#  Test of Barrow entropy using a model independent approach

A. Salehi111Email<EMAIL_ADDRESS>Department of Physics, Lorestan
University, Khoramabad, Iran

(August 28, 2024)

###### Abstract

Taking into consideration of a fractal structure for the black hole horizon,
Barrow argued that the area law of entropy get modified due to quantum-
gravitational effects. Accordingly, the corrected entropy takes the form
$S\sim A^{1+\frac{\Delta}{2}}$, where $0\leq\Delta\leq 1,$ indicates the
amount of the quantum-gravitational deformation effects. By considering the
modified Barrow entropy associated with the apparent horizon, the Friedmann
equations get modified as well. We show that considering a universe filled
with the matter and cosmological constant $\Lambda$, it is possible to
determine the amount of deviation from standard cosmology by reconstructing
the parameter $\delta$ in terms of curvature parameters $\\{q,Q,\Omega_{k}\\}$
as $\Delta=\frac{(Q-1-\Omega_{k})(1+\Omega_{k})}{(1+\Omega_{k}+q)^{2}}$. Here,
$q$ is the deceleration parameter and $Q$ is the third derivative of scale
factor . This relation provides some advantages. The first is that it
indicates that there is profound connection between quantum-gravitational
deformation effects and curvature effects, for $\Omega_{k}\simeq 0$ the pair
$\\{q,Q\\}$ can be regarded as deviation curvature factors which reflect the
amount of deviation of the model from the standard model. The second
interesting feature is that, since this pair are observational parameters
which can be directly measured in a model independent approach, they can be
regarded as powerful tools to enable us to put constraint on parameter
$\Delta$ and test the Barrow entropy model. Our analysis predicts the value
for $Q_{0}$ which is slightly deviates from 1 as $(Q_{0}-1)<0.001$. This can
be a relativity well target and criterion for theoretical and observational
measurements of parameter $Q_{0}$. Hence we can hope and wait the improvement
of the high redshift data in the future to support it.

###### pacs:

98.80.-k, 04.50.Kd, 04.25.Nx

The quantum phenomenon of Hawking radiation indicates that black hole has a
temperature proportional to its surface gravity and an entropy proportional to
its horizon area Shey -Bekenstein . This issue led to discovery of profound
connection between gravity and thermodynamics which was first addressed by
Jacobson Xin  who disclosed that the Einstein gravitational theory for the
spacetime metric can be extracted from the horizon entropy-area relation by
using the fundamental Clausius relation $\delta Q=T\Delta S$ Horava . The
investigations on the relation between Einstein field equations and first law
of thermodynamics in the setup of black hole spacetime, have been generalized
to the cosmological context to derive Friedmann equations with any spatial
curvature by applying the Clausius relation to apparent horizon of the FRW
universe Fischler -cai2 . See cc  for further studies of thermodynamical
aspects of gravity. Recently, Barrow 5  explained that quantum gravitational
effects might create complex and fractal properties on the black hole horizon.
In this perspective, the entropy of a black hole no longer follows the law of
area, but can instead be represented by exponential raising of the area
$\Delta$ as

$S_{B}=\left(\dfrac{A}{A_{0}}\right)^{1+\frac{\Delta}{2}}$ (1)

Where $A$ is the normal horizon region and $A_{0}$ is the Planck region.
Quantum gravitational disturbance is shown by the new represented symbol
$\Delta$. There are characteristic values for $\Delta$. For example, when
$\Delta=0$, we have the simplest horizon. On the other hand, when $\Delta=1$,
we see the phenomenon called maximum deformation. The authors of In this paper
we want to show that there is connection between parameter $\Delta$ which
reflects the quantum gravitational effects and deceleration parameter and jerk
parameter. which arise from curvature and its changes. Several studies have
been devoted to the possibility that quantum gravity might triggered by
curvature, but the relevant literature has so far focused exclusively on a
subclass of scenarios such that the quantum-gravity effects are independent of
(macroscopic) curvature. In this paper we want to show that there is
connection between parameter $\Delta$ which reflects the quantum gravitational
effects and curvature effects. which arise from curvature and its changes.
FLRW metric In the background of FRW universe, the line elements of the metric

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

where $a(t)$ is scale factor of the universe, $k=1,0,-1$ stand for closed,
flat and open geometries respectively, and $(t,r,\theta,\phi)$ are the co-
moving coordinates.In this we have adopted the convention of $a_{0}=1$, where
the subscript 0 denotes the value at present time (zero redshift). The Ricci
scalar curvature is given by

$\displaystyle R=-6\Big{(}\dot{H}+2H^{2}+\frac{k}{a^{2}}\Big{)}$ (3)

where, $H=\frac{\dot{a}}{a}$ is the Hubble parameter, dot denotes the
derivative with respect to the cosmic time $t$. The deceleration parameter $q$
and the third derivative of dimensionless scale factor $Q$ defined as

$\displaystyle
q=-\frac{\ddot{a}}{aH^{2}}=-\Big{(}1+\frac{\dot{H}}{H^{2}}\Big{)}$ (4)
$\displaystyle Q=\frac{\dddot{a}}{aH^{3}}=\frac{\ddot{H}}{H^{3}}-3q-2$ (5)

One can obtain $q$ and $Q$ in terms of $R$ and its time derivation $\dot{R}$
as

$\displaystyle q=\frac{R}{6H^{2}}+1+\Omega_{k}$ (6) $\displaystyle
Q=-\frac{\dot{R}}{6H^{3}}-\frac{R}{6H^{2}}+3\Omega_{k}+3$ (7)

Where, $\Omega_{k}=\frac{k}{a^{2}H^{2}}$. The equations (6) and (7) indicate
that there is connection between parameters $(q,Q)$ and curvature $R$ and its
time derivation $\dot{R}$. Hence in the rest of the paper we may call the as
curvature parameters. In principal by considering the modified Barrow entropy
associated with the apparent horizon, the Friedmann equations get modified as
well. In this paper we show that it is possible to determine the amount of
deviation from standard cosmology by reconstructing the parameter $\Delta$ in
terms of curvature parameters $\\{q,Q\\}$.

