# Fractional Holographic Dark Energy

Oem Trivedi 1, Ayush Bidlan 2, Paulo Moniz 3<EMAIL_ADDRESS><EMAIL_ADDRESS>,
<EMAIL_ADDRESS>ORCID:0000-0001-7170-8952

(1International Centre for Space and Cosmology, Ahmedabad University,
Ahmedabad 380009, India
2Department of Physics, Sardar Vallabhbhai National Institute of Technology,
Surat, Gujarat 395007, India
3Departamento de Física, Centro de Matemática e Aplicações (CMA-UBI),
Universidade da Beira Interior, Rua Marquês d’Avila e Bolama, 6200-001
Covilhã, Portugal
)

###### Abstract

Holographic dark energy theories present a fascinating interface to probe
late-time cosmology, as guided by contemporary ideas about quantum gravity. In
this work, we present a new holographic dark energy scenario designated
“Fractional Holographic Dark Energy”(FHDE). This model extends the
conventional framework of HDEs by incorporating specific features from
fractional calculus recently applied, e.g., in cosmological settings. In this
manner, we retrieve a novel form of HDE energy density. We then show how FHDE
can provide a consistent picture of the evolution of the late-time universe
even with the simple choice of the Hubble horizon as the IR (infrared) cutoff.
We provide detailed descriptions of the cosmological evolution, showing how
the fractional calculus ingredients can alleviate quite a few issues
associated with the conventional HDE scenario. Concretely, we compute and plot
diagrams using the Hubble horizon cutoff for HDE. The density parameters for
DE and dark matter (DM), the deceleration parameter, and the DE EoS parameter
indicate how the universe may evolve within our “FHDE” model, fitting within
an appropriate scenario of late-time cosmology.

##  I Introduction

The discovery of the universe’s late-time acceleration is a defining moment in
cosmology [1]. This unexpected revelation unveiled a significant discrepancy
between our understanding of gravity and the observed expansion history of the
cosmos. Since then, the pursuit of understanding this phenomenon has fueled a
vibrant field of research encompassing a diverse array of theoretical
frameworks. The conventional approach to address late-time cosmic acceleration
centres on introducing a Cosmological Constant within the framework of General
Relativity [2, 3]. This term acts as a form of “dark energy” (DE), a
mysterious entity contributing a repulsive force counteracting attractive
gravity and driving accelerated expansion. However, the $\Lambda$CDM model [4,
5] faces challenges in explaining the origin and fine-tuning required for the
specific value of the cosmological constant.

Alternative explanations for late-time acceleration venture beyond the realm
of General Relativity. Modified gravity theories propose alterations to the
gravitational field equations themselves, potentially leading to an
explanation for the observed expansion without resorting to dark energy [6,
7]. Additionally, scalar field models posit the existence of a new dynamical
field, quintessence or phantom energy, whose dynamics may drive the late-time
acceleration [8, 9, 10, 11]. The quest for a more fundamental understanding of
gravity has also led to the exploration of late-time acceleration in the
context of quantum gravity theories. String theory frameworks like Braneworld
cosmology, loop quantum cosmology, and asymptotically safe cosmology offer
intriguing possibilities by incorporating quantum mechanical effects into the
description of spacetime [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]. Despite
the plethora of proposed solutions, fundamental questions regarding the nature
of dark energy and late-time acceleration persist. Moreover, a critical
current challenge lies in reconciling conflicting measurements of the Hubble
constant, known as the Hubble tension [23, 24, 25]. This discrepancy
highlights potential shortcomings in our current understanding of the
Universe’s expansion history and the underlying physics governing it. It
underscores the need for advancements in gravitational physics to bridge the
gap between theoretical frameworks and observational data. The recent findings
from the Dark Energy Spectroscopic Instrument (DESI) [26, 27, 28] further
emphasize the need for a more nuanced understanding of dark energy. These
observations hint at potential deviations from the $\Lambda$CDM model,
suggesting that alternative explanations for late-time cosmic acceleration
deserve continued exploration.

