# An attempt to add Barrow entropy in $\displaystyle f(R)$ gravity

P. S. Ens<EMAIL_ADDRESS>Instituto de Física, Universidade Federal de
Mato Grosso,
78060-900, Cuiabá, Mato Grosso, Brazil  A. F. Santos
<EMAIL_ADDRESS>Instituto de Física, Universidade Federal de
Mato Grosso,
78060-900, Cuiabá, Mato Grosso, Brazil

###### Abstract

In this work, a way to consider together two originally different corrections
to the Friedmann equations is presented. The first is the Barrow entropy,
which imposes a fractal structure on the black hole horizon area. While the
second is the well-known $\displaystyle f(R)$ gravity, which comes from a
generalization of the Einstein-Hilbert action. Using the ideas of gravity-
thermodynamics conjecture, these two models are combined. Then the modified
Friedmann equation is obtained. Choosing a particular $\displaystyle f(R)$
model, an application is investigated. The state parameter and the density
parameters for matter and dark energy are calculated. With these results, the
dynamic evolution of the universe is discussed.

##  I Introduction

Gravity is one of the fundamental forces in the universe. However, there is no
complete theory applicable to all energy regimes that describes this
interaction. Classically, the theory of general relativity is the best
gravitational theory that responds very well to various observational tests
[1]. Although attempts at a quantum version have been built and discussed for
a long time, a completely consistent theory of quantum gravity has not been
obtained. At low energies, some effective theories have been investigated [2,
3, 4, 5, 6]. Another study that plays an important role in providing
fundamental information about the quantum aspect of gravity is the black hole
thermodynamics. Early works by Hawking and Bekenstein showed that a black hole
radiates at a given temperature and that its entropy is proportional to the
area of the black hole’s event horizon [7, 8, 9]. This leads to the
possibility that the black hole thermodynamics can be investigated in the
context of quantum gravity. Furthermore, since the beginning of discussions
about possible relations between general relativity and thermodynamics, ideas
have emerged on how to obtain equations to better describe the different
periods throughout the expansion of the universe. One of them is the
holographic principle, which consists of studying the behavior of a system
based on the characteristics of its surface [10]. Theories like this aim to
change the way entropy is calculated [8, 11]. From these discoveries, other
speculations have been analyzed, such as the relationship between
thermodynamics and Einstein equations, which is known as the gravity-
thermodynamics conjecture.

The gravity-thermodynamics conjecture is a formalism that allows obtaining
Einstein equations from a thermodynamic approach, using the relationship
between entropy and the area of the event horizon of a black hole [12, 13, 14,
15]. Taking the first law of thermodynamics on the apparent horizon, the
Friedmann equations can be derived. This procedure is applicable in general
relativity as well as in a variety of modified gravity theories. However, when
the analysis is extended to a modified theory, the entropy relation usually
changes [16, 17, 18, 19, 20]. Since the Bekenstein-Hawking entropy-area law is
a non-extensive measure, various generalized statistical mechanics have been
proposed to study cosmic evolution and, in general, the gravitational
phenomena. Inspired by this, some models have been developed, such as Tsallis
entropy [21, 22], Rényi entropy [23], among others. Recently, a different
proposal, known as Barrow entropy [24], has been considered. In this work,
Barrow entropy and its consequences for the evolution of the universe will be
studied.

In order to build his model, Barrow was inspired by the geometrical structure
of the COVID-19 virus [24]. In this model, quantum-gravitational effects can
introduce intricate fractal features into the black-hole structure. This
structure leads to a fractal horizon surface. Then a new black hole entropy
relation is defined, i.e. $\displaystyle S\sim A^{f(\Delta)}$, where
$\displaystyle f(\Delta)=1+\Delta/2$. The $\displaystyle\Delta$ parameter
quantifies the quantum-gravitational deformation. In this context, some
applications have been investigated. For example, Baryon asymmetry has been
studied [25], inflation driven by Barrow holographic dark energy has been
considered [26], early and late periods of the universe from a new generalized
entropy have been analyzed [27], Barrow holographic dark energy has been
formulated [28], the cosmology using Barrow entropy has been proposed [29],
the generalized second law of thermodynamics with Barrow entropy has been
investigated [30], among others. In this context, the main objective of this
paper is to analyze the dynamic evolution of the universe considering the
Barrow entropy in $\displaystyle f(R)$ gravity.

