# Modified cosmology from quantum deformed entropy

S. Jalalzadeh  0000-0003-4854-2960<EMAIL_ADDRESS>Departamento de
Física, Universidade Federal de Pernambuco, Recife-PE, 50670-901, Brazil  H.
Moradpour  0000-0003-0941-8422<EMAIL_ADDRESS>Research Institute for
Astronomy and Astrophysics of Maragha (RIAAM), P.O. Box 55134-441, Maragheh,
Iran  P.V. Moniz  0000-0001-7170-8952<EMAIL_ADDRESS>Departamento de Física and
Centro de Matemática e Aplicações (CMA - UBI), Universidade da Beira Interior,
6200 Covilhã, Portugal

###### Abstract

In Ref. [1], Jalalzadeh established that the thermodynamical entropy of a
quantum-deformed black hole with horizon area $A$ can be written as
$S_{q}=\pi\sin\left(\frac{A}{8G\mathcal{N}}\right)/\sin\left(\frac{\pi}{2\mathcal{N}}\right)$,
where $\mathcal{N}=L_{q}^{2}/L_{\text{P}}^{2}$, $L_{\text{P}}$ being the
Planck length and $L_{q}$ denoting, generically, the q-deformed cosmic event
horizon distance $L_{q}$. Motivated by this, we now extend the framework
constructed in [1] towards the Friedmann and Raychaudhuri equations describing
spatially homogeneous and isotropic universe dynamics. Our procedure in this
paper involves a twofold assumption. On the one hand, we take the entropy
associated with the apparent horizon of the Robertson–Walker universe in the
form of the aforementioned expression. On the other hand, we assume that the
unified first law of thermodynamics, $dE=TdS+WdV$, holds on the apparent
horizon. Subsequently, we find a novel modified cosmological scenario
characterized by quantum-deformed (q-deformed) Friedmann and Raychaudhuri
equations containing additional components that generate an effective dark
energy sector. Our results indicate an effective dark energy component, which
can explain the Universe’s late-time acceleration. Moreover, the Universe
follows the standard thermal history, with a transition redshift from
deceleration to acceleration at $z_{\text{tran}}=0.5$. More precisely,
according to our model, at a redshift of $z=0.377$, the effective dark energy
dominates with a de Sitter universe in the long run. We include the evolution
of luminosity distance, $\mu$, the Hubble parameter, $H(z)$, and the
deceleration parameter, $q(z)$, versus redshift. Finally, we have conducted a
comparative analysis of our proposed model with others involving non-extensive
entropies.

††journal: Physics of the Dark Uiverse

##  1 Introduction

Entropy is a fundamental quantity, constituting a significant concept. It is
desirable that a generalized definition of entropy may apply to all physical
systems; otherwise, it may mean that we do not grasp what physical entropy is.
In fact, as quantum mechanics, quantum field theory, and quantum gravity have
developed, entropy has not been cast as a universal property or a generic
form, but instead, it has been presented within many contexts, depending on
the particular physical system under investigation. For instance, one of the
key findings in theoretical physics is that a black hole (BH) is associated
with blackbody radiation, which has a distinct temperature and
Bekenstein–Hawking entropy [2, 3]. Unlike classical thermodynamics, where the
entropy is directly related to the system’s volume and represents an extensive
quantity, the Bekenstein–Hawking entropy is proportional to the horizon’s
area.

Interestingly enough, it has also been shown that the field equations of
general relativity can be obtained as a thermodynamic equation of state [4], a
result also confirmed in modified gravity [5]. Parallel to this achievement
signalling a deep connection between gravity and thermodynamics and hence
statistical mechanics, it has also been addressed that quantum aspects of
gravity may trigger gravity to show intrinsic non-extensive features [6, 7].
Indeed, the long-range nature of gravity is another similar reason to study
cosmology in the framework of generalized statistics and corresponding
thermodynamics [8, 9, 10, 11, 12, 13]. Employing those statistics as mentioned
earlier may also provide an origin for dark matter and MOND theory [14, 15].
The applications are not limited to these areas, and it seems there is hope to
solve the problems of BH physics, astrophysics, and high-energy physics using
the generalized statistics framework. [16, 17, 18, 19, 20, 21, 22].

Recent literature has proposed several definitions of entropy based on non-
additive statistics, such as the Tsallis [23, 24], Rényi [25], and Barrow [26]
entropies, in addition to the Bekenstein–Hawking one, due to the elusive
nature of entropy. Moreover, the authors of Refs. [27, 28] proposed a
fractional-fractal entropy that could encode the random fractal features of a
BH horizon surface that resulted from fractional quantum gravity effects. The
Sharma–Mittal entropy [10], the Kaniadakis entropy [29, 30, 31], the entropy
in the setting of Loop Quantum Gravity [32, 33], the quantum cosmology [34,
35, 36, 37] and the entropy in fractional quantum gravity-cosmology [28, 27,
38] constitute some more well-known entropies.

Furthermore, a novel entropy formula for a BH was recently proposed by one of
us [1], based on the quantum deformation (or q-deformation) approach to
quantum gravity. Very briefly, let us explain this entropy.

According to various quantum gravity proposals, the BH horizon area, $A_{n}$,
may be quantized, and the appropriate eigenvalues are given by [39]

$A_{n}=\gamma{\,}L_{\text{P}}^{2}{\,}n,~{}~{}~{}~{}n=1,2,3,...~{},$ (1)

where $\gamma$ is a model-dependent dimensionless constant of order one, and
$L_{\text{P}}=\sqrt{G}$ is the Planck length. Since Bekenstein’s
groundbreaking work, the literature has been enriched with a multitude of
contributions that reinforce the conjecture of the area spectrum (1). These
contributions are diverse and multifaceted, encompassing various
considerations such as information theory (as outlined in [40, 41]), arguments
from string theory [42], or the periodicity of time [43, 44, 37]. Furthermore,
contributions extended from the quantization of a dust collapse using
Hamiltonian methods [45, 46], can also be mentioned. However, a BH can be
considered physically embedded in the Universe, characterized by event horizon
$A_{\text{U}}$. As a result, the Schwarzschild event horizon of a BH cannot be
larger than the cosmic event horizon of the Universe, $A_{\text{U}}$. The
usual quantization methods with a spectrum given by Eq. (1) must consistently
meet this reasoning. It has to be adjusted to be bounded from above. Quantum
deformation of a model (using quantum groups) is one such method, allowing
retrieval of the dimension of a Hilbert space into a finite value, assuming
that the deformation parameter is a root of unity [47, 48, 49, 50, 51, 52,
53].

In this work, we propose to extend the quantum-deformed (q-deformed) entropy
idea, applying the first law of thermodynamics to the Universe’s horizon,
adapting to a cosmological scenario the thermodynamical entropy of a quantum-
deformed BH with horizon area $A$ presented in [1]. This allows us to
construct a q-deformed modified version of the Friedmann and Raychaudhuri
equations. In particular, there will be newly added components whose dynamical
implications motivate our investigation. To make clear our reasoning, let us
add the following. The quantum-deformed black hole is a model constructed from
the quantum Heisenberg–Weyl group. Quantum groups provide us with more complex
symmetries than the classical Lie algebras, which are included in the former
as a specific case. This indicates that quantum groups may be appropriate for
describing symmetries of physical systems that lie beyond the scope of Lie
algebras. Moreover, q-deformed models have a significant advantage because
their corresponding Hilbert space is finite-dimensional when $q$ is a root of
unity [54]. This implies that using quantum groups with a deformation
parameter of the root of unity is useful for constructing models with a finite
number of states. These models can be used to explore applications in quantum
gravity and quantum cosmology that adhere to the holographic principle and
UV/IR mixing to solve the CC problem [36].

Last but not least, within modified gravity settings, it may be deemed
necessary to elucidate the principal rationale behind the development of
alternative models for the standard model of cosmology despite its achievement
and conformity with observational data. The standard model of cosmology,
though successful in its own right, is plagued by a fundamental and unresolved
difficulty known as the cosmological constant (CC) problem. This enigma has
long baffled cosmologists and remains a significant obstacle in advancing the
field.

Numerous theoretical physicists expressed their reluctance to acknowledge the
CC as a feasible justification for the accelerated expansion of the universe
due to the fact that the anticipated value of CC from particle physics is
$\rho_{\Lambda}\simeq M_{\text{P}}^{4}\simeq(10^{18}~{}\text{GeV})^{4}$, a
value that differs significantly from the astronomical limit for CC, which is
$\rho_{\Lambda}\simeq(10^{-3}~{}\text{eV})^{4}$ –roughly $10^{123}$ times less
than expected. The CC is regarded as the zero-point energy with a UV cutoff
scale, such as the Planck scale or the supersymmetry breaking scale, from an
effective field theory (EFT) perspective, whereas from a cosmological
perspective, it is an IR scale problem that affects the entire universe’s
large-scale structure. As a result, the CC issue appears to contravene our
preconceived notion of separating UV and IR scales, which is the foundation of
EFT. The CC can be interpreted as both the zero point energy and the scale of
the observed Universe, which contradicts the concept of local quantum fields.
This suggests a mixing between the local UV and global IR physics. Some
physicists argue that the CC problem is essentially a quantum gravity and
quantum cosmology problem [55, 56, 57, 36]. Therefore, a candidate theory for
quantum gravity must provide a classical continuum spacetime geometry at
macroscopic scales with a global IR cutoff (CC) while also incorporating
quantum corrections at the local UV scale. This approach would allow for a
better understanding of the complex nature of the CC problem. It is commonly
assumed that there exists a solution to the problematic “old” CC problem. This
solution would result in the vacuum energy being exactly zero and radiatively
stable.