##  I Modified Friedmann Equations based on Barrow entropy

In this section we present derivation of modified Friedmann equations based on
Barrow entropy by using the gravity-thermodynamics conjecture. Modification of
Friedman equations based on Barrow entropy was first explored by Emmanuel .
Then another approach was presented by SheBFE . Although these two approaches
seem different, we show that they are equivalent. The author of Emmanuel
deduced that in an expanding universe, during a time interval $dt$ the heat
flow through the horizon is easily found to be ca

$\delta Q=-dE=A(\rho_{m}+p_{m})Hr_{A}dt.$ (8)

Where $A=4\pi r_{A}^{2}$ and $r_{A}^{2}$ is radius of apparent horizons. From
the thermodynamical viewpoint the apparent horizon is a suitable horizon
consistent with first and second law of thermodynamics wang1 -sheyECFE .
Assuming that the Universe is bounded by the apparent horizon of radius

$\tilde{r}_{A}={1}/{\sqrt{H^{2}+k/a^{2}}}$ (9)

and considering that for the universe horizon the temperature associated with
the horizon is given by Padmanabhan

$T_{h}=\frac{1}{2\pi\tilde{r}_{A}}$ (10)

We describe the content of the Universe as a perfect fluid of energy-momentum
tensor $T_{\mu\nu}=(\rho_{m}+p_{m})u_{\mu}u_{\nu}+p_{m}g_{\mu\nu}$, where
$\rho_{m}$ and $p_{m}$ are the energy density and pressure, respectively. The
energy-momentum tensor is conserved, $\nabla_{\mu}T^{\mu\nu}=0$, which implies
the continuity equation, $\dot{\rho_{m}}+3H(\rho_{m}+p_{m})=0$.
Differentiating the Barrow entropy(1), yields

$\displaystyle
dS_{h}=(2+\Delta)\left(\frac{4\pi}{A_{0}}\right)^{1+\Delta/2}{\tilde{r}_{A}}^{1+\Delta}\dot{\tilde{r}}_{A}dt.$
(11)

Inserting these relations into the first law of thermodynamics, and
substituting $\dot{{r}}_{A}$ using (9), we finally result to

$\displaystyle-(4\pi)^{(1-\Delta/2)}A_{0}^{(1+\Delta/2)}(\rho_{m}+p_{m})=$
$\displaystyle
2(2+\Delta)\frac{\dot{H}-\frac{k}{a^{2}}}{\left(H^{2}+\frac{k}{a^{2}}\right)^{\Delta/2}}.$
(12)

Lastly, integrating, for the validity region $0\leq\Delta\leq 1$ gives

$\displaystyle\frac{(4\pi)^{(1-\Delta/2)}A_{0}^{(1+\Delta/2)}}{6}\rho_{m}=$
$\displaystyle\frac{2+\Delta}{2-\Delta}\left(H^{2}+\frac{k}{a^{2}}\right)^{1-\Delta/2}$
$\displaystyle-\frac{{C}}{3}A_{0}^{(1+\Delta/2)},$ (13)

With C the integration constant. In the second approach which explored by
SheBFE , the temperature associated with the horizon is given by Cai1

$T_{h}=-\frac{1}{2\pi\tilde{r}_{A}}\left(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\right),$
(14)

Since the Universe is expanding, the work density associated with the volume
change of the expanding universe, is also given by $W=(\rho-p)/2$ Hay2 . Using
the gravity-thermodynamics conjecture, the Friedmann equation can be obtained
by considering the Universe as a thermodynamic system bounded by the apparent
horizon and applying the first law of thermodynamics

$dE=T_{h}dS_{h}+WdV,$ (15)

Note that the $dE$ which defied here is different from that defined in the
equation (8) of the first approach. In equation (8) the $dE$ is just the
energy flux crossing the apparent horizon, and the apparent horizon radius is
kept fixed during an infinitesimal internal of time $dt$. Also, since, it was
assumed that for infinitesimal internal $\dot{r_{A}}=0$, then the definition
of temperature in two approach are different, in other word in the first
approach, the term related to $\dot{r_{A}}=0$ has bee omitted . However, in
the equation (15) of the second, it was assumed that the first law of
thermodynamics on the apparent horizon in the form, $dE$ is the change in the
energy inside the apparent horizon due to the volume change $dV$ of the
expanding Universe where $E=\rho V$ is the total energy of the Universe of
3-dimensional volume $V=\frac{4\pi}{3}\tilde{r}_{A}^{3}$ with the area of
apparent horizon $A=4\pi\tilde{r}_{A}^{2}$ and $T_{h}$ and $S_{h}$ are,
respectively, the temperature and entropy associated with the apparent
horizon. Taking differential form of the total matter and energy, we find

$dE=4\pi\tilde{r}_{A}^{2}\rho
d\tilde{r}_{A}+\frac{4\pi}{3}\tilde{r}_{A}^{3}\dot{\rho}dt$ (16)

By combining with conservation equation, we obtain

$dE=4\pi\tilde{r}_{A}^{2}\rho d\tilde{r}_{A}-4\pi
H\tilde{r}_{A}^{3}(\rho_{m}+p_{m})dt.$ (17)

Finally, combining Eqs. (14), (17) and (11) with the first law of
thermodynamics (15) and using continuity relation, after some algebraic
calculations, we obtain

$-\frac{2+\Delta}{2\pi
A_{0}}\left(\frac{4\pi}{A_{0}}\right)^{\Delta/2}\frac{d\tilde{r}_{A}}{\tilde{r}_{A}^{3-\Delta}}=\frac{d\rho_{m}}{3}.$
(18)

After integration, we find the first modified Friedmann equation in Barrow
cosmology,

$\left(H^{2}+\frac{k}{a^{2}}\right)^{1-\Delta/2}=\frac{8\pi G_{\rm
eff}}{3}\rho_{m}+\frac{\Lambda}{3},$ (19)

where $\Lambda$ is a constant of integration which can be interpreted as the
cosmological constant, and $G_{\rm eff}$ stands for the effective Newtonian
gravitational constant,$G_{\rm
eff}\equiv\frac{A_{0}}{4}\left(\frac{2-\Delta}{2+\Delta}\right)\left(\frac{A_{0}}{4\pi}\right)^{\Delta/2}$.
Eq. (19), can be rewritten as

$\left(H^{2}+\frac{k}{a^{2}}\right)^{1-\Delta/2}=\frac{8\pi G_{\rm
eff}}{3}(\rho_{m}+\rho_{\Lambda}).$ (20)

Where, $\rho_{\Lambda}={\Lambda}/(8\pi G_{\rm eff})$. The second Friedmann
equation, can be obtained by the continuity equation with the first Friedmann
equation (19). SheBFE

$\displaystyle(2-\Delta)\frac{\ddot{a}}{a}\left(H^{2}+\frac{k}{a^{2}}\right)^{-\Delta/2}+(1+\Delta)\left(H^{2}+\frac{k}{a^{2}}\right)^{1-\Delta/2}$
$\displaystyle=-8\pi G_{\rm eff}(p_{m}+p_{\Lambda}),$ (21)

where $p_{\Lambda}=-{\Lambda}/(8\pi G_{\rm eff})$. In the limiting case where
$\Delta=0$ ($G_{\rm eff}\rightarrow G$), Eq. (21) reduces to the second
Friedmann equation in standard cosmology.
It is easy to show that equations (I) and (I) of the first approach are
equivalent to the equations (20) and (21) of the second approach. Note tat
equation (I)can be simplified as