In this context, applying the holographic principle to cosmology emerges as a
promising avenue for exploration [29, 30]. This principle, arising from string
theory and quantum gravity considerations, proposes an intriguing relationship
between a system’s entropy and its boundary surface area rather than its
volume [31]. This captivating concept offers a potential link between the
enigmatic nature of dark energy and the underlying structure of spacetime at
the quantum level [32, 33, 34]. Building upon the pioneering work in [32], our
research delves deeper into the theoretical framework of holographic dark
energy (HDE). In particular, we explore the connection between short-distance
cutoffs arising from quantum field theory and long-distance cutoffs imposed by
black hole formation constraints. Specifically, in the work of Cohen et al
[33], the authors showed that holographic considerations led to the inequality

$L^{3}\Lambda^{3}\leq S^{3/4},$ (1)

where S is the system entropy while L and $\Lambda$ denote the IR and UV
cutoffs, respectively, and we are working in units such that we take the
Planck mass as $m_{p}=1$. Using the Bekenstein-Hawking definition for the
entropy, $S\propto A$, together with the idea that the vacuum energy density
$\Lambda^{4}$ corresponds to the energy density for dark energy, i.e.,
$\rho_{DE}\sim\Lambda^{4}$, we can then arrive at the following expression

$\rho=3c^{2}L^{-2}.$ (2)

Quite a few other works have explored holographic dark energy from various
similar aspects in recent years [35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45,
46, 47, 48, 49, 50, 51, 52, 53, 54, 55]. In recent years, various other
alternative forms of HDE have been proposed. For example, Tsallis HDE models
are based on Tsallis’ corrections to the standard Boltzmann-Gibbs entropy
(obtained from applying these corrections to black hole physics), resulting in
the equation

$\rho_{\Lambda}=3c^{2}L^{-(4-2\sigma)},$ (3)

where $\sigma$ is the Tsallis parameter, which is considered to be positive
[56] and we recover simple HDE in the limit $\sigma\to 1$. On the other hand,
Barrow’s modification of the Bekenstein-Hawking formula led to the creation of
Barrow HDE models and such models are described by the energy density

$\rho_{\Lambda}=3c^{2}L^{\Delta-2},$ (4)

where $\Delta$ is the deformation parameter [57], which can have a maximum
value of $\Delta=1$ and in the limit of $\Delta\to 0$ one recovers the simple
HDE. There are many other models like the Renyi and Kaniadakis models [58, 59]
which have been forward in recent years based on various motivations, ranging
from thermodynamics to generalized uncertainty principles amongst other
approaches [60, 61, 62]. In our work, we would like to propose a new model of
dark energy which is based on considerations of fractional calculus. (and
recent applications in gravitational and (quantum) cosmological scenarios). In
Section II we motivate this scenario and provide the new energy density for
the HDE whereas in Section III, we investigate in detail the cosmological
evolution of our case study. We conclude and discuss our work in Section IV.

##  II Fractional Holographic Dark Energy

Given our interest in the implications of fractional calculus in either
classical or quantum cosmological scenarios it is appropriate to provide a
brief overview of their features. Fractional calculus extends differentiation
and integration to any real or complex orders applicable to the study of
complicated dynamical systems and various intricate physical phenomena.
Concretely, the fractional derivative generalizes the usual integer order of
differentiation to rational, real or complex numbers, with several types
having been defined, such as the the Liouville, Riemann, Caputo, and Riesz
fractional derivatives [63]. While no single such derivative type can fully be
applied to every situation, their suitability with reality depends on the
specific problem at hand. The interested reader can refer to various
interesting works by Ortigueira for more details [64, 65, 66, 67, 68, 69, 70].