The motivations for considering an alternative gravity theory to general
relativity arise from various observational data [31, 32, 33, 34] that lead to
the accelerated expansion of the current universe. One of the simplest and
most studied models of modified gravity is the $\displaystyle f(R)$ theory
[35]. In this modified theory, the gravitational action of general relativity
takes a general form by incorporating an arbitrary function of the Ricci
scalar into it. This is not the only modified theory of gravity, there are
several theories in the literature that generalize the general relativity,
such as $\displaystyle f(R,T)$ gravity [36], gravity theories with non-
minimally coupled scalar fields [37], among many others. Although there are
many theories of gravity, our interest here is to study $\displaystyle f(R)$
gravity combined with Barrow entropy using the ideas of the gravity-
thermodynamics conjecture and then describe the evolution of the universe from
there.

This paper is organized as follows. In section II, Barrow entropy and its
equations are introduced. Using the gravity-thermodynamics conjecture, the
Friedmann equation is derived for two different situations. First, the
Hawking-Bekenstein entropy is used. And then, consider Barrow entropy. In
section III, $\displaystyle f(R)$ theory is presented. The first law of
thermodynamics is discussed. In section IV, $\displaystyle f(R)$ theory and
Barrow entropy are put together. It is assumed that the energy density and
pressure of dark energy are composed of two parts, one due to the modified
gravity and the other due to Barrow entropy. In this context, a modified
Friedmann equation is obtained. In section V, an application is investigated
considering the generalized Carroll-Duvvuri-Trodden-Turner model. Then the
state parameter is calculated. In section VI, some concluding remarks are
made.

##  II Barrow entropy and its equations

In this section, Friedmann equations are obtained using the gravity-
thermodynamics conjecture [38]. Starting from the first law of thermodynamics,
two different cases are investigated. In the first case, the Bekenstein-
Hawking entropy is used, while in the second case the Barrow entropy is
considered. The first law of thermodynamics is given as

$TdS=dE-WdV,$ (1)

where $\displaystyle T$ is temperature, $\displaystyle S$ is entropy,
$\displaystyle E$ is the energy, $\displaystyle V$ is the volume and
$\displaystyle W=(\rho-p)/2$ is the work, with $\displaystyle\rho$ and
$\displaystyle p$ being the energy density and pressure, respectively. To
apply the first law of thermodynamics, temperature and entropy need to be
known. The standard choice is to use the Bekenstein-Hawking entropy, i.e.
$\displaystyle S=\frac{A}{4G}$, with $\displaystyle A=4\pi r^{2}$ the area of
the black hole horizon and $\displaystyle G$ the gravitational constant [39].
For the temperature is chosen $\displaystyle T=-\frac{1}{2\pi
r}\left(1-\frac{\dot{r}}{2Hr}\right)$, the horizon temperature [12, 40]. Here
$\displaystyle r$, the apparent horizon, would generally be $\displaystyle
r=\frac{1}{\sqrt{H^{2}+k/a^{2}}}$, but since we are dealing with the flat case
of FRW metric, implying $\displaystyle k=0$, we have $\displaystyle Hr=1$.
Thus the temperature can be written in a simpler form $\displaystyle
T=-\frac{1}{2\pi r}\left(1-\frac{\dot{r}}{2}\right)$. Then the first law of
thermodynamics becomes

$-\frac{1}{2\pi
r}\left(1-\frac{\dot{r}}{2}\right)\frac{dA}{4G}=dE-\frac{\rho-p}{2}dV.$ (2)

Imposing the matter conservation equation, i.e.
$\displaystyle\dot{\rho}+3H(\rho+p)=0$, Eq. (2) is written as

