# Friedmann equations of the fractal apparent horizon

R. Jalalzadeh<EMAIL_ADDRESS>Department of Physics, Faculty of
Science, University of Kurdistan, Pasdaran Street, P.O. Box 66177-15175
Sanandaj, Iran  S. Jalalzadeh<EMAIL_ADDRESS>Departamento de
Fisica, Universidade Federal de Pernambuco, Recife, PE 50670-901, Brazil
Center for Theoretical Physics, Khazar University, 41 Mehseti Street, AZ1096,
Azerbaijan  A. Sayahian Jahromi<EMAIL_ADDRESS>Zarghan Branch, Islamic
Azad University, Zarghan, Iran  H. Moradpour<EMAIL_ADDRESS>Research
Institute for Astronomy and Astrophysics of Maragha (RIAAM), University of
Maragheh, P.O. Box 55136-553, Maragheh, Iran

###### Abstract

From a fractal perspective, the entropy bound of gravitational systems
undergoes changes. Furthermore, in the cosmological setting, the conservation
law of a perfect fluid is also altered in such systems, affecting spatial
elements like volume, area, and radius. By applying the first law of
thermodynamics and deriving the Friedmann equations, we can gain insight into
the evolution of such a fractal cosmos. However, observations continue to
necessitate the existence of a dark energy source. To address this, in this
article, we have created a novel fractal $\Lambda$CDM cosmological model and
determined the fractal cosmological observables. We show that the spatial
fractal dimension is two, and the age of the Universe is 13.91 Gyr, by fitting
the model’s parameters to cosmological data.

###### keywords:

Friedmann equations, Fractals, Fractional derivative , Apparent horizon ,
Thermodynamics, Cosmology

††journal: Physics of the Dark Universe

##  1 Introduction

The laws of thermodynamics have shed light on the connection between gravity
and thermodynamics, especially as it relates to black holes (BH)
thermodynamics. Initially, the study of the thermodynamic aspects of gravity
was limited to static horizons and BH thermodynamics. However, further
attempts were made to explore dynamic cases. Based on these investigations, it
is possible to think of the gravitational field equations as thermodynamic
equations of state. Remarkably, thermodynamics principles may be used to
produce the Friedman equations in a homogeneous and isotropic cosmology. The
energy-momentum conservation law is a significant factor in this, as the first
rule of thermodynamics is a conservation law [1].

If one takes a look at cosmological models with horizons, there is a
resemblance to thermodynamic systems [1], which makes it possible to associate
entropy and temperature with them, just as with BHs. Because the laws of
thermodynamics have proven to be remarkably universal, cosmological models
that fulfil both the laws of thermodynamics and observational data are
generally regarded as more acceptable than models that only satisfy
observational tests.

The equivalency between the Friedmann equation and the first law of
thermodynamics on the apparent horizon in the context of general relativity,
and also in the Gauss–Bonnet and Lovelock gravity theories, was originally
established by Cai and Kim [2]. Later, this study was extended to include
$f(R)$ gravity and scalar-tensor gravity theories by Akbar and Cai [3]. A
mass-like function (with energy dimension) was later presented by Gong and
Wang [4]; this function corresponded to the Misner–Sharp–Hernandez mass on the
apparent horizon. In this way, the Friedmann equations were used to derive the
first law of thermodynamics.

Conversely, the cosmological (the Friedmann and its extensions) equations may
be obtained by extending the first law of thermodynamics to the Universe’s
horizon. Note that this procedure is still only a hypothesis and not a proven
theorem at this time. However, if the correct modified entropy relation is
used for each specific theory, this method seems to produce accurate findings
in many cases of modified gravitational theories. By applying this method, we
will be able to derive new, modified Friedmann equations that, up to a
specific limit–that is, the point at which the generalised entropy assumes the
standard form–include the standard equations. However, these modified
equations introduce new terminology that is used in the broader context. In
this manner, the Tsallis entropy [5], the Rényi entropy [6], the Barrow
entropy [7], the Sharma–Mittal entropy [8], the Kaniadakis entropy [9], and
the quantum deformed (q-deformed) entropy [10], have been used to construct
new modified cosmological models.

Finding a suitable boundary for the study of the thermodynamics of the cosmos
is an essential step that requires careful consideration. The apparent horizon
of the homogeneous and isotropic universe is the causal boundary that
satisfies the laws of thermodynamics. This claim has been supported by many
references, including [11, 12, 13, 14, 15, 16]. Notably, the Friedmann
equations result from the first law of thermodynamics being available on the
apparent horizon by employing the Bekenstein entropy. Our view of the Universe
is greatly affected by this relationship between thermodynamics and cosmology
[14, 15, 16].