$\displaystyle\left(H^{2}+\frac{k}{a^{2}}\right)^{1-\Delta/2}=(\frac{2-\Delta}{2+\Delta})\frac{(4\pi)^{(1-\Delta/2)}A_{0}^{(1+\Delta/2)}}{6}\rho_{m}$
(22)
$\displaystyle+\frac{{C}}{3}(\frac{2-\Delta}{2+\Delta})A_{0}^{(1+\Delta/2)},$

Equation (22) can be simplified as

$\left(H^{2}+\frac{k}{a^{2}}\right)^{1-\Delta/2}=\frac{8\pi
G_{eff}}{3}\rho_{m}+\Lambda.$ (23)

Where,

$\displaystyle
G_{eff}=(\frac{2-\Delta}{2+\Delta})\frac{(4\pi)^{(1-\Delta/2)}A_{0}^{(1+\Delta/2)}}{6}=$
(24)
$\displaystyle\frac{A_{0}}{4}\left(\frac{2-\Delta}{2+\Delta}\right)\left(\frac{A_{0}}{4\pi}\right)^{\Delta/2}$
(25)

and

$\displaystyle\Lambda=\frac{{C}}{3}(\frac{2-\Delta}{2+\Delta})A_{0}^{(1+\Delta/2)}$
(26)

We see that equation (23) which was obtained in the first approach is exactly
equivalent to the equation 20 or 19 in the second approach. It is also easy to
show that from equation (I), we can obtain equation (21). Note that equation
(I) can be rewritten as

$(2-\Delta)(\dot{H}-\frac{k}{a^{2}})(\left(H^{2}+\frac{k}{a^{2}}\right)^{-\Delta/2}=-8\pi
G_{\rm eff}(\rho_{m}+p_{m})$ (27)

Considering that $\dot{H}=\frac{\ddot{a}}{a}-H^{2}$. Hence, the equation (27)
can be rewritten as

$\displaystyle(2-\Delta)(\frac{\ddot{a}}{a})\left(H^{2}+\frac{k}{a^{2}}\right)^{-\Delta/2}$
$\displaystyle-(2-\Delta)\left(H^{2}+\frac{k}{a^{2}}\right)^{1-\Delta/2}=-8\pi
G_{\rm eff}(\rho_{m}+p_{m})$ (28)

Where, using equation (20) and noting that $p_{\Lambda}=-\rho_{\Lambda}$,
equation (I) will be simplified to the equation (21). Hence the two approaches
are equivalent. It is also interesting to note that, in the approach explored
by Emmanuel  by applying the first law of thermodynamics, the second Friedman
equation is extracted, then by integration of this equation the first Friedman
is also obtained, however in approach explored by SheBFE , by applying the
first law of thermodynamics the first Friedman equation is extracted then by
taking derivative of this equation and applying the conservation equation the
second Friedman equation is also obtained

##  II Reconstructing the parameters of the model in terms of geometrical
parameters

In this section we aim to reconstruct the parameters of the model in terms of
geometrical parameters. For simplicity, we define the following variables

$\Omega_{m}=\frac{8\pi G_{\rm
eff}\rho_{m}}{3H^{2}},\Omega_{\Lambda}=\frac{8\pi G_{\rm
eff}\rho_{\Lambda}}{3H^{2}}$ (29)

Employing the above variables we can derive the following dynamical system

$\frac{d}{dx}\left(\begin{array}[]{c}\Omega_{m}\\\ \Omega_{k}\\\
\Omega_{\Lambda}\\\
H\end{array}\right)=\left(\begin{array}[]{cccc}-1+2q&0&0&0\\\ 0&2q&0&0\\\
0&0&2+2q&0\\\
0&0&0&-1-q\end{array}\right)\left(\begin{array}[]{c}\Omega_{m}\\\
\Omega_{k}\\\ \Omega_{\Lambda}\\\ H\end{array}\right)$

. (30)

where $q$ is deceleration parameter and $x=\ln a$. Also we can use (29) to
rewrite the the first Friedman equation (20) as

$\Omega_{m}+\Omega_{\Lambda}={H}^{-\Delta}\left(1+\Omega_{k}\right)^{1-\frac{1}{2}\,\Delta}$
(31)

Hence, by differentiating equation (31) with respect to $x$ and using (30) ,
We can drive

$\displaystyle\Delta(1+q){H}^{-\Delta}+(1-2q){H}^{-\Delta}\left(1+\Omega_{k}\right)^{1-\frac{1}{2}\,\Delta}$
$\displaystyle+q(2-\Delta){H}^{-\Delta}\Omega_{k}\left(1+\Omega_{k}\right)^{-\frac{1}{2}\,\Delta}=3\Omega_{\Lambda}$
(32)

In 1970, Alan Sandage Sandage  interpreted cosmology as the search for two
numbers: $H_{0}$ and $q_{0}$ ,then Weinberg Winberg  drew attention to the
issue of extracting the value of constant spatial curvature $k$ and
deceleration parameter $q$ from the observations, without considering
cosmological constant and/or scalar field. In 1976 Harisson Harrison
challenged Sandage’s remark and proved that the third derivative of the scale
factor $Q$, is of great importance for observational cosmology, in a universe
with indeterminate (dust) matter density. He considered a universe containing
the cosmological constant $\Lambda$ and non-relativistic matter. In this case,
the Einstein equations reduce to the following Friedmann equations

$\displaystyle H^{2}+\frac{kc^{2}}{a^{2}}=\frac{8\pi
G}{3}\rho+\frac{\Lambda}{3}$ (33) $\displaystyle\dot{H}+H^{2}=-\frac{4\pi
G}{3}(\rho+3P)+\frac{\Lambda}{3}$ (34)

For zero-pressure model, the equations (33) and (34) can be combined as

$\displaystyle K=4\pi G\rho-H^{2}(q+1)$ (35)

Where $K=\frac{kc^{2}}{a^{2}}$, $\rho$ is the average mass density. The
verification of the cosmological equation (35) requires the measurement of the
quantities $K,H,\rho$ and $q$ Harrison . In principle, these quantities
$K,H,q$ can be determined, although in practice their precise determination is
difficult Sandage -Harrison . If every thing could be known about the matter
filling of the universe, then the cosmological constant will be derived as

$\displaystyle\Lambda=4\pi G\rho-3qH^{2}$ (36)

Hence, for a universe with known amount of matter, the general relativity (GR)
can be tested by measuring $H$ and $q$. However, the average density $\rho$
can not be determined, even in principal Sandage -Harrison . Here, the third
derivative of the scale factor is required to test the validity of the
equation (35). Since the parameters $(k,\Lambda,\rho)$ can be obtained in
terms of first three derivatives of the scale factor as follows.