With the current conditions, it is evident that any suggested substitute for
the standard model of cosmology must elude the CC problem. For example,
consider the replacement model with several free parameters. By adjusting the
free parameters of the model, we may achieve an even more precise fit than the
standard model of cosmology. However, the issue arises when we attempt to
explain these free parameters’ origin and physics in our model, which will
involve a comparable problem. In section 5 of the article, we will provide an
example to revisit this case.

Our paper structure is as follows. In the next section, we summarize with
minimal detail q-deformed BH entropy. Section 3 examines the applicability of
the previously mentioned procedure in cosmology. Consequently, this enables us
to advance a new modified scenario built from the q-deformed entropy. Section
4 examines the cosmological consequences of the additional components in the
q-deformed Friedmann and Raychaudhuri equations, concentrating on the dark
energy density and equation-of-state (EoS) parameters. In section 5, a
comparative analysis has been conducted between our proposed model and others
previously established in the field of entropic cosmology, with a particular
emphasis on examining the similarities and differences between ours and the
Barrow model in [58]. Finally, in Section 6, we will examine our findings.

##  2 Entropy of a q-deformed Schwarzschild BH

Let us briefly go through a few significant results about the q-deformation
quantization and entropy of the Schwarzschild BH with mass $M$. Following
Louko in Ref. [59], we start with the reduced action of a Schwarzschild BH
given by

$S=\int\Big{\\{}P_{M}\dot{M}-H(M)\Big{\\}}dt,$ (2)

where $H(M):=M$ is the reduced Hamiltonian, and $P_{M}$ is the canonical
conjugate momenta of the BH mass $M$. The solutions of the field equations are
$M=\text{const}.$ and $P_{M}=-t$. The constancy of mass $M$ follows Birkhoff’s
theorem, which states that the mass is the only time-independent and
coordinate-invariant solution. Furthermore, $P_{M}$ represents the asymptotic
time coordinate at the spacelike slice [60]. Thus, $M$ contains all the
pertinent information regarding the local geometry of the classical solutions.
On the other hand, the conjugate momenta, $P_{M}$, is equivalent to the
disparity of the asymptotic Killing times between the left and right
infinities on a constant $t$ hypersurface [59], following the convention where
the Killing time at the right (left) infinity increases towards the future
(past). As a result, it does not contain any information regarding the local
geometry. However, it instead contains information regarding securing the
spacelike hypersurfaces at the two infinities [59].

Moreover, it is customary to restrict the mass manually and the corresponding
momenta to the range $|P_{M}|<\pi M/M_{\text{P}}^{2}$, where
$M_{\text{P}}=1/\sqrt{G}$ is the Planck mass. This implies that the Minkowski
time on the asymptotic right-hand side of each classical solution only lies
within an interval of length $2\pi M/M_{\text{P}}^{2}$, centered around a
value that is diagonally opposite to the non-evolving left end of the
hypersurfaces in the Kruskal diagram. With respect to the time parameter $t$,
each classical solution is exclusively defined for an interval of $-\pi
M/M_{\text{P}}^{2}<t-t_{0}<\pi M/M_{\text{P}}^{2}$, where $t=t_{0}$ represents
the hypersurface whose two asymptotic ends are diagonally opposite. As
previously stated, in light of Euclidean quantum gravity [61] (where the
situation is similar to finite temperature quantum field theory, when the time
is Euclideanized [62]), we posit that the conjugate momentum, $P_{M}$, serves
as a temporal measure and, therefore, must exhibit periodicity with a period
inverse to the Hawking temperature $T_{\text{H}}=M^{2}_{\text{P}}/8\pi M$ [63,
64]. This guarantees the absence of a conical singularity in the two-
dimensional Euclidean section close to the black hole horizon. However, it is
important to note that the aforementioned identification results in a physical
phase space that constitutes a wedge extracted from the complete $(M,P_{M})$
plane. The $M$ axis and the $P_{M}=1/{T_{\text{H}}}$ line bound this wedge.
Therefore,

$P_{M}\sim P_{M}+\frac{1}{T_{\text{H}}}.$ (3)

The above boundary condition verifies that there is no conical singularity in
the $2D$ Euclidean section.

Also, it indicates that the phase space is a wedge cut out from the full
$(M,P_{M})$ phase space, bounded by the mass axis and the line
$P_{M}=1/T_{\text{H}}$ [65]. Hence, according to the references [59, 66], one
could make the following canonical transformation $(M,P_{M})\rightarrow(x,p)$,
which simultaneously opens up the phase space and also incorporates the
periodicity condition (3)

$x=\sqrt{\frac{A}{4\pi G}}\cos(2\pi
P_{M}T_{\text{H}}),\,\,\,\,\,\,p=\sqrt{\frac{A}{4\pi G}}\sin(2\pi
P_{M}T_{\text{H}}),$ (4)

where $A=16\pi M^{2}/M_{\text{P}}^{4}$ is the BH horizon area. From the above
canonical transformations, one immediately finds the horizon area in terms of
$(x,p)$

$A=4\pi L_{\text{P}}^{2}\Big{(}x^{2}+p^{2}\Big{)},$ (5)

where $L_{\text{P}}=1/M_{\text{P}}$ is the Planck length.

Let us define the ladder operators, $\\{a_{-},a_{+}\\}$, by

$a_{\pm}=\frac{1}{\sqrt{2}}\Big{(}\pm\frac{d}{dx}+x\Big{)}.$ (6)

The pairs of operators $a_{\pm}$ act on states as the following form

$a_{+}|n\rangle=\sqrt{n+1}|n+1\rangle,~{}~{}~{}~{}a_{-}|n\rangle=\sqrt{n}|n-1\rangle.$
(7)

This gives us the possibility to rewrite the area operator (5) of the BH in
terms of ladder operators

$A=4\pi L_{\text{P}}^{2}\Big{(}a_{+}a_{-}+a_{-}a_{+}\Big{)}.$ (8)

Therefore, the area of the event horizon and the mass spectrum [59] are

$A_{n}={8\pi
L_{\text{P}}^{2}}\Big{(}n+\frac{1}{2}\Big{)},~{}~{}~{}~{}M_{n}=\frac{M_{\text{P}}}{\sqrt{2}}\sqrt{n+\frac{1}{2}},$
(9)

where $n$ is an integer. Expressions (9) give the well-known result: Hawking
radiation takes place when the BH jumps from a higher state $n+1$ to a lower
state $n$, in which the difference in quanta is radiated away. Also, they show
that the BH does not evaporate completely, but a Planck-size remnant is left
over at the end of the evaporation process.

In 1974, Hawking [67] showed that due to quantum fluctuations, BHs emit
blackbody radiation, and the corresponding entropy is one-fourth of the event
horizon area, namely $A=16\pi G^{2}M^{2}$. Following Refs. [68] and [69], let
us assume that Hawking radiation of a massive BH, i.e., $M\gg M_{\text{P}}$
and $n\gg 1$, is emitted when the BH system spontaneously jumps from the state
$n+1$ into the closest state level, i.e., $n$, as described by (9). If we
denote the frequency of emitted radiation as $\omega_{0}$, then

$\omega_{0}=M_{n+1}-M_{n}\simeq\frac{M_{\text{P}}}{2\sqrt{2n}}\simeq\frac{M_{\text{P}}^{2}}{4M}.$
(10)

This agrees with the classical BH oscillation frequencies, which scale $1/M$.
We thus expect a BH to radiate with a characteristic temperature $T\propto
M_{\text{P}}^{2}/M$, matching the Hawking temperature.

The BH entropy can be expressed in terms of the following adiabatic invariant

$S_{\text{BH}}=8\pi\int_{M_{\text{P}}}^{M}\frac{dM}{\omega_{0}}=\frac{A}{4G}=4\pi
M^{2}G,$ (11)

where $A=4\pi R_{\text{S}}^{2}$ is the BH horizon area, and $R_{\text{S}}=2MG$
is the Schwarzschild radius. Because this spectrum is equally spaced, the
possible values for the area of a massive BH are equally spaced.

One can obtain the quantum deformed (q-deformed) extension of the BH horizon
area (8) by replacing the ordinary ladder operators (6) with their q-extended
ladder operators in which the Heisenberg–Weyl algebra turns into the quantum
Heisenberg–Weyl algebra, $U_{q}(h_{4})$. The quantum Heisenberg–Weyl algebra,
is the associative unital [70] $\mathbb{C}(q)$-algebra with generators
$\\{a_{+},a_{-},q^{\pm N/2}\\}$ with the following q-deformed commutation
relations [70]

$\begin{split}&a_{-}a_{+}-q^{\frac{1}{2}}a_{+}a_{-}=q^{\frac{N}{2}},~{}~{}~{}[N,a_{\pm}]=\pm
a_{\pm},\\\ &a_{\pm}^{\dagger}=a_{\mp},~{}~{}N^{\dagger}=N,\end{split}$ (12)

where $q$ is a primitive root of unity and is given by

$q=\exp\left(\frac{2\pi i}{\mathcal{N}}\right),$ (13)

where $\mathcal{N}$ is the q-deformation parameter. The q-deformation is
thought to be related to a fundamental dimensional constant. Furthermore, the
deformation parameter, $q$, must be a dimensionless function of such constant
as well as of any specific system-characterizing features. The natural length
scale of quantum gravity is the Planck length. Thus, the q-deformation
parameter should be a function of the gravitational constant, $G$, or the
Planck length squared [71] $\mathcal{N}=\mathcal{N}(L_{\text{P}}^{2})$. In
addition, one may expect that at the classical gravity limit,
$L_{\text{P}}\rightarrow 0$, $\mathcal{N}$ tends to infinity, and the theory
tums into a classical gravity. This means that the q-deformation is a pure
quantum gravity effect, and $\mathcal{N}\propto 1/L_{\text{P}}^{2}$. Regarding
$\mathcal{N}$ being a dimensionless parameter, we need another length scale in
which the ratio of the Planck length and the new length gives us the
q-deformation parameter, i.e., $\mathcal{N}=L_{q}^{2}/L_{\text{P}}^{2}$. In
section 4, we will show that this assumption leads us to an asymptotically de
Sitter cosmological model, where $L_{q}$ plays the role of de Sitter radius.
Moreover, we further take this new length scale as related to the
“cosmological” constant, as consistently induced by the q-deformation of the
model. Hence, we consider $L_{q}\equiv\sqrt{3/\Lambda_{q}}$, where
$\Lambda_{q}$ is q-cosmological constant, and
$\mathcal{N}=L_{q}^{2}/L_{\text{P}}^{2}$ which leads to