Within fractional calculus, a particularly enticing application is quantum
mechanics. Let us remind Brownian motion where the trajectories are self-
similar, non-differentiable lines with fractal dimensions differing from their
topological dimensions. In this context, Feynman and Hibbs re-established
quantum mechanics as a path integral over Brownian paths using the concept of
fractality [71]. In recent years, this led to the development of a fractional
path integral, establishing space-fractional quantum mechanics (FQM) as a path
integral over another type of steps, within Lévy flight paths, characterized
by the Lévy index $\alpha$. The Gaussian process or Brownian motion is
recovered in a specific limit where $\alpha=2$. A significant result of FQM is
the space-fractional Schrödinger equation (SE), where the second-order spatial
derivative of the ordinary SE is replaced by a fractional-order derivative,
specifically the quantum Riesz fractional derivative. For further reading, the
reader is directed to the monograph and influential works of Nick Laskin on
FQM [72, 73, 74, 75, 76, 77]. Additionally, Naber established the time-
fractional SE [78], replacing the first-order time derivative of the ordinary
SE with a fractional-order derivative, concretely the Caputo fractional
derivative, while keeping the spatial derivative unchanged. This subsequently
led to the development of the spacetime-fractional SE, wherein the
corresponding fractional derivatives replace both the second-order spatial and
first-order time derivatives. For further readings on the history of
fractional calculus, Levy paths and associated interests, one can have a look
at [79, 63, 80, 81, 82, 83, 84].

Regarding its application to cosmology and gravity, fractional calculus can
extend any gravitational or cosmological framework to another with non-integer
orders in the derivatives of either the classical or quantum regime. Among the
extended cosmological frameworks, fractional cosmology is particularly
promising for addressing open problems such as the Hubble tension,
synchronization, and cosmological constant problems. This approach offers an
effective means of investigating gravitational and cosmological phenomena like
the evolution of the universe, the dynamics of black holes, and gravitational
waves [85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98].

Classical cosmological models based on fractional derivatives have typically
followed two main approaches: Last-step modification, where fractional ones
replace ordinary derivatives in the field equations of the model after
obtaining the field equations; and First-step modification, where a fractional
derivative geometry is first established, followed by the proposal of a
fractional-order action. The first-step modification is considered more
fundamental. The fractional quantum cosmological framework has mainly explored
how one can derive the fractional Wheeler–DeWitt (WDW) equation, inspired by
the space-fractional Schrödinger equation, using the quantum Riesz fractional
derivative for a quantum gravity model. A direction where fractional cosmology
was found of considerable interest is elaborated in the next paragraph.

It is widely known that entropy can be modified in the context of various
theories. We have seen examples of entropies advanced within the Barrow and
Tsallis HDEs, previously discussed in [99, 100]. There has also been quite
some work on entropies and their cosmological implications besides these.
Being more concrete, this has been motivated by various approaches ranging
from loop quantum gravity and string theory to generalised uncertainty
principles and rainbow gravity [101, 102, 103, 104, 105, 106].

Recently, Jalalzadeh et al. [107] investigated the effects of fractional
quantum mechanics (FQM) on Schwazrschild black hole thermodynamics. The
authors used a space-fractional derivative of second order (Riesz derivative)
and obtained fractional black hole entropy from a modified Wheeler-DeWitt
equation. To obtain the modified Wheeler-DeWitt equation, they first used the
canonical quantization procedure for a Schawarzschild black hole, by which
they ended up at the corresponding Hamiltonian. After this, they included the
quantum Riesz derivative in the momentum operator of the Hamiltonian, which
led them to the fractional Wheeler-DeWitt equation. The fractional entropy was
then found out as

$S_{h}=CA^{\frac{2+\alpha}{2\alpha}},$ (5)

where $\alpha$ is a fractional parameter therein constrained to $1<\alpha\leq
2$ and the standard case is recovered for $\alpha=2$; $C$ is some constant
which can produce an appropriate combination of Planck units to reproduce the
standard Bekenstein-Hawking relation in the $\alpha=2$ case. This equation
implies that the entropy is a power-law function of its area and resembles the
Barrow and Tsallis entropies, although they have different motivations and
physical principles. Using the holographic inequality (1) and considering the
entropy (5), one ends up with

$\Lambda^{3}L^{3}\leq(CA^{\frac{2+\alpha}{2\alpha}})^{\frac{3}{4}}.$ (6)

From the consideration that $\rho_{DE}\sim\Lambda^{4}$, one can consider an
energy density for Holographic dark energy, based on fractional calculus
features and associated quantum effects, as