$\displaystyle K=H^{2}(Q-1)$ (37) $\displaystyle\Lambda=H^{2}(Q-2q)$ (38)
$\displaystyle 4\pi G\rho=H^{2}(Q+q)$ (39)

It is easy to obtain the following relation

$\displaystyle\frac{dq}{dx}=-Q+q+2q^{2}$ (40)

Hence, by differentiating equation (32) with respect to $x$ and using (30) and
(40) , we can drive

$\displaystyle(Q-1)=$
$\displaystyle\Omega_{k}-\Delta\,{q}^{2}\Omega_{k}\,\left(1+\Omega_{k}\right)^{-1}$
$\displaystyle+\left(1+2\,q+{q}^{2}+\Omega_{k}\right)\Delta$ (41)

Hence, the Barrow parameter $\Delta$ can be obtained as

$\displaystyle\Delta=\frac{(Q-1-\Omega_{k})(1+\Omega_{k})}{(1+\Omega_{k}+q)^{2}}$
(42)

It is interesting to note that the relation (42) is established throughout the
history of the Universe and is not limited to the present time, but we can use
the information related to the current values of these parameters to determine
the constant value of $\Delta$. Based on the Planck 2018 results Planck 2018 ,
where, $\Omega_{k}=0.001\pm 0.002$, $q_{0}=-0.527\pm 0.01$ and the results of
Nikodem  which have put constraint on parameter $Q_{0}$ as
$Q_{0}=1.01^{+0.08}_{-0.021}$, the deviation relation shows that
$\Delta=0.04^{+0.385}_{-0.068}$. In the the limiting case where $\Delta=0$,
the equation (II) reduces to

$\displaystyle\Omega_{k}=(Q-1)$ (43)

Consequently using equation (43), for $\Delta=0$, the equations (31) and (32)
gives

$\displaystyle\Omega_{\Lambda}=\frac{1}{3}(Q-2q)$ (44)
$\displaystyle\Omega_{m}=\frac{2}{3}(Q+q)$ (45)

Equations (43)-(45) are equivalent to equations (37)-(39) which previously
obtained by Harrison .

### Solution for case $\Omega_{k}=0$

Considering $\Omega_{k}=0$, equation (42) gives the Barrow parameter $\Delta$
as

$\displaystyle\Delta=\frac{Q-1}{(q+1)^{2}}$ (46)

The first interesting result of equation (46) is that it indicates that there
is profound connection between quantum-gravitational deformation effects
$\Delta$ and geometrical cosmology parameters $(q,Q)$. Another interesting
result is that this equation indicates that the geometrical parameters $(q,Q)$
can be regarded as deviation curvature factors which reflect the amount of
deviation of the model from the standard model. The equation (46) indicates
that for $Q=1$ the standard model $\Delta=0$ is recovered. This is an expected
result because $Q=1$ is correspond to $\Lambda CDM$ model. In fact almost all
current cosmological observations can be summarized by a simple case $(Q=1)$
Maciej ,Alam ,Kun . Also VisserVisse , have investigated in some details the
condition $(Q=1)$. In principal the case $Q=1$ is a third order ODE which is
hold only for $\Lambda CDM$ model (It is easy to find that for flat case of
$\Lambda CDM$ model, $Q=1+2\Omega_{r}$ , hence neglecting radiation density
energy $\Omega_{r}=0$, $Q=1$). This relation can also be immediately deduced
from Eq (37). In principle the parameter $Q$ as one of the pairs statefinder
diagnostic $\\{r\equiv Q,s\\}$ can distinguish $\Lambda CDM$ form other
cosmological models(For more discussion one can see (Sahni2003  and Alam ).
The relation $Q=1$ exactly gives the evolution of the universe for $\Lambda
CDM$ model. It gives the expected thermal history of the universe from matter
dominated to dark energy dominated. To show this, we start from equation
(40).By integration, it can be rewritten as

$\displaystyle\int dx=\int\frac{dq}{-Q+q+2q^{2}}$ (47)

Inserting $Q=1$, the above integral gives
$x=\frac{1}{3}\ln(\frac{2q-1}{1+q})+\frac{1}{3}\ln(C)$. Here $C$ is constant
of integration. Since, $x=\ln a=-\ln(1+z)$, the above equation can be
simplified as

$\displaystyle\ln\frac{1}{(1+z)^{3}}=\ln\Big{(}C(\frac{2q-1}{1+q})\Big{)}$
(48)

Hence, the deceleration parameter $q$ will be obtained in terms of redshift
$z$ as

$\displaystyle q=\frac{C(1+z)^{3}+1}{2C(1+z)^{3}-1}$ (49)

By set $q_{0}$ for $z=0$ in this equation, we can obtain the constant $C$ in
terms of $q_{0}$ as $C=\frac{1+q_{0}}{2q_{0}-1}$. Also transition redshift
$z_{t}$ will be obtained as

$\displaystyle
z_{t}=-1+(\frac{-1}{C})^{\frac{1}{3}}=-1+(\frac{1-2q_{0}}{1+q_{0}})^{\frac{1}{3}}$
(50)

According to the equation (49), when $z\rightarrow\infty$, the deceleration
parameter $q\rightarrow\frac{1}{2}$ and when $z\rightarrow-1$, the
deceleration parameter $q\rightarrow-1$. Also if we get $q_{0}\simeq-0.53$
which is the result of Planck 2018, the equation (50) gives $z_{t}\simeq 0.65$
In Fig. 1, we have also plotted the evolution of deceleration parameter
against redshift $z$ according to the equation (49). As can be seen the
evolution is exactly the evolution of the deceleration parameter for $\Lambda
CDM$ model. Hence $Q=1$ is specified to flat(k=0) $\Lambda CDM$ by considering
$\Omega_{r}=0$.