$q=\exp\left(2\pi i\frac{L_{\text{P}}^{2}}{L_{q}^{2}}\right).$ (14)

Note that in our approach, $L_{q}$, similar to the Planck length, is a
fundamental dimensional constant of quantum gravity theory 111The CC, like the
gravitational constant, is also used as a coupling constant in loop quantum
gravity (LQG) and spinfoam frameworks [72, 73, 74, 75, 76]. In LQG, a
q-deformation has been derived as a mechanism to apply the theory’s dynamics
using $\Lambda$ and the deformation parameter, $q$, which is then given by
(14) [77]. . One can obtain the classical gravity limit of the theory by
$L_{q}\rightarrow\infty$ or a vanishing q-cosmological constant. The above
form of the deformation parameter realizes a holographic picture of quantum
mechanics [78]. Concretely, as shown in [78], the Hilbert space of q-deformed
neutral hydrogen gas in de Sitter space satisfies the strong holographic bound
with the above deformation parameter.

The above quantum deformation of the BH gives us the following eigenvalues for
the surface area and the mass of the BH [1]

$A_{n}={4\pi
L_{\text{P}}^{2}}\frac{\sin(\frac{\pi}{\mathcal{N}}(n+\frac{1}{2}))}{\sin(\frac{\pi}{2\mathcal{N}})},$
(15)
$M_{n}=\frac{M_{\text{P}}}{2}\sqrt{\frac{\sin(\frac{\pi}{\mathcal{N}}(n+\frac{1}{2}))}{\sin(\frac{\pi}{2\mathcal{N}})}},$
(16)

where $n=0,...,\mathcal{N}-1$. Note that for $\mathcal{N}\rightarrow\infty$
(or equivalently, $\Lambda_{q}\rightarrow 0$) the earlier eigenvalues will
reduce to (9). Also, there is a two-fold degeneracy at the horizon’s
eigenvalues further at the BH’s mass spectrum.

To summarize the consequences of the above eigenvalue relations, for
simplicity, let us consider $\mathfrak{N}$ is an odd natural number:

1) The area and the mass of the ground state $n=0$, as well as the state
$n=\mathcal{N}-1$, are [1]

$A_{0}=A_{\mathcal{N}-1}=4\pi
L_{\text{P}}^{2},\,\,\,\,M_{0}=M_{\mathcal{N}-1}=\frac{M_{\text{P}}}{2}.$ (17)

These show that the ground state’s area and mass are not deformed, and their
values are the same as the non-deformed spectrum obtained in (9). Besides, the
spectrum is bounded in which the most excited state, $n=\mathcal{N}-1$, has
the same mass and area as the ground state. In addition, for
$n\ll(\mathcal{N}-1)/2$ or $(\mathcal{N}-1)/2\ll n\leq\mathcal{N}-1$, Eqs.
(15) and (16) will reduce to the non-deformed spectrum obtained in (9).

2) The above surface area spectrum of the BH leads us to the following
q-deformed entropy [1]

$S_{q}=\pi\frac{\sin(\frac{\pi}{\mathcal{N}}(n+\frac{1}{2}))}{\sin(\frac{\pi}{2\mathcal{N}})}=\pi\frac{\sin(\frac{\pi
R_{S}^{2}}{2G\mathcal{N}})}{\sin(\frac{\pi}{2\mathcal{N}})},~{}~{}~{}~{}\frac{\pi
R_{S}^{2}}{2G\mathcal{N}}\leq\frac{\pi}{2},$ (18)

where $R_{S}$ is the Schwarzschild radius of the BH.

With keeping one eye on Eq. (6), the commutation relation of the q-deformed
$x$ and $p$ is

$[x,p]|n\rangle=i\frac{\cos\left(\frac{\pi
G\Lambda_{q}}{3}(n+\frac{1}{2})\right)}{\cos(\frac{\pi
G\Lambda_{q}}{6})}|n\rangle.$ (19)

According to Eq. (18), the q-entropy of a BH reaches its greatest value for
$n=(\mathcal{N}-1)/2$. In this case, regarding the second equality in Eq.
(18), the Schwarzschild horizon radius is $R_{S}^{2}=G\mathcal{N}=L_{q}^{2}$.
On the other hand, Eq. (19) shows that for $n=(\mathcal{N}-1)/2$, (or when the
BH radius equals the de Sitter radius) $x$ and $p$ commute. As a result, the
classical state is the one with the most entropy or maximum radius. Note that
the effective Schwarzschild horizon radius is different from $R_{S}$ and is
given by

$R_{\text{eff}}=L_{\text{P}}\sqrt{\frac{\sin(\frac{\pi
R_{S}^{2}}{2\mathcal{N}L_{\text{P}}^{2}})}{\sin(\frac{\pi}{2\mathcal{N}})}}.$
(20)

##  3 q-deformed Friedmann and Raychaudhuri equations

Our starting point is the Friedmann–Lemaître–Robertson–Walker (FLRW) line
element in comoving coordinates

$ds^{2}=h_{ab}dx^{a}dx^{b}+R^{2}d\Omega^{2}_{(II)},~{}~{}~{}a=0,1,$ (21)

where $d\Omega^{2}_{(II)}$ is the line element of the standard 2-sphere,
$R(r,t)=ra(t)$ is the areal radius, and
$h_{ab}=\text{diag}{(-1,\frac{a(t)^{2}}{1+kr^{2}})}$. Moreover, $x^{0}=t$,
$x^{1}=r$, and as usual, the open, flat, and closed universes correspond to
$k=-1,0,1$, respectively

Several authors [79, 80, 81, 82, 83, 84] have taken the de Sitter event (and
apparent) horizon thermodynamic formulae to encompass, constituting a larger,
upper bound regarding a generic FLRW space’s non-static apparent horizon,
which differs from the event horizon. For the sake of clarity, let us
emphasize that it is widely proposed in dynamical spacetimes that the apparent
horizon represents a causal horizon associated to gravitational temperature
and entropy (and surface gravity). If this is correct, the same might be said
about cosmic horizons. It was suggested in Refs. [80, 85, 86, 82] that the
FLRW thermodynamical event horizon needs to be better defined (except for the
case of de Sitter space). The FLRW apparent horizon’s Hawking radiation was
calculated by the authors of [87, 88]. It was rederived in references [89, 90]
by applying the Hamilton–Jacobi method [91, 92, 93] to the Parikh–Wilczek
approach, which was initially developed for BH horizons [94].

A substantial amount of research has been conducted on the bridging of
geometry and thermodynamics within FLRW spaces (see, for example, [95, 96] and
references therein). The thermodynamical properties of the FLRW apparent
horizon are described in Ref. [97]. Allow us to point to the Kodama vector,
Kodama–Hayward surface gravity, and corresponding Hawking temperature
calculations. The Kodama–Hayward temperature of the FLRW apparent horizon is
given by

$T_{\text{AH}}=\frac{\kappa_{\text{kodama}}}{2\pi}=-\frac{1}{2\pi
R_{\text{AH}}}\left(1-\frac{\dot{R}_{\text{AH}}}{2HR_{\text{AH}}}\right),$
(22)

where $H=\dot{a}/a$ is the Hubble parameter, $\kappa_{\text{Kodama}}$ is the
Kodama surface gravity of the apparent horizon, and $R_{\text{AH}}$ is the
radius of the apparent horizon given by

$R_{\text{AH}}=\frac{1}{\sqrt{H^{2}+\frac{k}{a^{2}}}}.$ (23)

Let us assume that the FLRW universe is sourced by a perfect fluid with
energy-momentum tensor

$T_{\mu\nu}=(\rho+p)u_{\mu}u_{\nu}+g_{\mu\nu}p,$ (24)

where $\rho$, $p$, and $u_{\mu}$ are the fluid’s total energy density,
pressure, and 4-velocity field, respectively. The perfect fluid could be a
combination of non-interacting dust (cold dark and baryonic matters),
$\rho_{\text{c}}$, and radiation, $\rho_{\text{rad}}$ 222We use a subscript
$x$ as one of “c” for the dust of baryons plus dark matter, “rad” for
radiation (photons plus relativistic neutrinos), “m” for baryons plus dark
matter plus radiation, and “DE” for the effective dark energy.

$\begin{split}&\rho=\rho_{\text{c}}+\rho_{\text{rad}},\\\
&p=p_{\text{rad}},~{}~{}~{}~{}p_{\text{c}}=0,\end{split}$ (25)

Regarding these considerations, the unified first law of thermodynamics on the
apparent horizon [83, 84] is given by

$T_{\text{AH}}\dot{S}_{\text{AH}}=\dot{M}_{\text{AH}}+\frac{p-\rho}{2}\dot{V}_{\text{AH}},$
(26)

where an overdot means time derivative, $S_{\text{AH}}$ is the entropy of the
apparent horizon, and

$V_{\text{AH}}=\frac{4\pi}{3}R_{\text{AH}}^{3},~{}~{}~{}M_{\text{AH}}=\rho
V_{\text{AH}},$ (27)

are the areal volume and the Misner–Sharp–Hernandez mass contained inside the
apparent horizon, respectively.