$-\frac{1}{2\pi
r}\left(1-\frac{\dot{r}}{2}\right)\frac{dA}{4G}=-A(\rho+p)dt+\frac{\rho+p}{2}dV,$
(3)

where $\displaystyle dE=Vd\rho+\rho dV=-A(\rho+p)dt+\rho dV$ has been used.
Using $\displaystyle dV=Adr=A\dot{r}dt$ leads to

$\frac{\dot{r}}{r^{2}}=4\pi G(\rho+p).$ (4)

In a flat universe, $\displaystyle\frac{\dot{r}}{r^{2}}=-\dot{H}$, then Eq.
(4) becomes

$\dot{H}=-4\pi G(\rho+p).$ (5)

This is the Friedmann equation. Applying it to the conservation relation and
performing the integration in time, results in

$H^{2}=\frac{8\pi G}{3}\rho+C,$ (6)

where $\displaystyle C$ acts as the cosmological constant.

It is important to note that, considering a fixed boundary
($\displaystyle\dot{r}=0$) with an energy flux through it [12], there is
another way to obtain the same Friedmann equation. In this case, the
temperature is defined as $\displaystyle T=\frac{1}{2\pi r_{A}}$, for a static
system, and the first law of thermodynamics is used as $\displaystyle dE=TdS$.

There are some proposals for obtaining new equations of motion. One of them is
to consider modifications in the black hole area, as proposed by Barrow [24].
In this model, the area takes into account a possible fractal structure from
the surface roughness, which depends on a parameter $\displaystyle\Delta$
representing its intricacy. This leads to a new form for entropy which is
given as

$S=\left(\frac{A}{4G}\right)^{1+\Delta/2}.$ (7)

From the Barrow entropy, the equation of motion or the modified Friedmann
equation can be obtained following the same procedure applied previously. Let
us consider the entropy defined in a more general way, i.e. $\displaystyle
S=\frac{f(A)}{4G}$. In our case, $\displaystyle
f(A)=A^{1+\frac{\Delta}{2}}/(4G)^{\frac{\Delta}{2}}$ is chosen to match Barrow
definition. Starting from the first law of thermodynamics it is found

$-\frac{1}{2\pi
r}\left(1-\frac{\dot{r}}{2}\right)\frac{df(A)}{4G}=dE-\frac{\rho-p}{2}dV.$ (8)

Since $\displaystyle df(A)=f^{\prime}(A)dA$, the last equation reads

$-f^{\prime}(A)\frac{1}{2\pi
r}\left(1-\frac{\dot{r}}{2}\right)\frac{dA}{4G}=dE-\frac{\rho-p}{2}dV.$ (9)

It should be noted that this equation is similar to Eq. (2). Then, following
the same procedure as before, it is obtained that

$f^{\prime}(A)\dot{H}=-4\pi G(\rho+p).$ (10)

This is the modified Friedmann equation due to Barrow entropy. To obtain the
second Friedmann equation, Eq. (10) is applied in the energy conservation
relation. Then

$\dot{\rho}=\frac{3}{4\pi G}f^{\prime}(A)H\dot{H}.$ (11)

Performing the integration in time leads to

$\frac{2}{2-\Delta}f^{\prime}(A)H^{2}=\frac{8\pi G}{3}\rho+\frac{\Lambda}{3},$
(12)

where $\displaystyle\Lambda=3\frac{2}{2-\Delta}f^{\prime}(A_{0})H_{0}^{2}-8\pi
G\rho_{0}$ is the integration constant. Here $\displaystyle\rho_{0}$ and
$\displaystyle H_{0}$ are the energy density of the fluid and the Hubble
parameter, respectively, defined as initial conditions of the system,
typically at present time, furthermore, $\displaystyle A_{0}=4\pi H^{-2}_{0}$
is the horizon area calculated from these conditions. It is important to note
that this form of Eq. (12) depends exclusively on the function $\displaystyle
f(A)$.