$\rho_{DE}=\gamma L^{\frac{2-3\alpha}{\alpha}},$ (7)

where $\gamma$ is some constant. One immediately notes that we can recover the
standard HDE (2) in the case $\alpha=2$, which is indeed the limit in which
the entropy (5) decays to the standard Bekenstein-Hawking case. To be more in
line with the original HDE proposal, we can set $\gamma=3c^{2}$, where $c$ is
again a constant of $\mathcal{O}(1)$ and after this consideration, we can
write our proposed HDE energy density as

$\rho_{DE}=3c^{2}L^{\frac{2-3\alpha}{\alpha}}$ (8)

This forms the basis of our new HDE hypothesis and we will now investigate if
this model can provide a consistent scenario of universal evolution in the
late-times. One would note that there is a similarity of this formula with the
Barrow and Tsallis cases (3), (4), but that is because the form of the
entropies in those cases are quite similar and are all of the forms of power
laws entropy-area relations. However, as we have made clear in this section,
the motivation for this HDE model is quite different than the one for the
Tsallis case (namely, arising from modifications in fundamental
thermodynamics) and Barrow case (namely, from studies of black hole
deformations).

##  III Cosmological evolution

We start our analysis by indicating we will be dealing with a flat FLRW
universe, where we shall consider that the contributions of dark energy and
dark matter to the universal energy density are dominant. Indeed, this is a
reasonable approximation often used in cosmology literature. It means that our
Friedmann equation takes the form

$H^{2}=\frac{\rho}{3}=\frac{\rho_{DE}+\rho_{m}}{3}.$ (9)

The continuity equations for dark matter and dark energy have the usual form
as well

$\dot{\rho_{m}}+3H\rho_{m}(1+w_{m})=0,$ (10)
$\dot{\rho_{DE}}+3H\rho_{DE}(1+w_{DE})=0,$ (11)

where $w_{m}$ and $w_{DE}$ refer to the equation of state parameters (EoS) for
dark matter and dark energy. Note that we are not considering an interacting
dark sector here. Furthermore, we shall consider that dark matter has a
pressureless form, meaning that $w_{m}\sim 0$. Before proceeding further, we
must choose the cutoff scale $L$. The early suggestion was considering a
cutoff scale given by $L\to H^{-1}$, termed the Hubble horizon cutoff. This
choice aimed to alleviate the fine-tuning problem by introducing a natural
length scale associated with the inverse of the Hubble parameter $H$, but it
was found that this particular scale resulted in the dark energy EoS parameter
approaching zero, and it also failed to contribute significantly to the
current accelerated expansion of the universe. An alternative idea was then to
consider the utilization of the particle horizon as the length scale

$L_{p}=a\int_{0}^{t}\frac{dt}{a}.\quad$ (12)

This alternative resulted in an equation of state parameter higher than
$-1/3$, but despite this modification, the challenges of explaining the
present acceleration remained unresolved. Another option considered was the
future event horizon as the length scale

$L_{f}=a\int_{a}^{\infty}\frac{dt}{a}.$ (13)

Although the desired acceleration regime can be achieved in this case,
concerns are raised regarding causality. Another option for the cutoff scale
was the Granda-Oliveros cutoff, which took into account the derivative of $H$
into their definition of $L$ as well [37]

$L=(\alpha H^{2}+\beta\dot{H})^{-\frac{1}{2}}.$ (14)

The most generalized cutoff choice is the Nojiri-Odintsov cutoff [53, 35,
108], where we could have the form

$L=L(H,\dot{H},\ddot{H},...L_{p},L_{f},\dot{L_{p}},\dot{L_{f}}...).$ (15)