Inserting the Kodama–Hayward temperature (22), the areal volume, and the
Misner–Sharp–Hernandez mass defined in (27) into the unified first law of
thermodynamics (26) gives us

$\dot{S}_{\text{AH}}=8\pi^{2}HR_{\text{AH}}^{4}H(\rho+p).$ (28)

If we postulate that the covariant conservation equation is satisfied by the
energy-momentum tensor of the universe’s perfect fluid composition (24), then
we find

$H\left(\rho_{i}+p_{i}\right)=-\frac{\dot{\rho}_{i}}{3},~{}~{}~{}~{}i=\text{c},\text{rad}.$
(29)

Inserting this relation in the r.h.s of Eq. (28) simplifies it into

$\frac{1}{8\pi^{2}R_{\text{AH}}^{4}}\dot{S}_{\text{AH}}=-\frac{1}{3}\dot{\rho}.$
(30)

Now, our assumption is to take the entropy associated with the apparent
horizon in the form of q-deformed entropy (18) and replace the BH horizon
radius, $R_{S}$, with the apparent horizon radius $R_{\text{AH}}$. This gives
us the q-deformed entropy of the apparent horizon

$S_{q}=\pi\frac{\sin(\frac{\gamma
R_{\text{AH}}^{2}}{G})}{\sin(\gamma)},~{}~{}~{}~{}~{}0\leq\frac{\gamma
R_{\text{AH}}^{2}}{G}\leq\frac{\pi}{2},$ (31)

where
$\gamma=\pi/(2\mathcal{N})=\frac{\pi}{2}\left(\frac{L_{\text{P}}}{L_{q}}\right)^{2}=\frac{\pi}{6}G\Lambda_{q}$.
Let us first discuss some of the outcomes of this expression for the entropy
before delving further.

  1. 1.

Eq. (31) is non-classical by definition. According to the noncommutative
perspective of quantization [1, 36, 98, 99, 100, 101], $\hbar$ and
$\Lambda_{q}$ are both quantization parameters, and the classical limit of the
model can be realized by establishing the $\hbar\rightarrow 0$ limit. On the
other hand, $\Lambda_{q}\rightarrow 0$ gives us quantum gravity without a
q-cosmological constant.

  2. 2.

Our cosmological model did not account for a traditional CC, but our
subsequently implemented quantum deformation led to the emergence of an
effective one, strongly linked to the natural number $\mathcal{N}$ defined by
(13). This constant directly results from the assumption of a finite number of
states in Hilbert spaces [36].

  3. 3.

Regarding our discussion on the entropy of q-deformed BHs in the previous
section, the interval $0\leq\frac{\gamma R_{\text{AH}}^{2}}{G}\leq{\pi}$ is
divided to $0\leq\frac{\gamma R_{\text{AH}}^{2}}{G}\leq\frac{\pi}{2}$, and
$\frac{\pi}{2}\leq\frac{\gamma R_{\text{AH}}^{2}}{G}\leq{\pi}$. As the first
interval, $0\leq\frac{\gamma R_{\text{AH}}^{2}}{G}\leq\frac{\pi}{2}$
represents an expanding universe, the second interval realizes a contracting
universe. These universes coexist simultaneously at the q-deformed quantum
cosmology level and have an entangled quantum state [1]. By observation, the
state collapses it into expanding or contracting universe. Here, we assumed
(regarding the cosmological observations), the state has collapsed into the
expanding universe.

  4. 4.

The apparent horizon radius, $R_{\text{AH}}$, is quantized according to Eq.
(18). As a result, $R_{\text{AH}}=L_{\text{P}}$ gives the smallest value of
the apparent horizon radius, whereas $L_{q}$ is the larger scale within our
knowledge of physics models and to ensure a finite Hilbert space. However,
because we are interested in the late-time evolution of the universe, we
assumed the apparent horizon radius was zero at the Big Bang for simplicity.

Implying the q-deformed entropy (31) for FLRW spaces into (30), gives us

$\cos(\frac{\gamma}{G}R_{\text{AH}}^{2})\frac{\dot{R}_{\text{AH}}}{R_{\text{AH}}^{3}}=-\frac{4\pi
G}{3}\frac{\sin(\gamma)}{\gamma}\dot{\rho}.$ (32)

Integration of the above differential equation yields

$\frac{\cos(\frac{\gamma}{G}R_{\text{AH}}^{2})}{R_{\text{AH}}^{2}}+\frac{\gamma}{G}\left\\{\operatorname{Si}\left(\frac{\gamma}{G}R_{\text{AH}}^{2}\right)-\operatorname{Si}\left(\frac{\pi}{2}\right)\right\\}=\\\
\frac{\sin(\gamma)}{\gamma}\frac{8\pi G}{3}\rho,$ (33)

where

$\operatorname{Si}(x)=\int_{0}^{x}\frac{\sin(y)}{y}dy,$ (34)

is the integral sine function. Note that, regarding
$0\leq\frac{\gamma}{G}R_{\text{AH}}^{2}\leq\pi/2$, the constant of integration
is chosen in such a way that the above equation becomes trivial for
$\frac{\gamma}{G}R_{\text{AH}}^{2}=\frac{\pi}{2}$. This means that, as we
expected, the FLRW space would be instead asymptotic de Sitter, i.e.,
$1/R_{\text{AH}}^{2}=\frac{\Lambda_{q}}{3}$.

One can simplify Eq. (33) by considering the value of
$\gamma=\frac{\pi}{2}\left(\frac{L_{\text{P}}}{L_{q}}\right)^{2}\simeq
10^{-123}$. This shows that $\sin(\gamma)/\gamma\simeq 1$. Hence, (33) reduces
to

$\frac{\cos(\frac{\pi\Lambda_{q}}{6}R_{\text{AH}}^{2})}{R_{\text{AH}}^{2}}=\frac{8\pi
G}{3}\rho\\\
+\frac{\pi\Lambda_{q}}{6}\left\\{\operatorname{Si}\left(\frac{\pi}{2}\right)-\operatorname{Si}\left(\frac{\pi\Lambda_{q}}{6}R_{\text{AH}}^{2}\right)\right\\}.$
(35)

Equation (35) is the q-deformed Friedmann equation based on the q-deformed
entropy. Therefore, assuming the unified first law of thermodynamics applied
at the apparent horizon of an FLRW universe, plus supposing that the apparent
horizon area includes quantum deformed characteristics, we derive the
corresponding modified Friedmann equation of an FLRW universe with any spatial
curvature.

The Raychaudhuri equation, often known as the second Friedmann equation, may
be obtained directly using Eq. (28) and the Friedmann equation (35). Inserting
the q-entropy (31) into (28) and using

$\dot{R}_{\text{AH}}=-R_{\text{AH}}^{3}\left(\frac{\ddot{a}}{a}-\frac{1}{R_{\text{AH}}^{2}}\right)H,$
(36)

we obtain

$-\frac{\ddot{a}}{a}+\frac{1}{R_{\text{AH}}^{2}}=\frac{4\pi
G}{\cos(\frac{\gamma}{G}R_{\text{AH}}^{2})}\frac{\sin(\gamma)}{\gamma}(\rho+p).$
(37)

This equation, combined with the Friedmann equation (35), gives the q-deformed
Raychaudhuri equation

$\cos(\frac{\pi\Lambda_{q}}{6}R_{\text{AH}}^{2})\frac{\ddot{a}}{a}=-\frac{4\pi
G}{3}(\rho+3p)\\\
+\frac{\pi\Lambda_{q}}{6}\left\\{\operatorname{Si}\left(\frac{\pi}{2}\right)-\operatorname{Si}\left(\frac{\pi\Lambda_{q}}{6}R_{\text{AH}}^{2}\right)\right\\}.$
(38)

Note that at the limit classical limit of quantum geometry,
${\Lambda_{q}}\rightarrow 0$ (or $\mathcal{N}\rightarrow\infty$), Eqs. (35)
and (38) reduce to the standard Friedmann and Raychaudhuri equations

$\begin{split}&H^{2}+\frac{k}{a^{2}}=\frac{8\pi G}{3}\rho,\\\
&\frac{\ddot{a}}{a}=-\frac{4\pi G}{3}(\rho+3p),\\\
&\rho=\rho_{\text{c}}+\rho_{\text{rad}},~{}~{}~{}p=p_{\text{rad}},\end{split}$
(39)

where $\rho_{\text{c}}$ is the energy density of cold matter, and
$\rho_{\text{rad}}$, and $p_{\text{rad}}$ are the energy density and the
pressure of the radiation.