Moreover, the role of horizon entropy cannot be overstated, as it forms the
foundation of investigations into thermodynamics. Thermodynamics relies
significantly on entropy, and changes to the Bekenstein entropy can similarly
affect the Friedmann equations. This idea has been explored in various
theories that propose alternative versions of the Bekenstein entropy, as
evidenced by references such as [1, 17, 18, 19, 20, 21, 22]. These theories
offer new perspectives on the thermodynamics of the cosmos and contribute to
the ongoing scientific discourse in this field.

Fractional calculus aims to expand classical calculus by including
differentiation and integration principles and techniques that cover non-
integer orders. In traditional calculus, differentiation and integration are
limited to integer orders and do not account for non-integer or irrational
orders. Fractional calculus has several applications in gravity and cosmology,
which have been extensively researched. Its effectiveness in addressing
gravitational forces and cosmological models makes it an essential tool for
scientific inquiry [23, 24, 25, 26, 27, 28, 29, 30, 31, 32]. These
applications include the stochastic gravitational wave background in quantum
gravity [33], gravitational-wave luminosity distance [34], CMB spectrum and
inflation [35, 36], cosmology with fractional action [37, 38], discrete
gravity [39], chaotic inflation with a non-minimal coupling field [40],
phantom cosmology with a conformal coupling field [41], cosmology with
Ornstein–Uhlenbeck like fractional differential equation [42], fractional
action cosmology where the order parameter is variable [43], and wormholes in
fractional action cosmology [44].

Additionally, new metrics and models of dark energy have been considered in
emergent, logamediate, and intermediate models of the Universe [30].
Considering space-time is fractal, Refs. [45, 4] have established several
exact solutions for cosmological models that do not match the standard one.
Riesz’s fractional derivative is used in Ref. [26] to determine the non-
boundary and tunnelling wavefunctions for a closed de Sitter universe. The
authors of Ref. [35] utilised fractional quantum cosmology to investigate the
pre-inflation era. The analysis of the thermodynamics of fractional BHs is
done in Ref. [46]. Furthermore, the Friedmann equations are modified using
fractional calculus to explore the Universe’s dynamics without dark energy
(DE) and cold dark matter (CDM) [47]. Furthermore, Refs. [48, 49, 47, 50] also
discuss fractional calculus in the context of quantum cosmology and modified
Newtonian dynamics (MOND).

Fractional quantum gravity is a theoretical framework that explores the
possibility of describing quantum gravity in terms of fractional derivatives
[51]. One of the intriguing concepts within this framework is the existence of
fractional-fractal BHs [46]. These BHs are thought to have properties that are
different from the classical BHs predicted by general relativity. As shown in
Ref. [51], these BHs possess a fractal structure on the apparent horizon that
is characterized by self-similarity across different scales. Additionally, the
entropy ($S$)-area($\mathcal{A}$) relationship of a fractal horizon of a
fractional-fractal BH is described by $S\sim\mathcal{A}^{\frac{d}{2}}$ in
which $d$ denotes the fractal dimension of the apparent horizon[46, 52].
Recently, an attempt has been made to determine the corresponding Friedmann
equations for the Friedmann–Lemaître–Robertson–Walker (FLRW) universe by using
this entropy-area proposal and the first law of thermodynamics on the apparent
horizon [53]. The authors of the mentioned article utilised the ordinary
energy-momentum tensor and the standard volume, area, and radius to handle
their calculations.

It is worth noting that, despite the mathematical similarity of fractional-
fractal entropy of BHs to that of the Tsallis and Barrow entropies [54, 55],
they originated from different physics [35, 26, 46, 52, 10, 23]. Moreover, the
continuity equation of a perfect fluid in fractional-fractal cosmology is
different from the conservation equation for ordinary perfect fluid [52].
Additionally, the proper volume, horizon area, and radius in fractional-
fractal cosmology stored the effects of fractionality and rescaled compared to
their ordinary counterparts [35, 26, 46, 52, 10, 23].

Henceforth, the foremost objective of this article is to ascertain the
modified Friedmann equations by employing fractal entropy and the fractal
extension of the conservation law of matter fields, along with the fractal
considerations on the apparent horizon’s volume, area, and radius.

The following is an outline of this article. In Section 2, we apply the first
law of thermodynamics of FLRW cosmology, Cai–Kim temperature, and fractal
continuity equation to the fractal volume and surface of the apparent horizon
to obtain a fractal extension of the Friedmann equations. In Section 3, we use
the obtained fractal Friedmann equations to analyse the corresponding
cosmology. We obtain the fractal deceleration parameter, the age of the
Universe, and the transition redshift. By defining fractal density parameters,
we compare our equations with cosmological data in section 4. We show that the
Universe’s age can considerably surpass the customary standard flat
$\Lambda$CDM model. In the final section, we draw our conclusions.

##  2 Fractal Friedmann equations

Numerous studies and references, such as [15, 14, 56], have been dedicated to
exploring the intricacies of the thermodynamics of FLRW spaces. To
comprehensively understand the FLRW apparent horizon’s thermodynamical
properties, one can refer to a comprehensive review presented in Ref. [56].