These equations can be written in the same structure as the usual Friedmann
equations evidencing the presence of dark energy originating from the new
entropy. Then Eq. (10) is given as

$\dot{H}=-4\pi G\left(\rho+p-\frac{\dot{H}}{4\pi G}(1-f^{\prime}(A))\right),$
(13)

and Eq. (12) becomes

$H^{2}=\frac{8\pi G}{3}\left(\rho+\frac{\Lambda}{8\pi G}+\frac{3H^{2}}{8\pi
G}\left(1-\frac{2}{2-\Delta}f^{\prime}(A)\right)\right).$ (14)

As a result, the components of dark energy are

$\rho_{DE}=\frac{1}{8\pi
G}\left(\Lambda(\Delta)+3H^{2}\left(1-\frac{2}{2-\Delta}f^{\prime}(A)\right)\right),$
(15) $p_{DE}=-\frac{1}{8\pi
G}\biggl{(}\Lambda(\Delta)+2\dot{H}\left(1-f^{\prime}(A)\right)+3H^{2}\left(1-\frac{2}{2-\Delta}f^{\prime}(A)\right)\biggr{)},$
(16)

with $\displaystyle\rho_{DE}$ and $\displaystyle p_{DE}$ being the Barrow’s
holographic dark energy density and pressure, respectively [41]. It should be
emphasized that for Eq. (14) as for $\displaystyle\rho_{DE}$ and
$\displaystyle p_{DE}$, it is considered specifically $\displaystyle
f(A)=A^{1+\frac{\Delta}{2}}/(4G)^{\frac{\Delta}{2}}$.

It is important to note that, although Eqs. (10)-(12) are enough to study the
dynamics of the universe, the manipulations of Eqs. (13)-(14) that lead to the
definitions of Eqs. (15)-(16) are necessary to have the conservation relations
that are given as $\displaystyle\dot{\rho}_{m}+3H(\rho_{m}+p_{m})=0$ and
$\displaystyle\dot{\rho}_{DE}+3H(\rho_{DE}+p_{DE})=0$. Consequently, also
making it clear that these two sectors are not interacting, a similar result
is shown in references [42, 43, 44, 45].

As in the Bekenstein-Hawking case, a static system with an energy flux
crossing its boundary can be considered, as in [41]. Assuming $\displaystyle
T=\frac{1}{2\pi r_{A}}$ and the relation $\displaystyle-dE=TdS$, it is
possible to arrive at the same equation of motion Eq. (14) and dark energy
components Eqs. (15) and (16).

In the next section, the $\displaystyle f(R)$ gravity in the context of the
gravity-thermodynamics conjecture is investigated.

##  III $\displaystyle f(R)$ gravity and the first law of thermodynamics

Here the $\displaystyle f(R)$ gravity is considered. Then the effective energy
density and pressure are obtained. Using these quantities, the first law of
thermodynamics is investigated. In such a discussion, a correction for the
black hole entropy due to $\displaystyle f(R)$ gravity is analyzed.

Let’s start with the action that describes this modified gravitational theory.
This model generalizes the Einstein-Hilbert action, replacing the Ricci scalar
$\displaystyle R$ by a function of the Ricci scalar $\displaystyle f(R)$ ,
i.e.,

$S=\int f(R)\sqrt{-g}d^{4}x.$ (17)

Field equations are derived from the variational principle [35]. Varying the
action with respect to the metric, the field equations are obtained as

$R_{\mu\nu}f^{\prime}(R)-\frac{1}{2}g_{\mu\nu}f(R)-\left(\nabla_{\mu}\nabla_{\nu}+g_{\mu\nu}\Box\right)f^{\prime}(R)=-\kappa
T_{\mu\nu},$ (18)

where $\displaystyle\kappa=8\pi G$. Taking a perfect fluid as matter content
and a flat FRW universe, the field equations are given as