##  4 q-deformed cosmology

In the remainder of this paper, for convenience, we focus on the spatially
flat universe, namely $k=0$. Then, $1/R_{\text{AH}}=H^{2}$, and one can
rewrite the q-deformed Friedmann and Raychaudhuri equations (35) and (38) as

$H^{2}=\frac{8\pi G}{3}(\rho+\rho_{\text{DE}}),$ (40) $\dot{H}=-{4\pi
G}(\rho+p+\rho_{\text{DE}}+p_{\text{DE}}),$ (41)

where we have introduced an effective q-dark energy sector, with energy
density, $\rho_{\text{DE}}$, and pressure, $p_{\text{DE}}$, respectively, with
the following form

$\rho_{\text{DE}}=\frac{3}{4\pi
G}\sin^{2}\left(\frac{\pi{\Lambda_{q}}}{12H^{2}}\right)H^{2}+\\\
\frac{\Lambda_{q}}{16G}\left\\{\operatorname{Si}\left(\frac{\pi}{2}\right)-\operatorname{Si}\left(\frac{\pi\Lambda_{q}}{6H^{2}}\right)\right\\},$
(42) $p_{\text{DE}}=-\frac{1}{4\pi
G}\sin^{2}\left(\frac{\pi\Lambda_{q}}{12H^{2}}\right)\left[2\dot{H}+3H\right]\\\
-\frac{\Lambda_{q}}{16G}\left[\operatorname{Si}\left(\frac{\pi}{2}\right)-\operatorname{Si}\left(\frac{\pi\Lambda_{q}}{6H^{2}}\right)\right].$
(43)

These give us the EoS parameter for the q-dark energy sector

$\omega_{\text{DE}}=-1-\frac{8\sin^{2}\left(\frac{\pi\Lambda_{q}}{12H^{2}}\right)\dot{H}}{12\sin^{2}\left(\frac{\pi\Lambda_{q}}{12H^{2}}\right)H^{2}+\Lambda_{q}\pi\left[\operatorname{Si}\left(\frac{\pi}{2}\right)-\operatorname{Si}\left(\frac{\pi\Lambda_{q}}{6H^{2}}\right)\right]}.$
(44)

The equations are more easily expressed in terms of dimensionless variables.
Defining the density parameters of the matter (radiation plus cold dark and
baryonic matters), the q-deformation parameter, $\Lambda_{q}$, and effective
q-dark energy

$\Omega^{\text{(m)}}=\frac{8\pi
G\rho_{\text{m}}}{3H^{2}},~{}~{}~{}\Omega^{(q)}=\frac{\Lambda_{q}}{3H^{2}},~{}~{}~{}\Omega^{\text{(DE)}}=\frac{8\pi
G\rho_{\text{DE}}}{3H^{2}},$ (45)

then, the continuity equation (29) and q-deformed Raychaudhuri equations are
expressed as

$\frac{\cos(\frac{\pi\Omega^{(q)}}{2})}{3\Omega^{\text{(i)}}}\frac{d\Omega^{(i)}}{d\tilde{N}}=\sum_{j}(1+\omega_{j})\Omega^{\text{(j)}}-(1+\omega_{i}),~{}~{}~{}i=\text{c},\text{rad},$
(46)

and

$\frac{\cos(\frac{\pi\Omega^{(q)}}{2})}{H}\frac{dH}{d\tilde{N}}=-\frac{3}{2}\sum_{j}(1+\omega_{j})\Omega^{\text{(j)}},$
(47)

where $\tilde{N}=\ln(a/a_{0})$ is the e-folding factor. The equivalent
expression of the above equation can be obtained by using the definition of
the q-cosmological constant’s density parameter

$\frac{\cos(\frac{\pi\Omega^{(q)}}{2})}{\Omega^{(q)}}\frac{d\Omega^{(q)}}{d\tilde{N}}=3\sum_{j}(1+\omega_{j})\Omega^{\text{(j)}}.$
(48)

In addition, the q-deformed Friedmann (40) takes the following form

$\Omega^{\text{(DE)}}=1-\Omega^{\text{(m)}}=1-\cos(\frac{\pi\Omega^{(q)}}{2})+\\\
\frac{\pi\Omega^{(q)}}{2}\left\\{\operatorname{Si}\left(\frac{\pi}{2}\right)-\operatorname{Si}\left(\frac{\pi\Omega^{(q)}}{2}\right)\right\\},~{}~{}~{}i=\text{c},\text{rad}.$
(49)

This allows us to eliminate $\Omega^{(q)}$ (and $\Omega^{(DE)}$) in terms of
$\Omega^{\text{(m)}}$.

Eqs. (46) and (47) allow us to calculate the density parameters of the
radiation, the cold matter, and the q-deformation

$\begin{split}&\Omega^{\text{(rad)}}=\Omega^{\text{(rad)}}_{0}\left(\frac{H_{0}}{H}\right)^{2}(1+z)^{4},\\\
&\Omega^{\text{(c)}}=\Omega^{\text{(c)}}_{0}\left(\frac{H_{0}}{H}\right)^{2}(1+z)^{3},\\\
&\Omega^{(q)}=\Omega^{(q)}_{0}\left(\frac{H_{0}}{H}\right)^{2},\end{split}$
(50)

where $z=a_{0}/a-1$ is the redshift, and also $\Omega^{\text{(rad)}}_{0}$,
$\Omega^{\text{(c)}}_{0}$, and $\Omega^{(q)}_{0}$ are the density parameters
of the radiation, the cold matter, and q-deformation parameter at the present
epoch. Inserting these into the Hamiltonian constraint (49) gives us

$\cos(\frac{\pi\Omega^{(q)}_{0}}{2E^{2}})E^{2}=\frac{\pi\Omega^{(q)}_{0}}{2}\Big{\\{}\operatorname{Si}\left(\frac{\pi}{2}\right)-\\\
\operatorname{Si}\left(\frac{\pi\Omega^{(q)}_{0}}{2E^{2}}\right)\Big{\\}}+\Omega^{\text{(rad)}}_{0}(1+z)^{4}+\Omega^{\text{(c)}}_{0}(1+z)^{3},$
(51)

where $E={\frac{H}{H_{0}}}$.

Evaluating (51) at the present epoch gives us the relation between the density
parameters, which we now write as

$\cos(\frac{\pi\Omega^{(q)}_{0}}{2})-\frac{\pi\Omega^{(q)}_{0}}{2}\Big{\\{}\operatorname{Si}\left(\frac{\pi}{2}\right)-\\\
\operatorname{Si}\left(\frac{\pi\Omega^{(q)}_{0}}{2}\right)\Big{\\}}=\Omega^{\text{(m)}}_{0}.$
(52)

It would be beneficial to delve deeper into this equation. Fig. 1 illustrates
the profound influence of the q-deformation parameter on our universe. A
significant value of $\Omega^{(q)}_{0}$ (which implies a smaller current
Hubble distance of $L_{q}$) decreases the relevance of the baryonic and dark
matters. Conversely, a q-deformation parameter of zero elevates cold matter to
the position of the primary driving force behind the Universe’s evolution.
Consequently, very small q-deformation can significantly impact cosmology
during later stages.

Figure 1: Plot of the density parameter of the matter (radiation, plus cold
matter) versus the density parameter of the q-cosmological constant. We plot
this figure using Eq. (51).

It is clear that the above expression is not analytically solvable. Hence, we
use the global series expansion of
$\cos(x)+x\operatorname{Si}(x)=\sum_{n=0}^{\infty}\frac{(-1)^{n+1}x^{2n}}{(2n)!(2n-1)}$
to simplify Eq. (51) and obtain $\Omega^{(q)}_{0}$ in terms of density
parameter of the matte at the present epoch. Using this series expansion, Eq.
(52) up to $\mathcal{O}((\Omega^{(q)})^{4})$ simplifies to

$1+\frac{1}{2}\left(\frac{\pi\Omega^{(q)}_{0}}{2}\right)^{2}-\operatorname{Si}(\frac{\pi}{2})\left(\frac{\pi\Omega^{(q)}_{0}}{2}\right)-\Omega^{\text{(m)}}_{0}=0.$
(53)

It is essential to note that we have not used any approximations until this
point. However, the q-deformed Friedmann equation (40) and Raychaudhuri
equation (41) do not have analytical solutions, making it complicated to
calculate different quantities, like the age of the universe or distance
modulus. Although numerical methods can be used to perform these calculations
without approximation, our primary goal in this paper is to enable readers to
follow the calculations easily. In the early universe and during the
radiation-dominated epoch, Eqs. (40) and (41) are equivalent to the Friedmann
equation and Raychaudhuri equations in the $\Lambda$CDM model with exceptional
approximation; please see Fig. 2. Nonetheless, the differences between the
standard model and our model are only noticeable at small redshifts.

Figure 2: Plot of $H/H_{0}$ (radiation$+$cold matter) versus the scale factor
in flat $\Lambda$CDM and q-deformed cosmology. The black line (dash) shows the
evolution of the Hubble parameter in q-deformed ($\Lambda$CDM) cosmology. We
plot these figures using Eq. (51) for q-deformed cosmology and the standard
equation of $H/H_{0}$ in the standard model of cosmology, respectively. We
used $\Omega^{(q)}_{0}=0.43$, $\Omega^{\text{(c)}}_{0}=0.3$ in q-cosmology and
$\Omega^{(\Lambda)}_{0}=0.4$ in $\Lambda$CDM models.

Applying the above procedure for Eq. (51) results333It should be noted that
the approximation (54) is inappropriate for negative redshifts, and therefore
the higher order of approximation must be used.

$\Omega^{(q)}=2\Omega^{(q)}_{0}\Bigg{\\{}\beta\Omega^{(q)}_{0}+f(z)+\\\
\sqrt{\left(\beta\Omega^{(q)}_{0}+f(z)\right)^{2}-4\left(\beta\Omega^{(q)}_{0}+\Omega^{\text{(m)}}_{0}-1\right)}\Bigg{\\}}^{-1},$
(54)

where $\beta=\frac{\pi}{2}\operatorname{Si}(\frac{\pi}{2})$, and

$f(z)=\sum_{j}\Omega^{\text{(j)}}_{0}(1+z)^{3(\omega_{j}+1)},~{}~{}~{}i=\text{c},\text{rad}.$
(55)

Also, in the first approximation, one can simplify Eq. (51) to find an
explicit expression of the Hubble parameter

$\left(\frac{H}{H_{0}}\right)^{2}=\Big{\\{}f(z)+\\\
\frac{\pi}{2}\operatorname{Si}(\frac{\pi}{2})\Omega^{(q)}_{0}\Big{\\}}\left\\{1-\frac{\pi^{2}\left(\Omega_{0}^{(q)}\right)^{2}}{8\left(f(z)+\frac{\pi}{2}\operatorname{Si}(\frac{\pi}{2})\Omega^{(q)}_{0}\right)^{2}}\right\\},$
(56)

where the higher-order terms are omitted. This equation is tantamount to the
Friedmann equation in the standard flat $\Lambda$CDM cosmology, supplemented
with a correction term articulated in the last term. In addition, the induced
effective density parameter of the CC at the present epoch is
$\frac{\pi}{2}\operatorname{Si}(\frac{\pi}{2})\Omega^{(q)}_{0}$.