An additional issue with all these cutoffs relates to the classical stability
of these models against perturbations [109], an issue seen in all classes of
HDE energy densities with various cutoffs. In general, one can have HDE models
where the density could be of the form (15) being general functions of $H$,
particle horizon and event horizon scales. Let us note that regarding all the
forms of HDEs that we have discussed, whether it is Tsallis, Barrow, others or
our own herewith, it could just be said to be such that it would approach the
form in (15). Although such a comment may be mathematically of interest, one
in general endeavours to construct, in a justified, proper manner, a
particular form for the function in (15), even if one wants to allow for the
generality of the cutoff; This leads to the various HDE theories being
motivated by different physical and mathematical considerations. In our work
here, we would like to consider the Hubble horizon cutoff, and our reasoning
for making this choice is two-fold. On the one hand, we are now interested in
the simplest way fractional HDE (FHDE, as we can refer to it later on) can
produce universal evolution. In that regard, the Hubble Horizon cutoff
provides a suitable choice. On the other hand, the Hubble horizon cutoff has
been shown not to work within various frameworks [109, 110] and so if FHDE can
provide an observationally consistent scenario then, it would represent a step
forward in the ongoing appraisal of the Hubble cutoff.

With this in mind, the definition (5) with the Hubble Horizon cutoff $L\to
H^{-1}$ gives

$\rho_{DE}=3c^{2}H^{\frac{3\alpha-2}{\alpha}}.$ (16)

We also define the fractional density parameters for DE and DM as follows,
using (5):

$\Omega_{DE}=\frac{\rho_{DE}}{3H^{2}}\to
c^{2}H^{\frac{\alpha-2}{\alpha}},\quad\Omega_{m}=\frac{\rho_{m}}{3H^{2}}.$
(17)

We also note, using (9), that

$\Omega_{DE}+\Omega_{m}=\Omega_{DE}(1+y)=1,$ (18)

where $y=\frac{\Omega_{DE}}{\Omega_{m}}$ and using this, alongside (9), (10),
we can write

$\frac{\dot{H}}{H^{2}}=-\frac{3}{2}(1+w_{DE}+y)\Omega_{DE}.$ (19)

We can also utilize (11) and (16) to obtain

$\frac{\dot{H}}{H^{2}}=-\frac{3\alpha}{3\alpha-2}(1+w_{DE}).$ (20)

Equating (19) and (20) make it possible to write that

$w_{DE}=-1+\frac{(3\alpha-2)(1-\Omega_{DE})}{2\alpha-\Omega_{DE}(3\alpha-2)},$
(21)

where we employed the fact that $y=\frac{1-\Omega_{DE}}{\Omega_{DE}}$. We can
then use this parametrization for the dark EoS in (20). We thus find

$\frac{\dot{H}}{H^{2}}=-\frac{3\alpha(1-\Omega_{DE})}{2\alpha-\Omega_{DE}(3\alpha-2)}.$
(22)

From this, one can directly get the deceleration parameter as

$q=-1-\frac{\dot{H}}{H^{2}}=-1+\frac{3\alpha(1-\Omega_{DE})}{2\alpha-\Omega_{DE}(3\alpha-2)}.$
(23)

Beyond this, one can then define the derivative of $\Omega_{DE}$ using (17) as

$\Omega_{DE}^{\prime}=\frac{\alpha-2}{\alpha}\left(\frac{\Omega_{DE}\dot{H}}{H^{2}}\right),$
(24)

with overprime denoting differentiation with respect to $\ln{a}$. Using (22)
it is then easy to write that

$\Omega_{DE}^{\prime}=\frac{3(\alpha-2)(1-\Omega_{DE})\Omega_{DE}}{2\alpha-\Omega_{DE}(3\alpha-2)}.$
(25)

We will now use the relation, $d\ln{a}=-\frac{dz}{1+z}$ to find

$(1+z)\frac{d\Omega_{DE}}{dz}=\frac{3(\alpha-2)(1-\Omega_{DE})\Omega_{DE}}{2\alpha-\Omega_{DE}(3\alpha-2)}.$
(26)

We can solve this differential equation to arrive at

$\left(\frac{\Omega_{DE}}{\Omega_{DE0}}\right)^{2\alpha}\left(\frac{1-\Omega_{DE0}}{1-\Omega_{DE}}\right)^{2-\alpha}=(1+z)^{3(\alpha-2)},$
(27)

where $\Omega_{DE0}$ refers to $\Omega_{DE}$ at z=0. We shall now consider
$\Omega_{DE0}\sim 0.69$, which is in line with recent observations of the
universe [26, 27, 28] and using this, we shall numerically solve (27).