The line element of the FLRW universe is given by

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

where the curvature constant $k=-1,0,1$ addresses the open, flat, and closed
universes, respectively. The coordinate $r$ that moves with the Universe’s
expansion is not equivalent to the areal radius. Instead, the areal radius
(proper radius) depends on the time coordinate, $t$, and is given by
$\tilde{r}=a(t)r$. The apparent horizon (as the proper casual boundary) is
located at [11, 12, 13, 14, 15, 16]

$\displaystyle\tilde{r}_{A}=a(t)r_{A}=\frac{1}{\sqrt{H^{2}+\frac{k}{a(t)^{2}}}}.$
(2)

Indeed, $r_{A}$ is a marginally trapped surface, and it is calculated by
solving

$\displaystyle\partial_{\alpha}\tilde{r}\partial^{\alpha}\tilde{r}=0.$ (3)

In fractal cosmology, the effective area of the apparent horizon has a random
fractal structure [26, 52]. As a result, the effective areal radius, $R$, the
corresponding surface area $\mathcal{A}$, and volume, $\mathcal{V}$ are given
by [52]

$\begin{split}&R=(4\pi)^{\frac{2-\alpha}{4\alpha}}\left(\frac{\tilde{r}_{A}}{L}\right)^{\frac{2+\alpha}{2\alpha}}L,\\\
&\mathcal{A}=4\pi R^{2},\\\ &\mathcal{V}=\frac{4\pi}{3}R^{3},\end{split}$ (4)

where $\alpha$ is Lévy’s fractional parameter [57], and $L$ is a fractional
length scale that must be introduced to maintain dimensional consistency.
Notably, this scale corresponds to the grid size of the fractal structure
[52]. Also, if we assume the matter content of the model universe is a perfect
fluid with energy density $\rho$ and pressure $p$. As a result, the
Misner–Sharp–Hernandez mass of a sphere of radius $R$ is

$M_{\text{MSH}}=\rho\mathcal{V}=M_{0}\left(\frac{\tilde{r}_{A}}{L}\right)^{\frac{6+3\alpha}{2\alpha}},$
(5)

where $M_{0}$ is the grid mass, given by

$M_{0}=\frac{(4\pi)^{\frac{6+\alpha}{4\alpha}}}{3}\rho L^{3}.$ (6)

Note that for $\alpha=2$, these definitions reduce to their ordinary
counterparts in the standard model of cosmology.

It is vital to have a better comprehension of the subject at hand, and this
can only be done by realising the relevance of Eq. (5). Let us take a closer
look at its meaning. Suppose that within a volume of size $l$, we can locate
$N_{0}$ particles with a mass of $m_{0}$. We find $N_{1}=k_{N}N_{0}$ particles
if we increase the volume to a bigger size of $R_{1}=k_{r}l$. In a structure
that exhibits self-similarity, the values of $k_{r}$ and $k_{N}$ remain
consistent even when the scale is altered. Therefore, in a structure with a
size of $R_{n}=k_{r}^{n}l$, we would observe $N_{n}=k_{N}^{n}N_{0}$ particles.
Consequently, we can express the relationship between the number $N$ (or mass
$M=m_{0}N$) and the length $l$ in the form $N(R)=CR^{D}$ or
$M(R)=m_{0}N(R)=Cm_{0}R^{D}$, where $D$ denotes the fractal dimension, depends
on the rescaling factors $k_{r}$ and $k_{N}$, and $C=N_{0}/r_{0}^{D}$ gives
the prefactor $C$.

The argument presented above, along with Eq. (5), demonstrates that the
fractal dimension of the fractal cosmology is linked to Lévy’s fractional
parameter. Specifically, the relationship is given by the equation

$D=\frac{3}{2}\left(1+\frac{2}{\alpha}\right).$ (7)

Note that when $\alpha=2$, $D$ equals the ordinary dimension of three. It is
reasonable to suppose that the grid size in the fractional BH [46] is equal to
the Planck length $L=L_{\text{Pl}}$. The Planck length became less important
in later phases of cosmology, and the grid size was determined by the size of
cosmic dust distributed in a fractal pattern. Therefore, if we assume that we
can apply Eqs. (4) in fractal cosmology, we can relax the value of $D$ from
$\alpha$ and consider it as an independent parameter. When the dimension is
less than 3, we refer to it as a fractal structure. A linear distribution of
matter occurs when $D=1$, surface distribution when $D=2$, and space-filling
distribution when $D=3$ [58].

