-3-540-89674-6_27.
  * [35] Huth M, Kwiatkowska M.  Quantitative Analysis and Model Checking.  In: Logic in Computer Science. 1997 pp. 111–122. doi:10.1109/LICS.1997.614940.
  * [36] Baier C, Kwiatkowska M, Norman G.  Computing Probability Bounds for Linear Time Formulas over Concurrent Probabilistic Systems.  _Electronic Notes in Theoretical Computer Science_ , 1999.  21(19). doi:10.1016/S1571-0661(05)80595-X.
  * [37] Bulling N, Jamroga W.  What Agents Can Probably Enforce.  _Fundamenta Informaticae_ , 2009.  93(1-3):81–96. doi:10.3233/FI-2009-0089.
  * [38] Huang X, Su K, Zhang C.  Probabilistic Alternating-Time Temporal Logic of Incomplete Information and Synchronous Perfect Recall.  In: Proceedings of AAAI. 2012.
  * [39] Chen T, Forejt V, Kwiatkowska M, Parker D, Simaitis A.  PRISM-games: A Model Checker for Stochastic Multi-Player Games.  In: Proceedings of TACAS, volume 7795 of _Lecture Notes in Computer Science_. Springer, 2013 pp. 185–191. doi:10.1007/978-3-642-36742-7_13.
  * [40] Birkhoff G.  Lattice Theory (2nd edition).  American Mathematical Society, 1948.
  * [41] Davey BA, Priestley HA.  Introduction to Lattices and Order.  Cambridge University Press, 1990. ISBN:10:0521367662, 13:978-0521367660.
  * [42] Cousot P, Cousot R.  Abstract Interpretation: A Unified Lattice Model for Static Analysis of Programs by Construction or Approximation of Fixpoints.  In: Conference Record of the Fourth ACM Symposium on Principles of Programming Languages. 1977 pp. 238–252.  10.1145/512950.512973.
  * [43] Clarke E, Grumberg O, Long D.  Model Checking and Abstraction.  _ACM Transactions on Programming Languages and Systems_ , 1994.  16(5):1512–1542. doi:10.1145/186025.186051.
  * [44] Schobbens PY.  Alternating-Time Logic with Imperfect Recall.  _Electronic Notes in Theoretical Computer Science_ , 2004.  85(2):82–93. doi:10.1016/S1571-0661(05)82604-0.
  * [45] Hansson H, Jonsson B.  A Logic for Reasoning about Time and Reliability.  _Formal Asp. Comput._ , 1994.  6(5):512–535.  10.1007/BF01211866.
  * [46] Godefroid P, Jagadeesan R.  Automatic Abstraction Using Generalized Model Checking.  In: Proceedings of CAV, volume 2404 of _Lecture Notes in Computer Science_. Springer, 2002 pp. 137–150.  10.1007/3-540-45657-0_11.
  * [47] Jamroga W.  Some Remarks on Alternating Temporal Epistemic Logic.  In: Dunin-Keplicz B, Verbrugge R (eds.), Proceedings of FAMAS. 2003 pp. 133–140.
  * [48] Ågotnes T.  A note on Syntactic Characterization of Incomplete Information in ATEL.  In: Proceedings of Workshop on Knowledge and Games. 2004 pp. 34–42. ID:17497178.
  * [49] Jamroga W, van der Hoek W.  Agents that Know how to Play.  _Fundamenta Informaticae_ , 2004.  63(2–3):185–219.
  * [50] Ågotnes T.  Action and Knowledge in Alternating-time Temporal Logic.  _Synthese_ , 2006.  149(2):377–409. doi:10.1007/s11229-005-3875-8.
  * [51] Jamroga W, Ågotnes T.  Constructive Knowledge: What Agents Can Achieve under Incomplete Information.  _Journal of Applied Non-Classical Logics_ , 2007.  17(4):423–475. doi:10.3166/jancl.17.423-475.
  * [52] Schnoor H.  Strategic planning for probabilistic games with incomplete information.  In: Proceedings of AAMAS. 2010 pp. 1057–1064. doi:10.1145/1838206.1838349.
  * [53] Jamroga W, Dix J.  Model Checking Abilities of Agents: A Closer Look.  _Theory of Computing Systems_ , 2008.  42(3):366–410. doi:10.1007/s00224-007-9080-z.