Fig. 1: The evolution of deceleration parameter against redshift $z$ according
to equation (49)

Therefore, it is expected that models whose structure is similar to the
$\Lambda CDM$ model will have their Q closer to 1. This usually can be
achieved in the models such as Chevallier–Polarski–Linder (CPL), $\omega CDM$,
$XCDM$ Phenomenologically Emergent Dark Energy (PEDE). There are some previous
studies which have also reported $Q\simeq 1$ (i.e. the parameter $Q$ should be
near $1$). Moreover, there are some observational studies which have supported
this expectation, $Q=1$. The Refs Maciej ,Alam  find $Q=1$ and $Q=1.02$,
respectively. Also, There are also some previous studies which have obtained
values close to 1. For example see Herrera-Zamorano , Escamilla , Capozz ,
Munoz  , Escamilla2 . Harrison .
In order to examine the sensitivity of the deviation parameter $\Delta$ with
changes of each of these parameters, we introduce the following partial
differential equations

$\displaystyle d_{Q}=\frac{\partial\Delta}{\partial Q},\ \
d_{q}=\frac{\partial\Delta}{\partial q}$ (51)

Hence, using equations (46) and(51), we can find

$\displaystyle d_{Q}=$ $\displaystyle=\frac{1}{(q+1)^{2}}$ (52) $\displaystyle
d_{q}=$ $\displaystyle-2\frac{Q-1}{(q+1)^{3}}=-2\frac{\Delta}{q+1}$ (53)

Hence, the absolute value of the ratio of these changes would be

$\displaystyle\Big{|}\frac{d_{q}}{d_{Q}}\Big{|}=\Big{|}\frac{\Delta(q+1)}{2}\Big{|}$
(54)

Since, $|q+1|<2$, one can deduced that

$\displaystyle\Big{|}\frac{d_{q}}{d_{Q}}\Big{|}<\Delta$ (55)

Due to this fact that $\Delta\ll 1$, it can be concluded that $d_{q}\ll
d_{Q}$. In other word, $\Delta$ is more sensitive to changes of $Q$. Hence,
parameter $Q$ can be regarded as deviation parameter which determines the
deviation of the model from $\Lambda CDM$. The more the difference of the $Q$
value is greater than 1, the greater the deviation of the model from the
Standard model. Since, $0\leq\Delta\leq 1$, so applying this condition on
equation (46), a bound is obtained for parameter $Q$ as

$\displaystyle 1\leq Q\leq 2+2q+q^{2}$ (56)

Since, for accelerating Universe, $q_{0}<0$, the condition (56)gives
interesting result. It indicates that upper bound for $Q_{0}$ in Barrow
cosmology is $Q_{0}<2$. This bound is further restricted if we substitute the
observational value of $q_{0}$. For $q_{0}\simeq-0.5$, which is the result of
example for Planck 2018 , the condition is restricted to $1<Q_{0}<1.25$. This
condition has interesting result as indicates that observational studies on
Kaniadakkis cosmology must give the value for $Q_{0}$ which be closer to 1. It
also indicates that as $q_{0}$ gets closer to 1, the region becomes more
limited, where for $q_{0}=-1$ this region becomes a points as $Q_{0}=1$. This
interesting result indicates that $\Delta=0$ is correspond to
$(q_{0}=-1,Q_{0}=1)$ and these values are hold for $\Lambda CDM$ model.
Here we aim to find parameters of the model in terms of geometrical parameters
$(H,q,Q)$. Equation (32) for $\Omega_{k}=0$ can be rewritten as

$\displaystyle\Big{(}\Delta(1+q)+1-2q\Big{)}H^{-\Delta}=3\Omega_{\Lambda}$
(57)

Where, inserting the $\Delta$ from equation (46) into the equation(57), the
parameter, $\Omega_{\Lambda}$ will be obtained in terms of $(H,q,Q)$ as

$\displaystyle\Omega_{\Lambda}=\frac{1}{3}\Big{(}\frac{Q-q-2q^{2}}{q+1}\Big{)}H^{-(\frac{Q-1}{(q+1)^{2}})}$
(58)

Also, inserting $H^{-\Delta}$, from equation (31) into the equation(57)

$\displaystyle\Big{(}\Delta(1+q)+1-2q\Big{)}=\frac{3\Omega_{\Lambda}}{\Omega_{m}+\Omega_{\Lambda}}$
(59)

Where, inserting the $\Delta$ from equation (46) into the equation(59), we
arrive

$\displaystyle\frac{\Omega_{m}}{\Omega_{\Lambda}}=-\frac{(Q-4q-2q^{2}-3)}{(Q-q-2q^{2})}$
(60)

Combining equations (58) and (60), the parameter $\Omega_{m}$ also can be
obtained in terms of $(H,q,Q)$ as

$\displaystyle\Omega_{m}=-\frac{1}{3}\Big{(}\frac{Q-4q-2q^{2}-3}{q+1}\Big{)}H^{-(\frac{Q-1}{(q+1)^{2}})}$
(61)

The equations (46), (58) and (61) are equations that express the parameters of
the model,$(\Omega_{m},\Omega_{\Lambda},\Delta)$ in terms of geometrical
parameters $(H,q,Q)$.
According to the approach explored by Emmanuel , we can also find the
interesting parameters $\Omega_{DE}$ and $\omega_{DE}$. For this peropus, we
can can re-express equations (I) and (I) as

$\displaystyle H^{2}=\frac{8\pi G}{3}\left(\rho_{m}+\rho_{DE}\right)$ (62)
$\displaystyle\dot{H}=-4\pi G\left(\rho_{m}+p_{m}+\rho_{DE}+p_{DE}\right),$
(63)

where

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\rho_{DE}=\frac{3}{8\pi
G}\left\\{\frac{\Lambda}{3}+H^{2}\left[1-\frac{\beta(\Delta+2)}{2-\Delta}H^{-\Delta}\right]\right\\},$
(64) $\displaystyle p_{DE}=-\frac{1}{8\pi
G}\left\\{\Lambda+2\dot{H}\left[1-\beta\left(1+\frac{\Delta}{2}\right)H^{-\Delta}\right]\right\\}$
$\displaystyle+$ $\displaystyle
3\left\\{H^{2}\left[1-\frac{\beta(2+\Delta)}{2-\Delta}H^{-\Delta}\right]\right\\}$
(65)

where respectively are the energy density and pressure of the effective dark
energy sector and $\beta\equiv\frac{4(4\pi)^{\Delta/2}G}{A_{0}^{1+\Delta/2}}$.
Hence, by defining $\Omega_{DE}=\frac{8\pi G\rho_{DE}}{3H^{2}}$ and using
equation (64), we have

$\displaystyle\Omega_{DE}=\frac{8\pi
G\rho_{DE}}{3H^{2}}=\Omega_{\Lambda}+\left[1-\frac{\beta(\Delta+2)}{2-\Delta}H^{-\Delta}\right]$
(66)

Also the effective equation of state read

$\displaystyle\omega_{DE}=\frac{p_{DE}}{\rho_{DE}}=\frac{-1}{3}\frac{\left\\{3\Omega_{\Lambda}-2(1+q)\left[1-\beta\left(1+\frac{\Delta}{2}\right)H^{-\Delta}\right]+\left[1-\frac{\beta(2+\Delta)}{2-\Delta}H^{-\Delta}\right]\right\\}}{\Omega_{\Lambda}+\left[1-\frac{\beta(\Delta+2)}{2-\Delta}H^{-\Delta}\right]}$
(67)