On the apparent horizon of the FLRW cosmological model, the first law of
thermodynamics is expressed as [15, 16]

$\displaystyle dE=\mathcal{A}\Psi+Wd\mathcal{V},$ (8)

where $W=-\frac{1}{2}h_{ab}T^{ab}=\frac{\rho-p}{2}$ is the work density and

$\mathcal{A}\Psi=\mathcal{A}\underbrace{(T_{a}^{b}\partial_{b}\tilde{r}_{A}+W\partial_{a}\tilde{r}_{A})}_{\psi_{a}}dx^{a}.$
(9)

Here, $\psi_{a}$ denotes the projection of $T_{\mu\nu}$ on the direction
normal to the boundary. Considering that the Cai–Kim approach [16, 59] relies
on the fundamental assumption, and since $dE=\rho
d\mathcal{V}+\mathcal{V}d\rho$, one can use Eq. (9) or directly calculate $dE-
Wd\mathcal{V}$ to obtain

$\displaystyle\mathcal{A}\Psi=dE-Wd\mathcal{V}=\mathcal{V}d\rho.$ (10)

Moreover, the common result (based on using ordinary non-fractional spatial
elements) [17] is achieved if $\alpha=2$. The entropy of a fractal horizon is
reported as [46]

$\displaystyle S_{A}=\frac{\mathcal{A}}{4G},$ (11)

and obviously, $\alpha=2$ recovers the Bekenstein entropy, a desired result.
For the Cai–Kim temperature of the horizon (which is, in fact, the
corresponding Hawking temperature of the apparent horizon), calculations lead
to [16]

$\displaystyle T=\frac{1}{2\pi\tilde{r}_{A}}.$ (12)

As it has previously been noted, the area and volume of a fractional-fractal
system differ from those of a non-fractal one because the radius is also
modified under the shadow of fractionality as given by Eq. (4)
$\tilde{r}_{A}\rightarrow R$ [52]. The latter motivates us to modify the
Cai–Kim temperature as

$\displaystyle T^{\text{(fractal)}}=\frac{1}{2\pi R}.$ (13)

It is beneficial to note here that the radius temperature dependence of
Hawking radiation of BHs experiences the same modification in the presence of
fractionality [52]. Finally, since
$T^{\text{(fractal)}}dS_{A}=-\mathcal{A}\Psi$ [16], one reaches

$\displaystyle T^{\text{(fractal)}}dS_{A}=-\mathcal{V}d\rho,$ (14)

leading (after integrating) to the following equation

$\frac{(4\pi)^{1-\frac{D}{3}}}{L^{2}}\left(\frac{L}{\tilde{r}_{A}}\right)^{\frac{2D}{3}}=\frac{8\pi
G}{3}\rho,$ (15)

where $D$ is defined by (7). The modified first Friedmann equation can be
obtained by rearranging the previous equation and substituting the definition
of the effective areal radius provided in (4)

$\displaystyle
H^{2}+\frac{k}{a(t)^{2}}=\left(\frac{2GL^{2}\rho}{3}\right)^{\frac{D}{3}-1}\frac{8\pi
G}{3}\rho.$ (16)

Combining the time derivative of this equation with the corresponding
fractional-fractal continuity equation, i.e., [46]

$\displaystyle\dot{\rho}+DH(\rho+p)=0,$ (17)

one is finally able to get the second Friedmann equation as

$\displaystyle\frac{\ddot{a}}{a}=-\left(\frac{2GL^{2}\rho}{3}\right)^{\frac{D}{3}-1}\frac{4\pi
G}{3}(\rho+p).$ (18)

It is easy to check that for $D=3$, the prefactor
$\left(\frac{2GL^{2}\rho}{3}\right)^{\frac{D}{3}-1}$ in fractal Friedmann
equations (16) and (18) reduce to unity, and we recover the standard Friedmann
equations. Note that the difference between fractal Friedmann equations
obtained here with the results of Ref. [53] is due to $i$) fractal volume and
surface area are considered here, $ii$) the re-scaled Cai–Kim temperature is
taken into account (of course, it only affects the coefficient $D$), and
$iii$) the fractional-fractal continuity equation (9) is used instead of the
ordinary continuity equation employed in Ref. [53]. It should also be noted
that, as demonstrated by Eqs. (15) and (17), the fractal structure of the
apparent horizon has impacted not only the geometric aspect (the left-hand
side) of the Friedman equation but also the distribution of matter. Thus, both
components are subject to modification.

##  3 Fractal FLRW Cosmology

The cosmic fluid is composed of three non-interacting components: the
radiation (which includes relativistic particles) with energy density
$\rho_{\text{(rad)}}$, the cosmic dust (constituted of baryonic and cold dark
matter) with an energy density $\rho_{\text{(dust)}}$, and the vacuum energy
(or the cosmological constant) in which $\rho_{(\Lambda)}=\frac{\Lambda}{8\pi
G}$, where $\Lambda$ is the cosmological constant. Therefore

$\rho=\sum_{i}\rho_{\text{(i)}}=\rho_{\text{(rad)}}+\rho_{\text{(dust)}}+\rho_{(\Lambda)}.$
(19)

These components add to the Universe’s overall energy density; radiation and
cosmic dust were the main contributors in the early stages, but as the
Universe expands, vacuum energy becomes more dominant.