To analyze cosmological models, datasets obtained from SNIa observations are
particularly valuable as they serve as primary evidence for the Universe’s
accelerated expansion. To achieve optimal outcomes with SNIa data, we commence
with the observed distance modulus produced by SNIa detections and compare it
against the theoretical value. For this study, we utilize the Pantheon sample,
an up-to-date SNIa dataset that encompasses 1048 distance modulus $\mu$ at
various redshifts within the $0.01<z<2.26$ range [102]. By utilizing
luminosity distance, one can determine the distance modulus, which is
presented as

$\mu(z)=25+5\log_{10}\left((1+z)\int_{0}^{z}\frac{dz^{\prime}}{H(z^{\prime})}\right).$
(57)

Figure 3: Evolution of Luminosity Distance, $\mu(z)$, versus redshift $z$. The
black line denotes the best fit for the redshift evolution of the distance
modulus obtained from Eqs. (57) and (51). We used $\Omega^{(q)}_{0}=0.43$,
$\Omega^{\text{(c)}}_{0}=0.3$ and
$H_{0}=67.27~{}\text{Km}\cdot\text{sec}^{-1}\cdot\text{Mpc}^{-1}$, and the
absolute B-band magnitude of a fiducial SNIa is $M_{0}=-19.5$ [103]. The red
line (dots) denotes the redshift evolution of the distance modulus in the
standard flat $\Lambda$CDM model with $\Omega_{0}^{(\Lambda)}=0.7$. The
distance modulus measurements we use are taken from Ref. [102].

Cosmological observations show that the energy density parameter of the
radiation is negligible at the present epoch, $\Omega^{\text{(rad)}}_{0}\simeq
8.7\times 10^{-5}$. Hence, for simplicity, we assume that the matter content
of the model is only cold matter. Thus, one can safely ignore the radiation
density parameter in the field equations, i.e.,
$\Omega^{\text{(m)}}=\Omega^{\text{(c)}}$. As a result of this assumption,
$f(z)$ simplifies into

$f(z)=\Omega^{\text{(c)}}_{0}(1+z)^{3}.$ (58)

Figure 3 displays the Pantheon Survey as the standard Hubble diagram of SNIa
(absolute magnitude $M_{0}=-19.5$). By utilizing this dataset, we are able to
determine the optimal values for the density parameter of cold matter
$\Omega^{\text{(c)}}_{0}$ and $\Omega^{(q)}_{0}$

$\Omega^{(q)}_{0}=0.43,~{}~{}~{}\Omega^{\text{(c)}}_{0}=0.3,$ (59)

where the value of $H_{0}$ is set to be 67.27 (Km/s)/Mpc based on the Planck
2018 results [104]. Furthermore, in this figure, our model has been juxtaposed
against the standard flat cosmology. It is evident from the figure that the
two models display a remarkable degree of conformity with each other.

In addition, using the approximations
$\Omega^{\text{(m)}}=\Omega^{\text{(c)}}$, the energy density parameter of
q-dark energy and its EoS parameter, defined by Eqs. (45) and (44) simplify

$\begin{split}\Omega^{\text{(DE)}}&=1-\frac{\Omega^{(q)}}{\Omega^{(q)}_{0}}f(z),\\\
\omega_{\text{DE}}&=-1+\frac{\pi\left(\Omega^{(q)}\right)^{2}f(z)}{4\operatorname{Si}(\frac{\pi}{2})\Omega^{(q)}_{0}}\left\\{1+\frac{\pi\Omega^{(q)}}{4\operatorname{Si}(\frac{\pi}{2})}\right\\}.\end{split}$
(60)

By substituting the values of $\Omega^{\text{(c)}}_{0}$ and $\Omega^{(q)}_{0}$
into the above equations, we find the energy density parameter and EoS
parameter of q-dark energy at the present epoch

$\Omega^{\text{(DE)}}_{0}=0.7,~{}~{}~{}\omega_{\text{DE0}}=-0.91.$ (61)

Figure 4: The evolution of $H(z)$ (in units
$\text{Km}\cdot\text{sec}^{-1}\cdot\text{Mpc}^{-1}$) versus redshift $z$ with
error bars. The blue line denotes the dynamics of the Hubble parameter
obtained from Eq. (51). The values of $\Omega^{(q)}_{0}$,
$\Omega^{\text{(c)}}_{0}$ and $H_{0}$ are same as the Eq. 57, and we assumed
(according to the standard flat $\Lambda$CDM model)
$\Omega_{0}^{(\Lambda)}=0.7$. The dash-line denotes the evolution of the
Hubble parameter in the standard flat $\Lambda$CDM model with the same values
of $H_{0}$, $\Omega^{\text{(c)}}_{0}$ and $\Omega_{0}^{(\Lambda)}$. The Hubble
parameter measurements we use are taken from Table 1 of Ref. [105].

Furthermore, we were able to achieve a satisfactory level of agreement for the
Hubble parameter by utilizing the fitting parameters (59). By referring to the
data presented in Table 1 of [105], we plotted the Hubble parameter $H$ versus
the redshift $z$ in Fig. 4. Our model demonstrated a high level of consistency
with the standard model of cosmology, as evidenced by the figure. It is worth
noting that all entropic-force models tend to fit the modulus distance data
and Hubble parameter points in a similar fashion (cf. [106, 107, 108, 109,
110]). While more comprehensive data analysis (e.g., using a covariance
matrix) is necessary to authenticate these models further, such work falls
outside the scope of this article. Our present aim was to simply use the data
to compare and contrast the effectiveness of various entropic force models.

In summary, using quantum-deformed entropy, we could derive analytical results
for the observable parameters of the effective dark energy sector, namely the
density parameter and EoS parameter of the dark energy of the proposed
q-deformed cosmological scenario. We look at the following in more detail in
cosmological ramifications. To investigate the facts of the model, let us use
the present-time best-fit value of the density parameter of cold matter.

By using the obtained values in (59), one can determine the q-deformation
parameter, $\mathcal{N}$, and the q-deformed entropy of the apparent horizon
defined by Eq. (31). Regarding the definitions of $\mathcal{N}$ and
$\Omega^{(q)}_{0}$ in Eqs. (13) and (45) we obtain

$\mathcal{N}=\frac{\Omega^{(q)}_{0}}{H_{0}^{2}L_{\text{P}}^{2}}=1.70\times
10^{124}.$ (62)

It is noteworthy to acknowledge the estimations made by Refs. [36, 111], which
through general arguments in the context of quantum cosmology, approximated
the orders $\mathcal{N}\sim 10^{120}-10^{123}$, respectively. Furthermore, Eq.
(62) provides evidence that the argument of trigonometric functions found in
the deformation parameter, $q=\exp(2\pi
i/\mathcal{N})=\cos(2\pi/\mathcal{N})+i\sin(2\pi/\mathcal{N})$, is exceedingly
minute. However, as can be observed from Eq. (60) (also shown in Fig. 5), its
impact plays a crucial role in the late time cosmology. Additionally,
utilizing the aforementioned value of $\mathcal{N}$ in Eq. (31) allows us to
obtain the q-deformed entropy of the apparent horizon

$S_{q}(t_{0})=2.298\times 10^{124}.$ (63)

Thus, the implication of the diminutiveness of $\mathcal{N}$ is the
exceedingly high value of the entropy of the apparent horizon. Please note
that in order for the universe to be of significant size, the argument of the
trigonometric functions in $q$ has to be very small. Furthermore,
$\mathcal{N}$ is representative of the entropy of the apparent horizon, as
illustrated by Eq. (63). According to the holographic bound [112], the entropy
budget of the observable universe must not surpass that of the apparent
horizon. The total entropy of the observable universe is roughly
$S_{obs}\simeq 10^{102}-10^{103}$ [113], with supermassive black holes
occupying a dominating role at the cores of galaxies. This also offers a hint
regarding the minimum value of $\mathcal{N}$, which is greater than
$10^{123}$, as confirmed by our discovery in Eq. (63).

Figure 5: The evolution of the effective dark energy density parameter
$\Omega^{\text{(DE)}}$ (solid line) and the matter density parameter
$\Omega^{\text{(c)}}$ (dashed) respectively, as a function of the redshift
$z$.  Figure 6: The evolution of the effective (q-deformed) dark energy EoS
parameter, $\omega_{\text{DE}}$.

Fig. 5 depicts the evolution of the effective dark energy, as we found in Eq.
(60), and cold matter density parameters with redshift. It is simple to
demonstrate, and as this figure indicates, at redshift $z=0.377$, two density
parameters are identical, and we then have a universe dominated by dark
energy. Finally, at redshift $z=-1$, the contribution of the cold matter is
insignificant, resulting in a de Sitter universe. This demonstrates that the
model is a plausible cosmological model since it accommodates an early matter-
dominated phase in which structure could form and a recent acceleration phase
corresponding to observations.

In addition, in Fig. 6, we show the corresponding behavior of the effective
dark energy EoS parameter as a result of (60). This figure shows that the EoS
parameter begins at $\omega_{\text{DE}}=-1$ at high redshifts, achieves a
maximum of $\omega_{\text{DE}}=-0.7243$ at redshift $z=0.2024$, and returns to
$\omega_{\text{DE}}=-1$ at subsequent redshifts.