We, obtained $\Delta$ in terms of $\\{q,Q\\}$, also $\Omega_{\Lambda}$ and
$\Omega_{m}$ in terms of cosmographic parameters $\\{H,q,Q\\}$, according to
the above equations, $\omega_{DE}$ and $\Omega_{DE}$ can also be expressed in
terms of cosmographic parameters $\\{H,q,Q\\}$.
It is also possible to obtain the current value of parameters
$\Omega_{m},\Omega_{\Lambda},\Omega_{DE},\omega_{DE}$ in terms of the current
values $(H_{0},q_{0},Q_{0})$. Here, we aim to test the model for different
cosmographic sets. It is important to note that although, the present
observational data could provide strong constraint on the current values of
Hubble parameter $H_{0}$ and deceleration parameter $q_{0}$, but due to the
fact that constraining on $Q_{0}$ requires high redshift data and until today,
there is no enough high-quality data at high redshift,it is not possible to
put strong constrain on $Q_{0}$ and different studies have reported different
values for this parameter. Hence there is no very reliable set of
$\\{H_{0},q_{0},Q_{0}\\}$. However, it is possible to estimate this parameter
theoretically in different cosmological models. We can use the results of
Planck 2018 Planck 2018  for $H_{0}$ and $q_{0}$ as robust observational
measurements. According to the Planck results, $H_{0}=(67.4\pm 0.5)$ and
$q_{0}=-0.527\pm 0.01$. However, since no strong measurements on $Q_{0}$ and
$\Delta$ and wide ranges of values have been reported for this parameters. For
example authors of Alejandro , using same model, reported different values
$Q_{0}\simeq 1.268$ and $Q_{0}\simeq-7.746$ for different data. In the
Alejandro , they also reported $Q_{0}\simeq-13.695$ for another category of
the model. So there is no strong constrain on $Q_{0}$. We are also facing this
problem for the parameter $\Delta$. Recently, using Big Bang Nucleosynthesis
(BBN) data John  have imposed constraint on the exponent $\Delta$ and found
that Barrow exponent should be inside the bound $\Delta\leq 1.4\times
10^{-4}$, also the authors of Genly , found the value $\Delta=5.912\times
10^{-4}$ which is comparable withJohn . The authors of Mahnaz , found
$\Delta\simeq 10^{-4}$. In Leon  the entropic modified Friedmann equation
within the gravity-thermodynamics approach is confronted with a set of
cosmological probes, including Pantheon SNeIa and a BAO sample, and obtain
$\Delta\sim 10^{-4}$. In Feizi  the authors do not use any early-times probe,
but only SNeIa, CC and GRBs, and a different horizon from ours, and their
final estimation for the Tsallis parameter is $\delta\approx 0.16$, which
corresponds to $\Delta\approx-1.68$, clearly out of the physical boundary
required by Barrow theory. In S0  the holographic principle is applied and
they found the Tesallis parameter as $\delta\approx 1.07$ corresponding to
$\Delta\approx 0.14$. Similar results are obtained in S1 ; S2  using only
late-times data, with $\Delta\sim 0.09$. Finally, in Adhikary , the Barrow
entropy parameter was constrained as $\Delta\sim 0.03$. The authors ofNandhida
, using Barrow Holographic Dark Energy Model with GO Cut-of found
$\Delta=0.063\pm 0.029$
We see that different values from order $10^{-4}$ to order $1$ have been
reported for parameter $\Delta$. Hence same as $Q_{0}$, there is no
convergence in results.
Here, according to the analytical relations which have obtained, we want to
test the different results.
At first we call the relation (46). Since, for accelerating universe
$q_{0}<0$, hence, $(q_{0}+1)<1$ and ($(q_{0}+1)^{2}<1$ Hence according to the
equation (46)

$\displaystyle(Q_{0}-1)\leq\Delta$ (68)

This condition indicates that for those studies that reported $\Delta\simeq
10^{-4}$, the condition $(Q_{0}-1)<10^{-4}$ should be satisfied.. This
indicates that $Q_{0}$ should be very close to $1$. It also points out an
important point; Observational constraining on $Q_{0}$ should be performed
with high accuracy and sensitivity, so that the error of the order of
$10^{-4}$ is measured. Although, there are many studies that have reported
best fitted values for $Q_{0}$ which are close to 1, but in none of them, a
best fitted value less than $1.01$ has been reported. Among all the values
reported for $Q_{0}$, the closest value to 1 has been reported by Nikodem ,
which is $Q_{0}=1.01^{+0.08}_{-0.021}$. Considering this value and result of
Planck 2018Planck 2018  for $q_{0}$; $q_{0}=-0.527\pm 0.01$, the relation
gives $\Delta=0.04^{+0.385}_{-0.068}$. Although this result is consistence
with some of previous studies, however, the other parameters should be also
measured to test the validation of the result. Considering the Planck 2018
resultPlanck 2018  $H_{0}=(67.4\pm 0.5)$, the current values of parameters of
the model for $\Delta=0.04^{+0.385}_{-0.068}$, are reconstructed as

$\displaystyle\Omega_{m0}=.255,\ \ \Omega_{\Lambda_{0}}=0.573,$
$\displaystyle\Omega_{DE_{0}}=0.07,\ \ \omega_{DE0}\simeq-8.36$
$\displaystyle\beta\simeq 1.73$ (69)

As can be seen, the values obtained for $\Omega_{DE0},\omega_{DE0}$ are not
consistence with that expected by robust observational data and indicates that
the value obtained for $\Delta$ is not desired value. This is due to this fact
that parameter $\Delta$ should be very close to 0, and this requires the value
of $Q_{0}$ is very close to 1. If we check the model for upper bound of
$\Delta$ which reported by John ; $\Delta=1.4\times 10^{-4}$ and considering
the Planck results $q_{0}=-0.527\pm 0.01$,$H_{0}=(67.4\pm 0.5)$, the results
would be

$\displaystyle\Omega_{m0}\simeq 0.3151,\ \ \Omega_{\Lambda_{0}}\simeq 0.6842,$
$\displaystyle\Omega_{DE_{0}}\simeq 0.6830,\ \ \omega_{DE0}\simeq-1.0014,$
$\displaystyle\beta\simeq 1.00122$ (70)