$H^{2}=\frac{\kappa}{3f^{\prime}(R)}\left(\rho_{M}+\rho_{eff}\right),$ (19)
$2\dot{H}+3H^{2}=-\frac{\kappa}{f^{\prime}(R)}\left(p_{M}+p_{eff}\right),$
(20)

where the effective density and pressure are defined, respectively, as

$\rho_{eff}=\frac{1}{\kappa}\left(\frac{Rf^{\prime}(R)-f(R)}{2}-3H\dot{R}f^{\prime\prime}(R)\right),$
(21)
$p_{eff}=\frac{1}{\kappa}\left(\dot{R}^{2}f^{\prime\prime\prime}(R)+\ddot{R}f^{\prime\prime}(R)-\frac{Rf^{\prime}(R)-f(R)}{2}+2H\dot{R}f^{\prime\prime}(R)\right).$
(22)

These definitions of $\displaystyle\rho_{eff}$ and $\displaystyle p_{eff}$,
after some calculation considering
$\displaystyle\dot{f}^{\prime}(R)=\dot{R}f^{\prime\prime}(R)$ and that
$\displaystyle\frac{R}{2}-3\dot{H}=6H^{2}$, lead to a non-conservation energy
relation given by

$\dot{\rho}_{eff}+3H(\rho_{eff}+p_{eff})=\frac{3}{\kappa}H^{2}\dot{f}^{\prime}(R).$
(23)

Substituting Eq. (19) in Eq. (20) leads to

$dH=-\frac{\kappa}{2f^{\prime}(R)}\left(\rho_{M}+p_{M}+\rho_{eff}+p_{eff}\right)dt.$
(24)

Considering the relation $\displaystyle Hr=1$, the last equation is written as

$\frac{f^{\prime}(R)}{G}dr=A\left(\rho_{M}+p_{M}+\rho_{eff}+p_{eff}\right)dt.$
(25)

In this gravitational theory, the black hole entropy and its relation to the
horizon area is corrected [18, 19, 20]. In this context, the entropy is given
as

$S=\frac{Af^{\prime}(R)}{4G}.$ (26)

Using this modified black hole entropy, Eq. (25) becomes

$\frac{1}{2\pi r}dS-\frac{1}{2\pi
r}\frac{A}{4G}df^{\prime}(R)=A\left(\rho_{M}+p_{M}+\rho_{eff}+p_{eff}\right)dt.$
(27)

Now, taking the horizon temperature, i.e. $\displaystyle T=-\frac{1}{2\pi
r}\left(1-\frac{\dot{r}}{2}\right)$, the thermodynamics relation (27) reads

$TdS-T\frac{A}{4G}df^{\prime}(R)=-A\left(\rho+p\right)dt+\frac{A}{2}\left(\rho+p\right)dr.$
(28)

For convenience, it is defined $\displaystyle\rho=\rho_{M}+\rho_{eff}$ and
$\displaystyle p=p_{M}+p_{eff}$.

Adding the non-conservation relation for $\displaystyle f(R)$ gravity Eq. (23)
to that of matter, i.e. $\displaystyle\dot{\rho}_{M}+3H(\rho_{M}+p_{M})=0$, we
get

$d\rho=-3H(\rho+p)dt+\frac{3}{8\pi G}H^{2}\dot{f}^{\prime}(R)dt.$ (29)

Thus, the differential of the energy, which is given as $\displaystyle dE=\rho
dV+Vd\rho$, results in

$dE=\rho Adr-A(\rho+p)dt+\frac{1}{2\pi r}\frac{A}{4G}df^{\prime}(R).$ (30)

Using Eq. (30) in Eq. (28) we obtain

$TdS=dE-
WdV+T\left(\frac{4-\dot{r}}{2-\dot{r}}\right)\frac{A}{4G}df^{\prime}(R).$ (31)

This is the first law of thermodynamics generated in the context of
$\displaystyle f(R)$ gravity. However, it is important to note that, an extra
term $\displaystyle Td\bar{S}$ with $\displaystyle
d\bar{S}=-\left(\frac{4-\dot{r}}{2-\dot{r}}\right)\frac{A}{4G}df^{\prime}(R)$
arises. There are some discussions in the literature [46, 47] which argue that
this extra term may originate from the entropy generated internally by the
out-of-equilibrium system.