Figure 1: Evolution of $\Omega_{DE}$ against the redshift z for various values
of $\alpha$

In figure 1, we have plotted the density parameter $\Omega_{DE}$ against the
redshift z for various values of $\alpha$. One can retrieve that a suitable
evolution of the DE energy density is possible to identify as effects from
fractional features start to dominate. More concretely, it is so because, for
larger values of $\alpha$, like $\alpha=1.8$ in Figure 1, the evolution of DE
isn’t very dynamic. By a dynamic nature here, we refer to the DE density
increasing appreciably as the redshift becomes smaller. So, for larger values
of $\alpha$, one sees that the growth of DE (and thereby the dark sector) is
quite stagnant, while for smaller values, it is not the case. Indeed, as one
proceeds to smaller and smaller values of $\alpha$, like 1.5 and 1.1, we start
to see that the cosmological evolution becomes more and more dynamic,
dominating gradually as the redshift decreases, which is in line with our
expectation for FHDE, particularly when the fractional effects are most strong
with e.g., $\alpha=1.1$.

Figure 2: Evolution of $\Omega_{m}$ against the redshift z for various values
of $\alpha$

We similarly plotted for the dark matter density parameter $\Omega_{m}$ in
figure 2. We have again plotted here for various values of $\alpha$ as we did
in the first figure, and once again sees that the evolution of DM is quite
dynamic for smaller values of $\alpha$. In particular, one would note that
figures 1 and 2 inform about DE domination in the far future as one goes
towards the limit $z\to-1$.

Figure 3: Evolution of dark energy EoS parameter against the redshift z for
various values of $\alpha$

The dark energy EoS parameter (21) represents a significant quantity in
studying the DE evolution, and we have plotted for that in figure 3. We took
various values of $\alpha$, and we see that the EoS parameter starts to show a
behaviour more suitable and in line with observations for small values of
$\alpha$. In particular, for $\alpha=1.1$ the EoS parameter takes a value at
$z=0$ which is in close agreement with the recent constraints on $w$ by DESI,
see for example [26, 27, 28]. As one goes towards larger values of $\alpha$,
the evolution of the EoS parameter starts to deviate more and more from what
one would expect for DE. We note here that for both $\alpha=1.1$ and
$\alpha=1.5$, the EoS parameter remains in the quintessence regime across the
redshift range $2\leq z\leq-1$ and takes the value as $-1$ in the far future
limit $z\to-1$.

Figure 4: Evolution of deceleration parameter q against the redshift $z$ for
various values of $\alpha$

We have plotted similarly for the deceleration parameter $q$ (23) in figure 4.
We note here that in the case of smaller values of $\alpha$, one sees an
appropriate transition of the universe from the deceleration to the
acceleration phase. In particular, for the case $\alpha=1.1$ one sees the
transition redshift $z_{T}\sim 0.38$ which is well within the most recent
constraints on $z_{T}$ [111]. The case of $\alpha=1.3$ presents $z_{T}\sim
0.17$, which would be pretty hard to be realistic. We note that for larger
values of $\alpha$, with $\alpha=1.8$ here, one sees that the transition to
the acceleration phase of the universe seems to be pointed in happening in the
future $z<0$, which is, of course, unrealistic. This again shows that smaller
values of $\alpha$, where there is a stronger influence of fractional effects
present a viable HDE scenario and are consistent with a Hubble horizon cutoff,
which we find most satisfactory.