  * [54] Bulling N, Dix J, Jamroga W.  Model Checking Logics of Strategic Ability: Complexity.  In: Dastani M, Hindriks K, Meyer JJ (eds.), Specification and Verification of Multi-Agent Systems, pp. 125–159. Springer, 2010. doi:10.1007/978-1-4419-6984-2_5.
  * [55] Dima C, Tiplea F.  Model-checking ATL under Imperfect Information and Perfect Recall Semantics is Undecidable.  _CoRR_ , 2011.  abs/1102.4225.
  * [56] Bulling N, Jamroga W.  Comparing Variants of Strategic Ability: How Uncertainty and Memory Influence General Properties of Games.  _Journal of Autonomous Agents and Multi-Agent Systems_ , 2014.  28(3):474–518. doi:10.1007/s10458-013-9231-3.
  * [57] Pilecki J, Bednarczyk M, Jamroga W.  Synthesis and Verification of Uniform Strategies for Multi-Agent Systems.  In: Proceedings of CLIMA XV, volume 8624 of _Lecture Notes in Computer Science_. Springer, 2014 pp. 166–182. doi:10.1007/978-3-319-09764-0_11.
  * [58] Busard S, Pecheur C, Qu H, Raimondi F.  Improving the Model Checking of Strategies under Partial Observability and Fairness Constraints.  In: Formal Methods and Software Engineering, volume 8829 of _Lecture Notes in Computer Science_ , pp. 27–42. Springer.  ISBN 978-3-319-11736-2, 2014.  10.1007/978-3-319-11737-9_3.
  * [59] Huang X, van der Meyden R.  Symbolic Model Checking Epistemic Strategy Logic.  In: Proceedings of AAAI Conference on Artificial Intelligence. AAAI Press, 2014 pp. 1426–1432.
  * [60] Cermak P, Lomuscio A, Mogavero F, Murano A.  MCMAS-SLK: A Model Checker for the Verification of Strategy Logic Specifications.  In: Proc. of CAV’14, volume 8559 of _LNCS_. Springer-Verlag, 2014 pp. 525–532. doi:10.1007/978-3-319-08867-9_34.
  * [61] Jamroga W, Knapik M, Kurpiewski D, Mikulski L.  Approximate verification of strategic abilities under imperfect information.  _Artif. Intell._ , 2019.  277.  10.1016/j.artint.2019.103172.
  * [62] Kurpiewski D, Knapik M, Jamroga W.  On Domination and Control in Strategic Ability.  In: Proceedings of AAMAS. 2019 pp. 197–205. ID:169031456.
  * [63] Jamroga W, Knapik M, Kurpiewski D.  Fixpoint Approximation of Strategic Abilities under Imperfect Information.  In: Proceedings of AAMAS. IFAAMAS, 2017 pp. 1241–1249.
  * [64] Ågotnes T, Goranko V, Jamroga W, Wooldridge M.  Knowledge and Ability.  In: van Ditmarsch H, Halpern J, van der Hoek W, Kooi B (eds.), Handbook of Epistemic Logic, pp. 543–589. College Publications, 2015.
  * [65] Jamroga W.  Logical Methods for Specification and Verification of Multi-Agent Systems.  ICS PAS Publishing House, 2015.  ISBN:978-83-63159-25-2.
  * [66] Jamroga W, Dix J.  Model Checking ATLir is Indeed $\Delta_{2}^{P}$-complete.  In: Proceedings of EUMAS, volume 223 of _CEUR Workshop Proceedings_. 2006.
  * [67] Busard S.  Symbolic Model Checking of Multi-Modal Logics: Uniform Strategies and Rich Explanations.  Ph.D. thesis, Universite Catholique de Louvain, 2017. URL http://hdl.handle.net/2078.1/186372.
  * [68] Mogavero F, Murano A, Vardi M.  Reasoning About Strategies.  In: Proceedings of FSTTCS. 2010 pp. 133–144. doi:10.4230/LIPIcs.FSTTCS.2010.133.
  * [69] Mogavero F, Murano A, Perelli G, Vardi M.  Reasoning About Strategies: On the Model-Checking Problem.  _ACM Transactions on Computational Logic_ , 2014.  15(4):1–42. doi:10.1145/2631917.
  * [70] Berthon R, Maubert B, Murano A, Rubin S, Vardi MY.  Strategy logic with imperfect information.  In: Proceedings of LICS. 2017 pp. 1–12.  10.1109/LICS.2017.8005136.