Finally, we show the deceleration parameter in Fig. 7 defined by

$\mathfrak{q}=-1-\frac{\dot{H}}{H^{2}}=-1+\frac{3f(z)\Omega^{(q)}}{2\Omega^{(q)}_{0}}\left\\{1+\frac{\pi\Omega^{(q)}}{4}\right\\}.$
(64)

For the reader’s convenience, we have extended evolution into the far future,
i.e., $z\rightarrow-1$. This graph shows the transition from deceleration to
acceleration at $z_{\text{tran}}=0.5$, consistent with the cosmological
observations [114, 115].

Figure 7: Evolution of the deceleration parameter $\mathfrak{q}(z)$ versus
redshift. The black line exhibits the deceleration parameter of the q-deformed
cosmology, with $\Omega_{0}^{\text{(q)}}=0.43$, $\Omega_{0}^{\text{(c)}}=0.3$
and $z_{\text{tran}}=0.5$. The dash-line denotes the deceleration parameter of
the standard flat $\Lambda$CDM model with $\Omega^{(\Lambda)}_{0}=0.7$ (with
$z_{\text{tran}}=0.67$).  Figure 8: The age of the Universe as a function of
$\Omega^{\text{(cm)}}$. Regarding the Planck 2018 collaboration observations,
we assumed $H_{0}=67.27~{}\text{Km}\cdot\text{sec}^{-1}\cdot\text{Mpc}^{-1}$.

Also, Eq. (64) gives $\mathfrak{q}_{0}=-0.3973$ for the deceleration parameter
today, which is consistent with model-independent observations [114].

We close this section by calculating the Universe’s age according to the
scenario. Figure 8 illustrates the Universe’s age as a function of the density
parameter of the cold matter. There are two different estimates for the age of
the Universe. Based on early observations of the Universe, the first estimate
suggests that the Universe is 13.787 billion years old according to the
$\Lambda$CDM model as of 2021 [104]. The second estimate is based on
observations of the current universe and suggests that the Universe is younger
[116, 117]. Recent studies have lowered the uncertainty of the first type of
measurement to 20 million years. This was achieved through various studies
that produced similar results, including analyses of the microwave background
radiation by the Planck satellite, the WMAP Probe, and other space missions
[118].

Inserting the obtained Hubble parameter in (56) to the expression of the
Universe’s age

$t_{0}=\int_{0}^{\infty}\frac{dz}{(1+z)H(z)},$ (65)

we find,

$t_{0}=\frac{0.9479}{H_{0}}=13.779~{}\text{Gyrs}.$ (66)

The above value coincides with the value corresponding to the standard
$\Lambda$CDM scenario, namely $13.787\pm 0.020$ Gyrs [104].

##  5 Comparison with other models

The conventional Friedmann and Raychaudhuri equations involving ordinary
matter fields are inadequate for characterizing the dark energy epoch of the
present universe. It is in this context that we can fit the following comment.
Concretely, Jacobson [4] successfully obtained the Einstein field equations
from the entropy and horizon area proportionality by presuming the heat flow
across the horizon, and Padmanabhan [119] derived the Friedmann and
Raychaudhuri equations using the holographic equipartition law, which
indicates that the difference between the degrees of freedom at the surface
and in the bulk of an area of space causes cosmic space expansion. Verlinde
[120] has also proposed a new concept that defines gravity as an entropic
force arising in a system due to the statistical tendency to increase its
entropy444This theory is a significant breakthrough in the field of physics
and a promising avenue for future research. All the innovative ideas above
offer fresh insights into the quantum gravity puzzle and could even shed light
on the thermodynamic origins of space-time. These breakthroughs in the field
may significantly enhance our understanding of the universe.. Consequently,
alternative forms of entropy, especially non-extensive entropies, distinct
from the Bekenstein–Hawking variety, have been advocated to induce accelerated
expansion, as now found in the literature for thermodynamic interpretations of
modern cosmology. In this regard, various non-extensive entropies have been
proposed, such as Tsallis [23], Renyi [25], Sharma–Mittal [121], Kaniadakis
[122], Loop Quantum Gravity [32, 33], Tsallis–Cirto [24], Tsallis–Zamora
[109], Barrow [26], and fractional [27, 28] entropies.

Tsallis entropy is a generalized form of Gibbs/Shannon entropy applicable in
cases where entropy’s additive and extensive properties do not hold. Renyi
entropy, on the other hand, is a measure of the entanglement of quantum
systems and is commonly used in the context of BHs and cosmological horizons.
Tsallis-Cirto entropy, which has been motivated by the need to make BH entropy
extensive, is similar in nature. Regarding mathematical equivalence, Barrow
and fractional entropies are comparable to Tsallis–Cirto entropy. However,
they are primarily driven by the fractal structure of the horizon caused by
quantum fluctuations. The Barrow entropy, proposed as a toy model for the
potential effects of quantum gravitational spacetime foam, was not supported
by any concrete evidence. Barrow argued that quantum-gravitational effects
could introduce intricate, fractal features on the BH structure, similar to
the illustrations of the Covid-19 virus. In contrast, the fractal entropy of
black holes in fractional quantum gravity has been proven to be real, and it
has been shown that the structure of the BH surface horizon is a random
fractal surface distinct from Barrow’s inspiration. Furthermore, the
Sharma–Mittal entropy can extend the Tsallis and Renyi entropies, whereas the
Kaniadakis entropy is derived from the principles of special relativity.

Referring to the findings presented in Ref. [123], it can be observed that a
comprehensive overview of the majority of these entropies can be effectively
encapsulated in the subsequent form

$S_{\mathrm{g}}(\alpha_{+},\alpha_{-},\beta,\gamma)=\\\
\frac{1}{\gamma}\left[\left(1+\frac{\alpha_{+}}{\beta}~{}S_{\text{BH}}\right)^{\beta}-\left(1+\frac{\alpha_{-}}{\beta}~{}S_{\text{BH}}\right)^{-\beta}\right],$
(67)

where $\alpha_{+}$, $\alpha_{-}$, $\beta$, and $\gamma$ are real and positive
the parameters positive, and $S_{\text{BH}}$ is the Bekenstein–Hawking
entropy. As authors of [123] showed, by suitable choice of these parameters,
the generalized entropy (67) reduces to the

$\begin{split}&S_{g}=S_{\text{BH}}^{\beta},\hskip 56.9055pt\text{ (Tsallis--
Barrow--Fractional)},\\\
&(\alpha_{+}\rightarrow\infty,\alpha_{-}=0,\gamma=\left(\alpha_{+}/\beta\right)^{\beta}),\\\
&S_{g}=\frac{1}{\alpha}\ln{\left(1+\alpha~{}S_{\text{BH}}\right)},\hskip
56.9055pt\text{(Rényi)}\\\ &(\alpha_{-}=0,~{}\beta\rightarrow
0,~{}\frac{\alpha_{+}}{\beta}\rightarrow\mathrm{finite}),\\\
&S_{g}=\frac{1}{\gamma}\left[\left(1+\frac{\alpha_{+}}{\beta}~{}S\right)^{\beta}-1\right],~{}~{}~{}\text{(Sharma
--Mittal)},\\\ &(\alpha_{-}=0),\\\
&S_{g}=\frac{1}{K}\sinh{\left(KS\right)},\hskip
56.9055pt\text{(Kaniadakis)},\\\
&(\beta\rightarrow\infty,\alpha_{+}=\alpha_{-}=\frac{\gamma}{2}=K),\\\
&S_{g}=\frac{1}{(1-q)}\left[\mathrm{e}^{(1-q)S}-1\right],~{}~{}~{}\text{( Loop
Quantum Gravity)}\\\
&(\alpha_{-}=0,\beta\rightarrow\infty,\gamma=\alpha_{+}=(1-q)),\end{split}$
(68)

entropies. The entropy function described in Eq. (67) and the q-deformed
entropy (31) share several similar properties. Firstly, both of these
functions satisfy the generalized third law of thermodynamics. Secondly, they
both exhibit a monotonically increasing behavior concerning the
Bekenstein–Hawking entropy. Finally, these functions converge to the
Bekenstein–Hawking entropy under certain parameter limits. As previously
stated in the introduction, one of the main challenges such models face is the
presence of free parameters with varying origins. Upon reviewing the
generalized entropy introduced in (67), it is apparent that the existing
parameters have distinct origins. Even if the model were to provide a highly
accurate representation of observational data, comprehending the physics
behind these parameters would prove to be more challenging than the initial
issue of the cosmological constant problem in the standard model of cosmology.

If we use the generalized entropy (67) in process of section 3, the resulting
Friedmann equation will be [123]

$\frac{G\beta
H^{4}}{\pi\gamma}\Bigg{\\{}\frac{1}{\left(2+\beta\right)}\left(\frac{GH^{2}\beta}{\pi\alpha_{-}}\right)^{\beta}~{}_{2}F_{1}\Big{(}1+\beta,2+\beta,3+\beta,\\\
-\frac{GH^{2}\beta}{\pi\alpha_{-}}\Big{)}+\frac{1}{\left(2-\beta\right)}\left(\frac{GH^{2}\beta}{\pi\alpha_{+}}\right)^{-\beta}~{}_{2}F_{1}\Big{(}1-\beta,2-\beta,\\\
3-\beta,-\frac{GH^{2}\beta}{\pi\alpha_{+}}\Big{)}\Bigg{\\}}=\frac{8\pi
G\rho}{3}+\frac{\Lambda}{3},$ (69)

where ${}_{2}F_{1}(a,b,c,d)$ denotes the Hypergeometric function. Note that
the “cosmological constant”, $\Lambda$, appears in the above generalized
Friedmann equation as an integration constant. This is where our model truly
distinguishes itself, as it separates from the generalized entropy and its
various subset entropies as defined in equation (67). Let us go through this
in further depth. The CC seen in the previous equation is a constant of
integration that can be chosen arbitrarily. In contrast, the constant of
integration in the Friedmann equation (33) is fixed to a specific value.
Understanding this fixed value is simple. As the universe approaches very late
times and the scale factor tends towards infinity, the energy density of
matter fields in the right-hand side of the equation vanishes. Additionally,
the first term on the right-hand side of (33), which is
$\frac{\cos(\frac{\gamma}{G}R_{\text{AH}}^{2})}{R_{\text{AH}}^{2}}$,
identically vanishes as well. This boundary condition fixes the value of the
constant of integration to
$\frac{\pi\Lambda_{q}}{6}\operatorname{Si}\left(\frac{\pi}{2}\right)$.