Regarding the fractal Friedmann equation (16), the density parameters’
fractional-fractal extension can be defined similarly to the standard model of
cosmology

$\tilde{\Omega}^{\text{(i)}}=\left(\frac{L^{2}H^{2}}{4\pi}\right)^{1-\frac{D}{3}}\frac{8\pi
G\rho_{\text{(i)}}}{3H^{2}}=\left(\frac{L^{2}H^{2}}{4\pi}\right)^{1-\frac{D}{3}}\Omega^{\text{(i)}},$
(20)

where $\Omega^{\text{(i)}}=\frac{8\pi G\rho_{\text{(i)}}}{3H^{2}}$ denote the
usual density parameters of the standard model of cosmology. The fractional-
fractal density parameters closely resemble those in the standard model of
cosmology. However, they are modified by a prefactor
$\left(\frac{L^{2}H^{2}}{4\pi}\right)^{1-\frac{D}{3}}$. This prefactor takes
into account the effects of grid size $L$ and the fractal dimension $D$.
Utilising these fractal extensions of density parameters, the fractal
Friedmann (16) takes the following form

$\sum_{i}\tilde{\Omega}^{\text{(i)}}=\left(1-\Omega^{(k)}\right)^{\frac{D}{3}},~{}~{}~{}~{}\text{i}=\text{rad,
dust},\Lambda,$ (21)

where $\Omega^{(k)}=-\frac{k}{a^{2}H^{2}}$, is the standard curvature density
parameter.

Also, the second fractal Friedmann equation (18) gives us the deceleration
parameter

$q=-\frac{\ddot{a}}{aH^{2}}=\frac{1}{2}\left(1-\Omega^{(k)}\right)^{\frac{D}{3}}\sum_{i}(1+3\omega_{i})\tilde{\Omega}^{\text{(i)}}.$
(22)

Note that $\omega_{i}$ is the equation of state parameter of the i-component
of the cosmic fluid given by
$\omega_{i}=\frac{p_{\text{(i)}}}{\rho_{\text{(i)}}}$, where $p_{\text{(i)}}$
is the pressure of the i-component of the cosmic fluid.

In addition, employing the continuity equation (17), the fractional-fractal
Eqs. (21) and (22) can be rewritten as

$\left(\frac{H}{H_{0}}\right)^{2}=\frac{\Omega^{(k)}_{0}}{a^{2}}+\left(\sum_{i}\tilde{\Omega}^{\text{(i)}}_{0}a^{-D(1+\omega_{i})}\right)^{\frac{3}{D}}:=E^{2},$
(23)

and

$q=\frac{1}{2E^{\frac{2D}{3}}}\left(1-\frac{\Omega_{0}^{(k)}}{E^{2}a^{2}}\right)^{1-\frac{D}{3}}\sum_{i}(1+3\omega_{i})\frac{\tilde{\Omega}_{0}^{(i)}}{a^{D(1+\omega_{i})}},$
(24)

where subscript zero denotes the value of the corresponding quantity at the
present epoch. Utilising Eq. (23), one can easily find the age of the Universe
as

$t_{0}=\frac{1}{H_{0}}\displaystyle\int_{0}^{1}\frac{da}{a\left[\Omega_{0}^{\text{(k)}}a^{-2}+\left(\displaystyle\sum_{i}\tilde{\Omega}_{0}^{\text{(i)}}a^{-D(1+\omega_{i})}\right)^{\frac{3}{D}}\right]^{\frac{1}{2}}}.$
(25)

Note that based on the age-dating of field halo stars and GCs, it is strongly
suggested that the oldest stars in the Milky Way have ages that are comparable
to the age of the Universe in the standard $\Lambda$CDM model of cosmology
[60]. For instance, the age of the Population II halo in the Milky Way,
characterized by a deficiency in metals and a high velocity, is estimated to
be $14.46\pm 0.31$ Gyr, according to Bond et al. [61]. Also, utilizing new
stellar models, VandenBerg et al. [62] estimate the age to be $14.27\pm 0.80$
Gyr. These estimations of the star’s age slightly surpass ($\sim 2\sigma$) the
age of the Universe, based on CMB Planck18 data.

On the other hand, fractal cosmology has the potential to surpass the
estimated age of the Universe in the standard model of cosmology to overcome
the above difficulty in the explanation of age dating of field halo stars and
GCs. For example, let us consider a flat ($k=0$ or $\Omega^{(k)}=0$) fractal
cosmological model, and utilise the above age equation to compare the
dimensionless age of the Universe, $t_{0}H_{0}$, for three values of the
fractal dimension $D=1,2$ and 3. Fig. 1 shows the age of a fractal universe in
units of Hubble time versus $\tilde{\Omega}_{0}^{\text{(dust)}}$.