The values obtained for this case are in excellent agreement with
observations. In particular the value $\Omega_{m0}\simeq 0.3151$ is very close
to that obtained by Planck 2018  as $\Omega_{m0}\simeq 0.315\pm 0.007$.
According to the equation (46) for $\Delta=1.4\times 10^{-4}$ and
$q_{0}=-0.527\pm 0.01$ the current value $Q_{0}$ is

$\displaystyle(Q_{0}\simeq 1.00002)$ (71)

If we test the model for $\Delta=5.912\times 10^{-4}$ which obtained by Genly
, the following results would be obtained

$\displaystyle\Omega_{m0}\simeq 0.3144,\ \ \Omega_{\Lambda_{0}}\simeq 0.6830,$
$\displaystyle\Omega_{DE_{0}}\simeq 0.6776,\ \ \omega_{DE0}\simeq-1.0080,$
$\displaystyle\beta\simeq 1.00122$ (72)

These results are very close to results of Genly , who found

$\displaystyle\Omega_{m0}=311^{+0.006}_{-0.005},\
\omega_{DE0}\simeq-1.0001,\beta=0.920^{+0.042}_{-0.042}$ (73)

For this case the current value of $Q_{0}$ be

$\displaystyle(Q_{0}\simeq 1.00012)$ (74)

However, as we mentioned, so far, the measurement on $Q_{0}$ has not been done
with this high accuracy. The fact that values closer to 1 are not reported may
be due to this fact that, so far, the importance of very small changes of
order $10^{-4}$ of this parameter and how much it can affect the results has
not been paid attention to. So observational measurements and numerical
analysis on $Q_{0}$ are not done with this accuracy. As mentioned, another
reason could be the lack of high quality data required for constraining this
parameter. So this study indicates the parameter $Q$ is important parameter in
Barrow campanology which its current value is very close to 1. In Table(1), we
have obtained current values for parameters of the model, for different values
of $Q_{0}$ by fixing Planck 2018 results for $q_{0}$ and $H_{0}$. Due to the
high sensitivity of the model with small changes of this parameter, even very
small errors in estimation of this parameter lead to very different results
for the model. As we investigated, very different results was obtained for two
very close values $(Q_{0}\simeq 1.01)$ and $(Q_{0}\simeq 1.001).$

Table 1:  Reconstructing parameters for diffferent values of $Q_{0}$ by fixing $q_{0}=-0.527$ $Q_{0}$ | $\Delta$ | $\Omega_{m0}$ | $\Omega_{\Lambda 0}$ | $\Omega_{DE0}$ | $\omega_{DE0}$ | $\beta$
---|---|---|---|---|---|---
$1$ | 0 | 0.3155 | $0.6846$ | 0.6846 | -1 | 1
$1.00001$ | 0.00004 | 0.3152 | $0.6845$ | 0.6841 | -1.0006 | 1.00054
$1.0001$ | 0.000044 | 0.31467 | $.6834$ | .6793 | -1.0066 | 1.0055
$1.001$ | 0.0044 | 0.30876 | $0.67259$ | 0.63118 | -1.0725 | 1.05645
$1.005$ | 0.0223 | 0.2838 | $0.6263$ | 0.4014 | -1.6098 | 1.31601
$1.008$ | 0.0357 | 0.2664 | $0.5938$ | 0.2103 | -2.9622 | 1.5517
$1.01$ | 0.0446 | 0.2553 | $0.5730$ | 0.0726 | -8.36038 | 1.7318
$1.1$ | 0.446 | 0.0372 | $0.11499$ | -57.13 | -0.0831 | 242.77

### Solution for case $\Omega_{k}\neq 0$

For general case $(\Delta\neq 0,\Omega_{k}\neq 0)$, from equation(II),
$\Omega_{k}$ can be obtained as

$\Omega_{k}=-1+{\frac{-2\,\Delta\,q+Q\pm\Big{(}{Q}^{2}-4\,\Delta\,qQ-4\,\Delta\,{q}^{2}\Big{)}^{\frac{1}{2}}}{2(1+\Delta)}},$
(75)

By substituting $\Omega_{k}$ from equation (75) in equation (32),
$\Omega_{\Lambda}$ is obtained in terms of $(H,q,Q,\Delta)$. In equation (42),
we have obtained $\Delta$ in terms of the cosmographic parameters $(q,Q)$ and
$\Omega_{k}$, however if we want to obtain $\Delta$ only in terms of the
cosmographic parameters and eliminate the parameter $\Omega_{k}$, we need the
cosmographic parameter $X=\frac{\ddddot{a}}{aH^{4}}$. For this respect, by
taking derivative of both sides of equation (II) and using equation (30) we
can obtain

$\displaystyle-4\Delta
q^{3}+(-8\Delta\Omega_{k}-6\Delta)q^{2}+\Big{(}(-4\Delta-4)\Omega_{k}^{2}$
$\displaystyle+(5Q-6\Delta-4)\Omega_{k}+(2Q-2)\Delta+3Q\Big{)}q$
$\displaystyle+(1+\Omega_{k})(X+2Q+12+2\Delta Q)=0$ (76)

Where we have also used the following relation

$\displaystyle\frac{dQ}{dx}=X+(2+3q)Q$ (77)

From equations (II) and (75), the barrow parameter $\Delta$ can be obtained in
term of the cosmographic parameters $(q,Q,X)$ as

$\displaystyle\Delta=-\frac{1}{4(q+Q)^{3}}\Big{(}2AQ+\frac{1}{2}AqQ+2Aq+6A+\frac{1}{2}A$
$\displaystyle+$ $\displaystyle(qQ^{2}+XQ+12Q)(q+Q)^{3}\Big{)}$ (78)

Where

$\displaystyle A=$ $\displaystyle-4qQ-q^{2}Q-4q^{2}-12q-qX$
$\displaystyle+\Big{(}144q^{2}+12XQq^{2}+56q^{3}Q+2q^{3}QX$
$\displaystyle+q^{4}Q^{2}+8q^{4}Q+q^{2}X^{2}+144q^{2}Q+96q^{3}$
$\displaystyle+20q^{2}Q^{2}+12q^{3}Q^{2}+24Xq^{2}+8Xq^{3}+16q^{4}$
$\displaystyle+8qQ^{3}+4Q^{4}+4q^{2}Q^{3}+4qXQ^{2}+48qQ^{2}\Big{)}^{\frac{1}{2}}$
(79)

We have used Maple software to derive the above relations. By substituting
$\Delta$ from equation (II) into equation (75), the parameter $\Omega_{k}$ is
also obtained in terms of cosmographic parameters $(q,Q,X)$. Also, from
equation (32), the parameter $\Omega_{\Lambda}$ is obtained in terms of
$(H,q,Q,X)$, finally from equation (31), the parameter $\Omega_{m}$ is also
obtained in terms of $(H,q,Q,X)$. This means that it is possible to
reconstruct the parameter of the Barrow model
$(\Delta,\Omega_{k},\Omega_{m},\Omega_{\Lambda})$ in terms of directly
measurable parameters $(H,q,Q,X)$.