As an attempt to obtain an energy conservation relation in $\displaystyle
f(R)$ theory and determine the first law of thermodynamics without an extra
term that comes from an out-of-equilibrium system, let us write the modified
Friedmann equations, given in Eqs. (19) and (20), following Barrow definition.
Then these equations become

$H^{2}=\frac{\kappa}{3}\left(\rho_{M}+\rho_{DE}\right),$ (32)
$2\dot{H}+3H^{2}=-\kappa\left(p_{M}+p_{DE}\right),$ (33)

where

$\rho_{DE}=\frac{1}{\kappa}\left(\rho_{eff}+3H^{2}(1-f^{\prime}(R))\right),$
(34)
$p_{DE}=\frac{1}{\kappa}\left(p_{eff}+2\dot{H}(1-f^{\prime}(R))+3H^{2}(1-f^{\prime}(R))\right).$
(35)

It should be noted that Eqs. (34) and (35) describe holographic dark energy
due to the $\displaystyle f(R)$ theory. These definitions lead to
$\displaystyle\dot{\rho}_{DE}+3H(\rho_{DE}+p_{DE})=0$. Performing the same
procedure as before, one finds

$TdS=dE-WdV.$ (36)

Therefore, the usual first law of thermodynamics is obtained. Furthermore, the
entropy correction due to the $\displaystyle f(R)$ theory, Eq. (26), is not
used.

In the next section, Barrow entropy is taken to $\displaystyle f(R)$ gravity.
In this context, the dynamics of the universe is investigated.

##  IV Barrow entropy and $\displaystyle f(R)$ theory

Barrow’s proposal is elegant and carries good motivations. However, there are
obstacles that prevent direct application to explain the current behavior of
the universe and its history. One of them is the fact that in all cases this
modification always leads to a de-Sitter expansion [29]. Another problem is
the drastic restriction on the possible values of $\displaystyle\Delta$, when
considering the characteristics of the Big Bang [41, 48], as well as other
observations, such as those given in [49]. Likewise, there are also several
problems present in the $\displaystyle f(R)$ gravity models, for example,
general viability conditions such as $\displaystyle f^{\prime}(R)>0$ and
$\displaystyle f^{\prime\prime}(R)>0$ to avoid ghost scalar fields and complex
mass values for the scalaron field. For a review of $\displaystyle f(R)$
gravity and its problems see [46].

As both proposals present some problems, the main objective of this section is
to propose a way to merge these models, in an attempt to obtain results that
are not subject to the problems each one carries individually.

Looking at the definitions given in Eqs. (15), (16), (34) and (35), it is
possible to perceive similarity in how the terms originated from their
respective modifications act as dark energy. Taking this into account, a new
definition for holographic dark energy, which carries the Barrow and
$\displaystyle f(R)$ modifications, is considered, that is,

$\rho_{DE}=\frac{1}{\kappa}\left(\rho_{eff}+3H^{2}\left(1-\frac{2}{2-\Delta}f^{\prime}(A)f^{\prime}(R)\right)\right),$
(37)
$p_{DE}=\frac{1}{\kappa}\biggl{(}p_{eff}-2\dot{H}\left(1-f^{\prime}(A)f^{\prime}(R)\right)-3H^{2}\left(1-\frac{2}{2-\Delta}f^{\prime}(A)f^{\prime}(R)\right)\biggr{)},$
(38)

where Eqs. (37) and (38) satisfy the relation
$\displaystyle\dot{\rho}_{DE}+3H(\rho_{DE}+p_{DE})=0$. In order to obtain the
modified Friedmann equations, let us start with the first law

$TdS=dE-WdV,$ (39)

with $\displaystyle dE=\rho dV+Vd\rho$ and $\displaystyle W=(\rho-p)/2$. Then

$TdS=Vd\rho+\frac{\rho+p}{2}dV.$ (40)