Figure 1: The age of the Universe as a function of
$\tilde{\Omega}_{0}^{\text{(dust)}}$. The line, dashed line, and dash-doted
line show the age for $D=1,2$ and 3, respectively.

As we can see in Fig. 1, the lowest and greatest values of the age are
provided for $D=3$ and $D=1$, respectively, for a given value of
$\tilde{\Omega}_{0}^{\text{(dust)}}$. This demonstrates that the standard
model of cosmology provides a smaller estimate of the age of the Universe for
a given value of the dust density parameter at the present epoch.

##  4 Data-fitting

To have a better grasp of the influence of the fractal dimension on cosmic
dynamics, it is helpful to examine the simple flat ($k=0$ or $\Omega^{(k)}=0$)
universe model with cosmic dust and the cosmological constant.

Our objective, then, is to use the Markov Chain Monte Carlo (MCMC) approach to
restrict the parameters of the fractal FLRW model at the background level
using a recent dataset of SNeIa (Pantheon+) and Observational Hubble Data
(OHD). The fitting method employed in this approach is the maximum likelihood
estimation. Following this method, the total likelihood function, denoted as
$\mathcal{L}_{\text{tot}}\backsimeq e^{-\frac{\chi^{2}_{\text{tot}}}{2}}$, is
maximized by minimizing $\chi^{2}_{\text{tot}}$. For determining
$\chi^{2}_{\text{tot}}$, the following observational dataset is utilised: the
Hubble evolution data ($H(z)$, 37 points) [63], (where $z=1/a-1$ is the
redshift) and type Ia supernova data (SNIa, 1701 points from Pantheon$+$
sample) [64]. Consequently, the $\chi^{2}_{\text{tot}}$ for the initial
analysis can be written as

$\chi^{2}_{\text{tot}}(p)=\chi^{2}_{\text{OHD}}(p)+\chi^{2}_{\text{SN}}(p),$
(26)

where the statistical vector $p$ comprises free parameters of the model
$(\tilde{\Omega}^{\text{(dust)}}_{0},H_{0},D)$.

Here, we used the Hubble parameter data from Table 1 of Ref. [63] (obtained by
the BAO signal in the distribution of galaxies, the BAO signal in the
distribution of the Ly$\alpha$ forest alone, and the cross-correlation with
QSOs), a standardised compilation of 37 data points collected from the
recently accurate estimates $H(z)$ within the redshift range of $0.07\leq
z\leq 2.36$ to obtain the Hubble parameter data. The chi-square, $(\chi^{2})$,
for the OHD is calculated as follows

$\chi^{2}_{\text{OHD}}(p)=\sum_{i=1}^{37}\frac{\left[H_{\text{th}}(z_{i},p)-H_{\text{ob}}(z_{i})\right]^{2}}{\sigma^{2}_{i}},$
(27)

the term $\sigma_{i}$ represents the Gaussian error associated with the
measured value of $H_{i}$.

In Table 1, we presented the outcomes of our statistical analysis for a flat
$\Lambda$CDM and the fractal $\Lambda$CDM through the utilization of OHD data.
As it is evident, both models are consistent with the observational data.

Parameter |  $\Lambda$CDM | Fractal $\Lambda$CDM
---|---|---
$\tilde{\Omega}^{\text{(dust)}}_{0}$ | $0.258^{+0.056}_{-0.046}$ | $0.37^{+0.23}_{-0.16}$
$H_{0}$ | $70.1^{+4.2}_{-4.3}$ | $67.4^{+5.6}_{-6.7}$
$D$ | $3$ | $2.2^{+1.1}_{-1.4}$
$\chi^{2}_{\text{min}}$ | $18.08$ | $18.08$
Table 1: The best fitting values of free parameters of the model, and the
minimal value of $\chi^{2}_{OHD}$ for the fractal model as well as for the
flat $\Lambda$CDM using the $H(z)$ dataset.

Type SNIa observations are essential for gathering essential data that
contributes to a complete picture of the expansion of the Universe. The main
evidence in favour of the notion of the Universe’s accelerated expansion comes
from these findings. To extract the most accurate and reliable results from
the SNIa data, it is crucial to carefully compare the observed distance
modulus of SNIa detection with the corresponding theoretical value. In light
of this, we have employed the Pantheon+ sample, a recent dataset of SNeIa,
consisting of $1701$ data points of distance modulus $\mu_{\text{obs}}$ at
various redshifts $z_{i}$ within the range $0.001<z_{i}<2.26$. The
corresponding chi-squared statistic, $\chi^{2}$, is defined based on the
comparison between the observed and theoretical distance modulus as follows

$\chi^{2}_{\text{sn}}(p)=\sum_{i=1}^{1701}\frac{[\mu_{\text{th}}(z_{i},p)-\mu_{\text{ob}}(z_{i})]^{2}}{\sigma^{2}_{i}},$
(28)

and

$\mu_{\text{th}}(z_{i},p)=\\\
5\log_{10}{\Big{[}(1+z)\int^{z}_{0}\frac{dx}{E(x)}\Big{]}}+42.384-5\log_{10}h-19.37.$
(29)

Table 2 presents the parameter values obtained through constraining the
parameters of the $\Lambda$CDM and Fractal$\Lambda$CDM by pantheon$+$ data.