##  III summary and remarks

In this paper we reconstructed the parameters of the modified Friedman
equation due to the gravity-thermodynamics approach and assuming the Barrow
entropy in terms of the first four cosmographic parameters $(H,q,Q,X)$. Barrow
entropy arises from the fact that the black-hole surface may be deformed due
to quantum-gravitational effects, and its deviation from Bekenstein-Hawking
one is quantified by $\Delta$. Due to significant role of the parameter
$\Delta$ in describing the late-time universe and early-time behavior, most of
the recent studies on Barrow entropy have focused on imposing constraints on
the exponent $\Delta$ . We found that for a universe filled with mater and
cosmological constant $\Lambda$ this parameter could be obtained in terms of
$\Omega_{k}$ and two important cosmographic parameters $(q,Q)$ where for flat
universe or neglecting $\Omega_{k}$ it can be obtained only in terms of
$(q,Q)$. The connection between parameter $\Delta$ which reflects the quantum-
gravitational effects and cosmographic parameters which reflect the curvature
effects has interesting features which can be summarized as follows.
The first interesting feature is that this relation indicates that it is
possible to imposing constrain on parameter $\Delta$ by imposing constrain on
parameters current values of $(q,Q)$. This can be interesting because of this
facts that cosmographic parameters $(q_{0},Q_{0})$ could be measured directly
without need of any background cosmological model, in addition one of the main
purposes of observational cosmology is to get precise measurements of the
first cosmographic parameters $(H_{0},q_{0},Q_{0})$ that will provide a
crucial test for cosmological models Sandage ,Harrison . Hence a vast majority
of cosmological observational studies have focused on constraining on these
parameters. As a robust observational data, the Planck 2018 Planck 2018
providing a major source of information relevant to many cosmological
parameters indicates that $q_{0}=-0.527\pm 0.01$. Also Harrison  proved that
the third derivative of the scale factor $Q$, is of great importance for
observational cosmology, hence some of studies only have focused on
constraining cosmic jerk parameter $Q$ in different cosmological models. The
authors of Nikodem  have put constrain on parameter $Q_{0}$ in $f(R)$ gravity
and find $Q_{0}=1.01^{+0.08}_{-0.021}$. Considering this value and the value
of $q_{0}=-0.527\pm 0.01$ obtained from Planck 2018 results the deviation
parameter $\Delta$ is obtained as $\Delta=0.04^{+0.385}_{-0.068}$. We showed
that although this result is consistence with some of previous studies,
however, the other important parameters should be also measured to test the
validation of the result. We reconstructed important parameters
$\Omega_{DE_{0}},omega_{DE0}$ and $\beta$ according to the approach explored
by Emmanuel  and test the model by this parameter. For
$\Delta=0.04^{+0.385}_{-0.068}$, considering the Planck 2018 resultPlanck 2018
$H_{0}=(67.4\pm 0.5)$, the current values of parameters of the model for this
value, are reconstructed as $\Omega_{DE_{0}}=0.07,\omega_{DE0}\simeq-8.36$
which are not consistence with that expected by robust observational data and
indicates that the value obtained for $\Delta$ is not desired value. This is
due to this fact that parameter $\Delta$ should be very close to 0, and this
requires the value of $Q_{0}$ be very close to 1. We checked the model for
upper bound of $\Delta$ which reported by John ; $\Delta=1.4\times 10^{-4}$
and considering the Planck results $q_{0}=-0.527\pm 0.01$,$H_{0}=(67.4\pm
0.5)$, and

$\displaystyle\Omega_{m0}\simeq 0.315,\ \Omega_{\Lambda_{0}}\simeq
0.684,\Omega_{DE_{0}}\simeq 0.683,\omega_{DE0}\simeq-1.0014,$

Also $\beta$ was found as $\beta\simeq 1.00122$. The values obtained for this
case are in excellent agreement with observations. In particular the value
$\Omega_{m0}\simeq 0.3151$ is very close to that obtained by Planck 2018  as
$\Omega_{m0}\simeq 0.315\pm 0.007$. According to the equation (46) for
$\Delta=1.4\times 10^{-4}$ and $q_{0}=-0.527\pm 0.01$ the current value
$Q_{0}$ is $(Q_{0}\simeq 1.00002).$ We also test the model for
$\Delta=5.912\times 10^{-4}$ which obtained by Genly , the following results
would be obtained

$\displaystyle\Omega_{m0}\simeq 0.314,\ \ \Omega_{\Lambda_{0}}\simeq 0.683,$
$\displaystyle\Omega_{DE_{0}}\simeq 0.677,\ \ \omega_{DE0}\simeq-1.008$

For this case the parameter $\beta$ was found as $\beta\simeq 1.00122$. These
results are very close to results of Genly , who found

$\displaystyle\Omega_{m0}=311^{+0.006}_{-0.005},\
\omega_{DE0}\simeq-1.0001,\beta=0.920^{+0.042}_{-0.042}$

For this case the current value of $Q_{0}$ be $(Q_{0}\simeq 1.00012)$.
However, so far, the measurement on $Q_{0}$ has not been done with this high
accuracy. The fact that values closer to 1 are not reported may be due to this
fact that, so far, the importance of very small changes of order $10^{-4}$ of
this parameter and how much it can affect the results has not been paid
attention to. So observational measurements and numerical analysis on $Q_{0}$
are not done with this accuracy. Another reason could be the lack of hat high
redshift there is no strong constrain on this parameter.
So, we can conclude that this study reveals the importance of parameter $Q$ in
Barrow campanology, it also indicates that the current value of $Q_{0}$ is
very close to 1. Our theoretical approach indicates that, the works done by
John  and Genly  which found that Barrow exponent should be inside the bound
$\Delta\leq 1.4\times 10^{-4}$ have good agrement with observations. In other
word for this bound, the values which reconstructed for parameters of the
model are in excellent agreement wit observations. This bound indicates that
by considering Planck results Planck 2018 , for $q_{0}=-0.527\pm 0.01$ and
$\Omega_{k}=0.001\pm 0.002$ the relation (42) predicts the value for $Q_{0}$
which is slightly deviates from 1 as $(Q_{0}-1)<0.001$. This can be a
relativity well target and criterion for theoretical and observational
measurements of parameter $Q$, so we can hope and wait the improvement of the
high redshift data in the future to support it.

##  IV Data availability statement

The manuscript has no associated data or the data will not be deposited

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