Assuming $\displaystyle\rho=\rho_{M}+\rho_{DE}$ and $\displaystyle
p=p_{M}+p_{DE}$ and using the energy conservation relation, the last equation
becomes

$TdS=-\left(1-\frac{\dot{r}}{2}\right)A(\rho+p)dt.$ (41)

Taking the horizon temperature $\displaystyle T=-\frac{1}{2\pi
r}\left(1-\frac{\dot{r}}{2}\right)$ and entropy $\displaystyle S=A/4G$ we have

$-\frac{1}{2\pi
r}\left(1-\frac{\dot{r}}{2}\right)\frac{dA}{4G}=-\left(1-\frac{\dot{r}}{2}\right)A(\rho+p)dt.$
(42)

After some steps, it is written as

$\frac{\dot{r}}{r^{2}}=4\pi G(\rho+p).$ (43)

Using $\displaystyle r=H^{-1}$ and $\displaystyle\dot{r}=-H^{-2}\dot{H}$ lead
to

$\dot{H}=-4\pi G(\rho+p),$ (44)

that by bringing the definitions of densities and pressures, Eq. (44) is
rewritten as

$f^{\prime}(A)\dot{H}=-\frac{\kappa}{2f^{\prime}(R)}\left(\rho_{M}+p_{M}+\rho_{eff}+p_{eff}\right).$
(45)

Applying to the conservation equation and performing a temporal integration
results in

$\frac{2}{2-\Delta}f^{\prime}(A)H^{2}=\frac{\kappa}{3f^{\prime}(R)}\left(\rho_{M}+\rho_{eff}\right).$
(46)

It should be noted that Eqs. (45) and (46) are reduced to the usual form for
both Barrow and $\displaystyle f(R)$ individually when considering
$\displaystyle\Delta=0$ or $\displaystyle f(R)=R$, respectively.

Another analysis is possible. If the non-equilibrium case is desired, the non-
conservation relation is used, i.e.

$\dot{\rho}_{eff}+3H(\rho_{eff}+p_{eff})=\frac{3}{8\pi
G}\frac{2}{2-\Delta}f^{\prime}(A)H^{2}\dot{f}^{\prime}(R).$ (47)

In addition, entropy correction

$S=\frac{f(A)}{4G}f^{\prime}(R)$ (48)

is considered. This definition for entropy comes from the fact that while the
Barrow correction acts on how the horizon area is calculated [24], the
$\displaystyle f(R)$ correction acts on the physics of the system,
specifically as a correction of the gravitational constant [35], as
$\displaystyle G_{eff}=G/f^{\prime}(R)$. Furthermore, this form of entropy
also fits the proposal of dark energy components to result in a thermodynamic
relation that recovers all the characteristics of the Barrow and
$\displaystyle f(R)$ cases individually. With these ingredients, the
thermodynamic relation reads

$TdS=dE-
WdV+T\left(\frac{4-\dot{r}+\Delta\dot{r}/2}{2-\dot{r}-\Delta\left(1-\dot{r}/2\right)}\right)\frac{f(A)}{4G}df^{\prime}(R).$
(49)

As discussed earlier, the last term can originate from an out-of-equilibrium
system [46, 47].

As an application, in the next section, the Barrow-$\displaystyle f(R)$ field
equation will be solved and the state parameter in this context will be
analyzed.

##  V Application on the generalized Carroll-Duvvuri-Trodden-Turner model

To get an idea of how the addition of Barrow entropy can influence an
$\displaystyle f(R)$ model, Eq. (46) is numerically solved using the
generalized Carroll-Duvvuri-Trodden-Turner (CDTT) model [50, 51, 52]. The
$\displaystyle f(R)$ function that describes this model is defined as

$f(R)=R+\lambda R^{2}+\frac{\mu}{R},$ (50)

where $\displaystyle\lambda$ is a constant and $\displaystyle\mu$ is a
parameter with units of mass. This model is chosen because it presents two
distinct expansion periods. Initially dominated by the term
$\displaystyle\lambda R^{2}$, which causes an exponential expansion. Later, it
is dominated by $\displaystyle\mu/R$, which leads to asymptotic behavior with
a power-law expansion.