Figure 5: The $w_{d}-w_{d}^{\prime}$ plane for the redshift range $-1\leq
z\leq 2$, for various values of $\alpha$

A beneficial way to understand the nature of models of dark energy which show
a quintessence nature is by studying the nature of the $w_{d}-w_{d}^{\prime}$
plane of that model, which was introduced in [112]. We note here that $w_{d}$
refers to the EoS parameter for DE, while $w_{d}^{\prime}$ refers to its
derivative with respect to $\ln{a}$. The plane allows one to distinguish
between a “freezing” or a “thawing” form of quintessence regime for the model
at hand. For regions in the plane where $w_{d}<0$ and $w_{d}^{\prime}>0$, the
model is in the thawing regime, while for $w_{d}<0$ and $w_{d}^{\prime}<0$ the
model is in the freezing regime. As one sees in figure 5, the FHDE scenario
has a freezing form of quintessence throughout the range of $\alpha$. However,
the implication is only particularly applicable for $\alpha=1.1$ and $1.5$ as
these are regions for which the EoS remains in the quintessence regime, as
seen earlier in figure 3.

Figure 6: The evolution of the squared sound speed $v_{s}^{2}$ against
redshift z for various values of $\alpha$

Finally, another important quantity with regards to HDE models, in particular,
is the squared sound speed, given by

$v_{s}^{2}=\frac{\dot{p}}{\dot{\rho}}=w+\frac{\dot{w}\rho}{\dot{\rho}}.$ (28)

Using (22) and the relation $d\ln{a}=-\frac{dz}{(1+z)}$, it is easy to write

$v_{s}^{2}=w+\frac{(1+z)(2\alpha-\Omega_{DE}(3\alpha-2)}{(3\alpha-2)(1-\Omega_{DE}}\frac{dw}{dz}.$
(29)

One can then use (21) and (27) to arrive towards expressions for the sound
speed in terms of $\Omega_{DE}(z)$ and using that, we have plotted the sound
speed for various values $\alpha$ in figure 6. It should be noted that
classical instability has been a prevalent issue in HDE models [109], a
problem observed not only in conventional HDE but in various cutoffs and other
extended HDE densities. In a lot of cases, it so happens that the model is
always unstable towards classical perturbations, which is the case when the
sound speed is not in the interval $0\leq v_{s}^{2}\leq 1$. It is important to
note that the squared sound speed is usually between $0$ and $1$, as this
region makes the most physical sense. While it may be straightforward to
figure out why squared sound speeds cannot be negative, it is essential to
clarify that sound speeds greater than 1 signify superluminal speeds of
perturbations, thus pointing towards instability of the underlying theory
against classical perturbations. While superluminal perturbations have been
used as a possible way to explain issues in both the early and late-universe
[113, 114], one usually sees this behaviour only in models with non-canonical
Lagrangians and making the case for them even in those scenarios is quite
difficult. We note in our model that our model maintains classical stability
in the current epoch $z=0$ for smaller values of $\alpha$ where there is a
more significant influence of the fractional effects. At the same time, in the
far future, instability creeps in. We suspect this instability can be treated
if one considers cutoffs besides the Hubble horizon cutoff for the model.
Still, we shall not pursue that here and would leave that as an open direction
for work for the interested reader. We have not specified any value for the
constant $c$ in (8), so our results hold for any value of $c$.

##  IV Conclusions

In this paper, we have provided a new holographic dark energy scenario, which
we have named Fractional Holographic Dark Energy. The name is so because this
HDE scenario is motivated by corrections due to fractional calculus and
fractional quantum mechanics, thus incorporating ideas from an exotic
interface of mathematics and physics into gravitational (and cosmological)
settings. The key feature of our model is a new energy density form for
holographic dark energy, which we showed reduces to the conventional HDE
energy density in the limit $\alpha\to 2$; NB. The parameter $\alpha$ is
associated with fractional calculus features emerging in fractional (classical
and quantum) mechanics and its implications.

In particular, we derived this energy density from the holographic inequality,
incorporating effects from the corrections to the Bekenstein-Hawking entropy
with the assistance of the fractional Wheeler-De Witt equation. The energy
density smoothly extends the conventional HDE energy density and provides
outlooks to explore. In this line, we studied the cosmological evolution using
the Hubble horizon cutoff for HDE. Then, we plotted the density parameters for
DE and DM, the deceleration parameter, and the DE EoS parameter to give a
detailed account of how the universe evolves in this scenario. The results are
exciting, as our “FHDE” model provides a very appropriate description of late-
time cosmology. In particular, the Hubble horizon cutoff has been long shown
to have quite a few issues when employed to study HDE. Still, our model can
provide an acceptable evolution with this cutoff. Concretely, we showed that
for smaller values of $\alpha$, one sees acceptable evolution. We also
explored more subtleties of the model by studying its stability towards
classical perturbations using the squared sound speed and detailing the nature
of quintessence followed in it by considering the $w_{d}-w_{d}^{\prime}$ plane
evolution through a sizeable redshift range. Let us reiterate that we worked
in the Hubble horizon cutoff, thereby showing that the influence of fractional
effects can thus provide a fairly consistent HDE scenario and, in particular,
with a Hubble horizon cutoff.