Parameter |  $\Lambda$CDM | Fractal $\Lambda$CDM
---|---|---
$\tilde{\Omega}^{\text{(dust)}}_{0}$ | $0.352^{+0.054}_{-0.045}$ | $0.393^{+0.073}_{-0.070}$
$H_{0}$ | $69.14^{+0.61}_{-0.69}$ | $68.98^{+0.70}_{-0.70}$
$D$ | $3$ | $1.99^{+0.99}_{-0.98}$
$\chi^{2}_{\text{min}}$ | $758.58$ | $758.06$
Table 2: The best fitting values of free parameters of the model, and the
minimal value of $\chi^{2}_{SN}$ for the fractal $\Lambda$CDM and flat
$\Lambda$CDM using the SNe dataset. The units of $H_{0}$ is
$\text{Km}\cdot\text{s}^{-1}\cdot\text{Mpc}^{-1}$.

Finally, we perform an overall statistical analysis using joint analyses of
SNe$+$H(z) by the MCMC algorithm. Figure 2 displays the best-fit values for
the free parameters of the fractal $\Lambda$CDM model, i.e.,
$(\tilde{\Omega}^{\text{dust}}_{0},H_{0},D)$ with $68\%$ and $95\%$ confidence
level contours, and Fig. 3 includes the comparison of the fractal $\Lambda$CDM
model with the flat $\Lambda$CDM standard model of cosmology.

Figure 2: The $\%68$ and $\%95$ confidence level contours for the fractal
$\Lambda$CDM model using the OHD, pantheon$+$, and joint datasets. Figure 3:
The $\%68$ and $\%95$ confidence level contours for the fractal $\Lambda$CDM
and $\Lambda$CDM models using the SNe$+$H(z) joint dataset

The best-fit range (%99 confidence level) from observational data analysis for
the free parameters of the fractal model and the $\Lambda$CDM model is
available in Table 3.

Parameter |  $\Lambda$CDM | Fractal $\Lambda$CDM
---|---|---
$\tilde{\Omega}^{\text{(dust)}}_{0}$ | $0.285^{+0.026}_{-0.025}$ | $0.375^{+0.067}_{-0.067}$
$H_{0}$ | $69.76^{+0.56}_{-0.56}$ | $69.11^{+0.71}_{-0.71}$
$D$ | $3$ | $2.00^{+0.67}_{-0.61}$
$\chi^{2}_{\text{min}}$ | $796.53$ | $781.23$
Table 3: The best fitting values of free parameters of the model, and the
minimal value of $\chi^{2}_{tot}=\chi^{2}_{OHD}+\chi^{2}_{SN}$ for the fractal
model as well as for the flat $\Lambda$CDM using the SNe+$H(z)$ joint dataset.
The units of $H_{0}$ is $\text{Km}\cdot\text{s}^{-1}\cdot\text{Mpc}^{-1}$.

Using the constraint values of the model parameters from the Pantheon+ and
Hubble joint dataset, we next compared our model parameterization with the
$\Lambda$CDM model by studying the evolution of the Hubble parameter $H(z)$,
the deceleration parameter $q(z)$, and the distance modulus $\mu(z)$. Our
analysis is illustrated in Figs 4, 5, and 6. Fig. 4 showcases the relationship
between the Hubble parameter $H$ and the redshift $z=\frac{1}{a}-1$. The
Pantheon Survey is visually depicted in Fig. 6, which showcases the standard
Hubble diagram of SNIa, graphically presenting the relationship between the
observed distance modulus and the corresponding redshift. Furthermore, Fig. 6
also highlights the comparison between our proposed model and the widely
accepted $\Lambda$CDM model, revealing intriguing and significant similarities
which further strengthen the empirical evidence and support the validity of
our findings. In addition, Fig. 5 shows the behaviour of the deceleration
parameter concerning redshift for the best-fit values in Table 3. The
parameterization of our model closely matches the observed data. The
cosmological parameters of the fractal model and the flat $\Lambda$CDM at the
present time ($z=0$), the Universe’s age, and the corresponding transition
redshift ($z_{t}$) are summarised in Table 4. It should be emphasised that
both the fractal $\Lambda$CDM and the conventional flat $\Lambda$CDM models
agree with the latest cosmological findings, concerning their respective
estimates of the transition redshift $z_{t}$ and $q_{0}$.