The presence of the CC in the generalized Friedmann equation leads to the CC
problem. The challenge is explaining why the measured CC is not precisely zero
but has a very small nonzero value. In contrast, the q-deformed Friedmann
equation (33) does not suffer from this problem.

We can use Barrow’s entropy model as an illustrative example to provide a more
precise reference point. In the context of cosmology, various perspectives
have been explored regarding the impact of Barrow entropy on the universe’s
evolution. One such view involves modifying the area law, which results in a
novel holographic dark energy model based on Barrow entropy [58, 124]. Another
cosmological scenario that incorporates Barrow entropy was presented in Ref.
[125], where it was demonstrated that new additional terms emerge in the
Friedmann and Raychaudhuri equations, forming an effective dark energy sector.
While Ref. [125] argued that the modified cosmological equations rooted in
Barrow entropy (68) could account for the thermal history of the universe,
spanning from early deceleration to later acceleration during the dark-energy
epoch that follows the cold matter dominated epoch, regardless of the presence
of CC, $\Lambda$, it seems that this conclusion is only accurate if there is a
CC present [125, 126]. To illustrate this important point, let us rewrite the
Friedmann and Raychaudhuri equations with Barrow’s entropy [125, 126]

$\begin{split}&H^{2}=\frac{8\pi G}{3}(\rho+\rho_{\text{DE}}),\\\
&\dot{H}=-4\pi G(\rho+p+\rho_{\text{DE}}+p_{\text{DE}}),\end{split}$ (70)

where $\rho$ and $p$ are the energy density and pressure of the ordinary
matter, and

$\begin{split}\rho_{\text{DE}}&=\frac{3}{8\pi
G}\Big{\\{}\frac{\Lambda}{3}+H^{2}\Big{[}1-\frac{\Delta+2}{2-\Delta}\left(\frac{\pi}{G}\right)^{\frac{\Delta}{2}}H^{-\Delta}\Big{]}\Big{\\}},\\\
p_{\text{DE}}&=-\frac{1}{8\pi
G}\Big{\\{}\Lambda+2\dot{H}\Big{[}1-(1+\frac{\Delta}{2})\left(\frac{\pi}{G}\right)^{\frac{\Delta}{2}}H^{-\Delta}\Big{]}\\\
&+3H^{2}\Big{[}1-\frac{\Delta+2}{2-\Delta}\left(\frac{\pi}{G}\right)^{\frac{\Delta}{2}}H^{-\Delta}\Big{]}\Big{\\}},\end{split}$
(71)

where $\Delta=2(\beta-1)$, $(0\leq\Delta\leq 1)$ is the Barrow’s deformation
parameter. For $\rho=p=0$ and $\Lambda=0$, the solution of the above field
equations is $H=0$. On the other hand, for $\rho=p=0$ and $\Lambda\neq 0$ we
obtain

$H=\left(\frac{\Lambda}{3}\right)^{\frac{1}{2-\Delta}}.$ (72)

These findings validate that the accelerated phase can solely occur when the
CC is present. However, we now have two unknown parameters in the theory,
$\Lambda$ and $\Delta$. As previously discussed, the presence of a constant of
integration, $\Lambda$, reintroduces the cosmological constant problem and
warrants an explanation. Furthermore, it is imperative to provide a sound
justification for the particular numerical value of $\Delta$ that is deemed
valid based on cosmological observations.

On the other hand, the q-deformed Friedmann and Raychaudhuri equations (35)
and (38) for $\rho=p=0$ reduce to

$\cos(\frac{\pi\Lambda_{q}}{6H^{2}})H^{2}=\frac{\pi\Lambda_{q}}{6}\left\\{\operatorname{Si}\left(\frac{\pi}{2}\right)-\operatorname{Si}\left(\frac{\pi\Lambda_{q}}{6H^{2}}\right)\right\\},$
(73)

and

$\cos(\frac{\pi\Lambda_{q}}{6H^{2}})\frac{\ddot{a}}{a}=\frac{\pi\Lambda_{q}}{6}\left\\{\operatorname{Si}\left(\frac{\pi}{2}\right)-\operatorname{Si}\left(\frac{\pi\Lambda_{q}}{6H^{2}}\right)\right\\}.$
(74)

The solution of these equations is a de Sitter spacetime,
$H^{2}=\Lambda_{q}/3$, in which $\Lambda_{q}$ plays the role of the CC. As
previously described, our initial cosmological model did not include a
conventional CC. However, our subsequent implementation of quantum deformation
resulted in the emergence of an effective CC, which is strongly linked to the
natural number $\mathcal{N}$ defined by equation (13). This constant is
directly derived from the assumption of a finite number of states in Hilbert
spaces, as stated in [36].

##  6 Conclusions

In this paper, we developed a novel cosmological scenario assuming that
thermodynamics is intertwined with gravity. It is known, for instance, that
one may start with the first law of thermodynamics, apply it to the Universe
horizon, and end up with the Friedmann and Raychaudhuri equations. This method
utilizes the Bekenstein–Hawking entropy in the case of GR or the modified
entropy expression in the case of modified gravity.

We examined the compatibility of the apparent horizon’s entropy with
thermodynamic laws on the assumption that it is the q-deformed entropy of the
black hole-white hole pair. To achieve this, we first presupposed that the
apparent horizon is subject to the unified first law of thermodynamics and
that its entropy has the form (18). Then we demonstrated how the unified first
law of thermodynamics on the apparent horizon might be expressed as modified
Friedmann equations of an FLRW universe with arbitrary spatial curvature.

Our research shows that the q-deformed Friedmann and Raychaudhuri equations
lead to the existence of an effective dark energy component, which can explain
the Universe’s late-time acceleration. According to our model, the density
parameters of cold matter and the effective dark energy are equal at a
redshift of $z=0.377$, after which the effective dark energy dominates the
universe. Additionally, our model predicts a de Sitter universe in the long
run. The model predicts that the effective dark energy equation of state (EoS)
parameter changes over time. Initially, it starts at $\omega_{\text{DE}}=-1$
during high redshifts. Then, it reaches a peak at $\omega_{\text{DE}}=-0.7243$
at redshift $z=0.2024$. After that, it returns to its initial value of
$\omega_{\text{DE}}=-1$ during subsequent redshifts. Furthermore, we find a
transition from deceleration to acceleration at $z_{\text{tran}}=0.5$,
consistent with the cosmological observations. The obtained age of the
Universe is 13.779 Gyrs, which coincides with the value corresponding to the
standard $\Lambda$CDM scenario, namely $13.787\pm 0.020$ Gyrs. These findings
suggest that our model is a viable cosmological model, as it accommodates an
early phase of matter domination that allowed the formation of structures and
a recent acceleration phase that aligns with observations. Concretely, we
support this claim upon using suitable calculations and corresponding plots,
in particular of $H/H_{0}$ (radiation plus cold matter) versus the scale
factor, either in flat $\Lambda$CDM or our q-deformed cosmology, and the
evolution of luminosity distance, $\mu$, versus redshift $z$, the evolution of
$H(z)$ versus redshift $z$ (including error bars) and the evolution of the
deceleration parameter $q(z)$ versus redshift (see figures in section 4).
Additionally, we have conducted a thorough comparative analysis of our
proposed model with others involving non-extensive entropies.

In conclusion, the q-deformed horizon entropy cosmology scenario demonstrates
intriguing phenomenology and is consistent with cosmological observations. As
a result, it may be an intriguing option for the description of Nature.

Last but not least, let us add that it is imperative to acknowledge that the
current models possess both advantageous and disadvantageous features and are
not without flaws. For example, in entropic cosmology, the form of the driving
entropic force terms is determined by the definition of entropy. In the
original entropic force models, the Bekenstein entropy and the Hawking
temperature have been used to get the entropic force terms. However, such
models fail to account for both the Universe’s acceleration and deceleration,
and it has been shown in [127, 128] that they neither account for cosmic
fluctuations nor are compatible with the formation of structures. As we have
thoroughly discussed in section 5, using Barrow entropy-based models, with the
addition of the cosmological constant as an integration constant, can
effectively elucidate the shift from a decelerated universe to an accelerated
universe. Nevertheless, including the cosmological constant in the model leads
to a resurgence of the cosmological constant problem and the incorporation of
obscure parameters into the cosmological model, further complicating the
matter. Therefore, exploring alternative models that can account for these
phenomena is necessary and provides a more accurate description of the
Universe’s behavior. Ultimately, this will lead to a better understanding of
our Universe and its complex dynamics.

## Acknowledgements

S.J. acknowledges financial support from the National Council for Scientific
and Technological Development–CNPq, Grant no. 308131/2022-3. P.V.M.
acknowledges the FCT grants UID-B-MAT/00212/2020 at CMA-UBI as well as the
COST Action CA18108 (Quantum gravity phenomenology in the multi-messenger
approach).

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