Solving the field equation, i.e. Eq.(46), the scale factor is obtained, which
is then used to calculate the state parameter, that is defined as

$\omega=\frac{p_{M}+p_{DE}}{\rho_{M}+\rho_{DE}}.$ (51)

In addition, the density parameters,

$\Omega_{M}=\frac{8\pi
G}{3H^{2}}\rho_{M}\quad\quad\mathrm{and}\quad\quad\Omega_{DE}=\frac{8\pi
G}{3H^{2}}\rho_{DE},$ (52)

for matter and dark energy, respectively, are calculated. In Figure 1 the
evolution of the state and density parameters are compared using different
values of $\displaystyle\Delta$.

Figure 1: Left side: State parameter $\displaystyle\omega$ versus redshift
$\displaystyle z$. Right side: Matter ($\displaystyle\Omega_{M}$) and dark
energy ($\displaystyle\Omega_{DE}$) densities versus redshift. Here it is
considered gCDTT + Barrow models with $\displaystyle H_{0}=0.1$,
$\displaystyle\lambda=3\times 10^{3}$, $\displaystyle\mu=9.7\times 10^{-2}$,
$\displaystyle\Omega_{M}(0)=1$, $\displaystyle\Lambda=0$.

The initial conditions are defined so that the system always starts from an
inertial expansion, $\displaystyle\omega=-1/3$, to better visualize the
influence of the dark energy. By default, this $\displaystyle f(R)$ model
produces an initial acceleration that approaches to exponential,
$\displaystyle\omega=-1$, depending on the choosen values of
$\displaystyle\lambda$, and eventually it stabilizes itself to a power law,
$\displaystyle\omega=-2/3$, according to the parameter $\displaystyle\mu$. All
this can be observed by the behaviour of curve $\displaystyle\Delta=0$. Note
that the presence of the Barrow modification, i.e. $\displaystyle\Delta\neq
0$, induces an initial deceleration in the expansion, while changing the
asymptotic behaviour, progressively taking the expansion of the system from a
power law to an exponential one. In our simulations for values close to
$\displaystyle\Delta=0.04$ or higher, it remains exponential with the
difference being an initial deceleration more predominant while having a
sharper transition to the asymptotic state. Very large values of the
$\displaystyle\Delta$ parameter can bring the system collapse for the given
conditions.

##  VI Conclusions

From the works of Hawking and Bekenstein, black hole thermodynamics opened a
window to investigate the equations of the gravitational field in a different
way, i.e. using the thermodynamics laws. In this paper, gravity-thermodynamics
conjecture is used to derived the Friedmann equation considering Barrow
entropy. In Barrow model, it is considered the case that quantum-gravitational
effects can change actual horizon area of a black hole. In other words, this
model constructs a fractal horizon surface by increasing the black hole area.
Taking the ideas of gravity-thermodynamics conjecture, $\displaystyle f(R)$
gravity is considered and the first law of thermodynamics in this
gravitational model is analyzed. The main study developed in this work was to
join the Barrow entropy and $\displaystyle f(R)$ gravity, and from that,
analyze the dynamic evolution of the universe during different periods.
Corrections for the dark energy components are imposed due to the two models.
Our results show that the scalar factor and the state parameter exhibit the
expected behavior both for the individual cases and for the union of the
Barrow and $\displaystyle f(R)$ models. The results point to a significant
influence of the intricacity parameter $\displaystyle\Delta$. However,
restrictions related to its values in this new context will be studied in
future investigations.

## Acknowledgments

This work by A. F. S. is partially supported by National Council for
Scientific and Technological Development - CNPq project No. 313400/2020-2; P.
S. E. thanks CAPES for financial support.

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