We can propose multiple new directions of study based on our work here. For
example, it would be interesting to see how this model fits with the event
horizon or the particle horizon cutoff instead of the Hubble horizon cutoff.
One can also go towards more involved cutoffs like the Granda-Oliveros or the
generalized Nojiri-Odintsov cutoff and study this scenario there. It will be
interesting to study various thermodynamic aspects of this model, discussing
the validity of the generalized second law with this model in the multiple
cutoffs. One could also be tempted to study the far future scenarios of the
universe, including rips, with this model (like in [115]). Another point of
interest can be studying various cosmological singularities [116, 117] in this
scenario or studying these HDE models in non-GR cosmologies. These are all
exciting endeavours, but we shall keep these directions open for future
exploration as they currently lie beyond the scope of this work.

## Acknowledgements

The authors would like to thank Robert Scherrer and Sergei Odintsov for
various useful discussions on HDEs. PM acknowledges the FCT grant UID-B-
MAT/00212/2020 at CMA-UBI plus the COST Action CA23130 (Bridging high and low
energies in search of quantum gravity (BridgeQG)).

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Lack of personal confidence |
Incompatibility |
How do devices work |  | Topics that came up when any interview participant
---
explained their understanding of how home IoT devices worked
Data devices have |
Always listening? |
Never considered |
Not sure how secure it is |
Comfortable about use in home |
Not comfortable about use in home |
How much can they do with the data? |
Download of personal data |
About the devices |  | Explanations of how interview participants
---
viewed the devices and why they bought them
Just like any other white good |
Why bought? |
Parental relationship with devices |  | How parents view home IoT devices
---
with respect to their children’s use
| Devices aren’t different
---
to other technologies
Content concerns |
Destruction of device |
Kid’s filter is too restrictive |
Concerns for children |
Educating yourself to look after children |
Where to get support from |  | Where participants explained they would
---
get information about security, should they need it
TV programs |
Online - specific source |
YouTube |
Bank |
Government |
No idea |
Police |
Family |
Friends |
Online - generic |
Concerns |  | All concerns listed, by any participant,
---
about digital technology use
Privacy |
Security |
Unintended consequences |
Parental tracking |
Lack of parental control |
Restrictions |
Children believing everything online |
Child/restricted accounts |
Risks |  | The risks that participants saw in
---
relation to using home IoT devices
Data breach |
Time waste |
Physical damage |
Malfunction |
Inappropriate content |
Viruses |
Lack of accountability for online info |
Identity Theft (real or accidental) |
Unsolicited contact from strangers |
Scams/financial fraud |
”Hacking” |
Doxxing/identifying yourself |
Information theft |
Parent/child discussion | Types of family discussions described by participants
Discussing things learned at school |
Discussion as a means of solving problem |
Children learn from parents |
Safeguarding |
Child telling parent what to do |
Cyber security management in the home |
Knowledge |  | Instances in which the participant’s level of,
---
or means of acquiring, knowledge were explained
Tech negatively in the news |
Tech in the news |
How to do it |
Not that concerned |
Self-confidence |
Too difficult to think about |
Good level of security in the house |
Unsure of level of security in the house |
What is taught at school |
What school could cover |
Strategies | Examples of specific cyber security strategies
Turning off WiFi |
Password manager |
Awareness of malicious things online |
Smishing |
Old devices |
Biometrics |
2FA |
Data deletion |
Anti-malware |
Password reuse/weak passwords |
Personal data use |
Passwords |
Software updates |