Figure 4: Prediction of the $H(z)$ (in units $\text{Km}\cdot\text{s}^{-1}\cdot\text{Mpc}^{-1}$) versus redshift z with error bars. The red-solid line denotes the fractal$\Lambda$CDM model. The dark blue points denote the prediction of the $\Lambda$CDM.. Figure 5: Prediction of the deceleration parameter $q(z)$ (red-solid line) using the best-fit fractal model is compared to the prediction of the $\Lambda$CDM (black dotted line). Figure 6: Prediction of the $\mu(z)$ (red-solid line) using the best-fit fractional $\Lambda$CDM model is compared to the observational data (dark blue points -with vertical lines indicating the uncertainty-) and the prediction of the $\Lambda$CDM (black dotted line).  Parameter |  $\Lambda$CDM | Fractal $\Lambda$CDM
---|---|---
$q_{0}$ | $-0.572176^{+0.03}_{-0.03}$ | $-0.440145^{+0.09}_{-0.09}$
$t_{0}$ | $13.7^{+0.46}_{-0.43}$ | $13.9105^{+0.151}_{-0.152}$
$\tilde{\Omega}^{(\Lambda)}_{0}$ | $0.714814^{+0.02}_{-0.02}$ | $0.626793^{+0.06}_{-0.06}$
$z_{t}$ | $0.71^{+0.07}_{-0.07}$ | $0.82^{+0.151}_{-0.155}$
Table 4: present values obtained for the cosmological parameters, age of the
Universe, and the transition redshift using best-fit values of the free
parameters for the fractal $\Lambda$CDM and flat $\Lambda$CDM models.

It is noteworthy to observe that the model examined in this paper posits that
the fractal dimension $D$ is equivalent to 2. This outcome aligns with the
findings of numerous redshift surveys, which have consistently reported a
fractal dimension of approximately $D=2$ for the clustering of galaxies on a
large scale [65, 66, 67, 68, 69]. These results strengthen the validity of our
findings. According to Table (4) we obtain the age of the fractal Universe
$13.910\pm 0.15$ Gyr. This demonstrates that the temporal extent of a fractal
universe has the potential to considerably surpass the customary standard flat
$\Lambda$CDM model of cosmology, wherein the age of the Universe is estimated
to be $13.7\pm 0.4$Gyr in Table 4, and $13.800\pm 0.024$ Gyr according to the
Planck18 data [70]. The concept of fractal cosmology seems to possess the
inherent capacity to offer a more comprehensive and improved elucidation for
the aforementioned observations. Our model could be considered a viable
alternative to $\Lambda$CDM through these findings.

##  5 Conclusions

According to fractional quantum gravity, the event horizon of a BH possesses
the intriguing characteristic of being a surface that exhibits a random
fractal pattern, meaning it displays intricate and self-similar patterns at
various scales. Moreover, it is interesting to note that this property of a
fractal horizon extends beyond BHs and also applies to the event horizon of de
Sitter space-time [26]. This implies that the event horizon of de Sitter
space-time shares the same fascinating property of a random fractal surface.
These insightful findings and ideas build upon the notion of a fractal horizon
and successfully incorporate it into the broader cosmic framework. In light of
this, the Friedmann equations are derived from the extension of the fractal
entropy-area relationship of a fractional BH into the context of the apparent
horizon of an isotropic and homogeneous universe.

This was accomplished by employing the FLRW universe’s fractal apparent
horizon in conjunction with the first law of thermodynamics. Also, considering
the differences between fractal and non-fractional cosmological models
summarized as

  * 1.

The effective proper distance is re-scaled, affecting the corresponding volume
and the horizon’s area.

  * 2.

Regarding the effective proper radius and proper volume, the energy-momentum
conservation law of a perfect fluid is modified in fractal cosmology.

  * 3.

Rescaling in apparent horizon radius also alters the Cai–Kim temperature.

  * 4.

The fractal density parameter of the fluid is modified by a prefactor which
depends on the fractal dimension, Hubble parameter, and the grid size.

Our encountered equations show that dark energy is required to explain the
Universe’s present stage and associated observations. It is noteworthy that
when $D=3$, the standard Friedmann equations are restored. Our model offers an
alternate interpretation, speculating that the measurable parameters could
have been affected by fractal properties. In particular, changing the fractal
dimension may have a major impact on the age of the Universe as well as the
deceleration and transition redshift at the current epoch.

In conclusion, we have only just begun to explore and refine our fractal
model, which is a modified cosmology. We contribute to the advancement of
fractional models and may find exciting new findings at both quantum and
classical levels by examining a number of modern cosmological issues,
including the Hubble tension, gravitational wave propagation, gravitational
lensing, and density perturbation generation. However, it is prudent to defer
the above evaluations to later undertakings.

## Acknowledgements

S.J. acknowledges financial support from the National Council for Scientific
and Technological Development–CNPq, Grant no. 308131/2022-3.

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