# Observational constraints of the modified cosmology through Barrow entropy

Mahnaz Asghari<EMAIL_ADDRESS>Physics Department, College of
Sciences, Shiraz University, Shiraz 71454, Iran and Biruni Observatory,
College of Sciences, Shiraz University, Shiraz 71454, Iran Ahmad Sheykhi
<EMAIL_ADDRESS>Physics Department, College of Sciences, Shiraz
University, Shiraz 71454, Iran and Biruni Observatory, College of Sciences,
Shiraz University, Shiraz 71454, Iran

###### Abstract

Taking into account a fractal structure for the black hole horizon, Barrow
argued that the area law of entropy get modified due to quantum-gravitational
effects [Phys. Lett. B 808 (2020) 135643]. Accordingly, the corrected entropy
takes the form $S\sim A^{1+\mathrm{\Delta}/2}$, where
$0\leq\mathrm{\Delta}\leq 1$, indicates the amount of the quantum-
gravitational deformation effects. In this paper, based on Barrow entropy, we
first derive the modified gravitational field equations through the Clausius
relation. We then consider the Friedmann-Lemaître-Robertson-Walker (FLRW) as
the background metric and derive the modified Friedmann equations inspired by
Barrow entropy. In order to explore observational constraints on the modified
Barrow cosmology, we employ two different combinations of available datasets,
mainly "Planck + Pantheon \+ BAO" and "Planck + Planck-SZ + CFHTLenS +
Pantheon + BAO + BAORSD" datasets. According to numerical results, we observe
that "Planck \+ Pantheon + BAO" dataset predicts higher values of $H_{0}$ in
Barrow cosmology with a phantom dark energy compared to $\mathrm{\Lambda}$CDM,
so tensions between low redshift determinations of Hubble constant and cosmic
microwave background (CMB) results is slightly reduced. On the other hand, in
case of dataset "Planck + Planck-SZ + CFHTLenS + Pantheon + BAO + BAORSD"
there is a slight amelioration in $\sigma_{8}$ tension in Barrow cosmology
with a quintessential dark energy.

##  1 Introduction

The profound connection between thermodynamics and gravitational field
equations has arisen a lot of attentions, since the discovery of thermodynamic
properties of black holes [1, 2, 3]. It has been confirmed that the field
equations of general relativity is nothing but an equation of state for the
spacetime [4]. In other words, when the spacetime is regarded as a
thermodynamic system, the law of thermodynamics on the large scale can be
interpreted as the law of gravity. The thermodynamics-gravity conjecture has
now been well explored in the literatures [5, 6, 7, 8, 9, 10, 11, 12, 13]. The
investigation has been generalized to the cosmological background, where it
has been shown that the Friedmann equations describing the evolution of the
universe can be rewritten in the form of the first law of thermodynamics and
vise versa [14, 15, 16, 17, 18, 19, 20, 21]. Furthermore, one can apply this
remarkable connection in the context of braneworld scenarios [22, 23, 24, 25].

In the cosmological approach, it is possible to extract the Friedmann
equations of FLRW universe by applying the first law of thermodynamics
$\mathrm{d}E=T\mathrm{d}S+W\mathrm{d}V$ at the apparent horizon [20]. It was
argued that, in any gravity theory, one can consider the entropy expression
associated with the apparent horizon in the form of the black hole entropy in
the same gravity theory. The only change needed is to replace the black hole
horizon radius $r_{+}$ in the entropy expression, by the apparent horizon
radius $\tilde{r}_{\mathrm{A}}$ in the entropy expression associated with the
apparent horizon. While it is more convenient to apply the Bekenstein-Hawking
area law relation defined as $S_{BH}={A}/{(4G)}$ for the black hole entropy
[26, 3] (with the black hole horizon area $A=4\pi r_{+}^{2}$), it should be
noted that, there are several types of corrections to the area law entropy.
Two possible corrections occur due to quantum effects are known as logarithmic
corrections arise from loop quantum gravity [27, 28, 29, 30, 31, 32, 33, 34],
and also power-law corrections established in the entanglement of quantum
fields between inside and outside the horizon [35, 36, 37]. The other
appropriate correction concerning the area law of entropy comes from the fact
that Boltzmann-Gibbs (BG) additive entropy should be generalized to non-
additive entropy in divergent partition function systems like gravitational
systems [38, 39, 40, 41, 42, 43, 44]. Therefore, it has been argued that the
entropy of these systems should be in the form of non-extensive Tsallis
entropy $S\sim A^{\beta}$, where $\beta$ is the non-extensive or Tsallis
parameter [45]. The studies on the non-additive Tsallis entropy and its
applications in cosmology can be carried out in [46, 47, 48, 49, 50, 51, 52,
53, 54, 55].

Moreover, it is worth turning our attention to Barrow correction to area law
entropy due to quantum-gravitational effects [56]. Around one year ago, J. D.
Barrow considered an intricate and fractal geometry for the black hole
horizon, which leads to an increase in surface area. The modified entropy
relation based on Barrow corrections takes the form [56]

$S=\Big{(}\frac{A}{A_{0}}\Big{)}^{1+\mathrm{\Delta}/2}\;,$ (1)

where $A$ is the black hole horizon area, $A_{0}$ is the Planck area, and the
exponent $\mathrm{\Delta}$ which quantifies the quantum-gravitational
deformation, ranges as $0\leq\mathrm{\Delta}\leq 1$ [56, 57, 58]. The area law
entropy expression is restored by choosing $\mathrm{\Delta}=0$ which
accordingly corresponds to the simplest horizon structure, while
$\mathrm{\Delta}=1$ expresses the most intricate surface structure. There are
some investigations on Barrow entropy in the cosmological and gravitational
setups [59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75].

In order to rewrite the modified field equations from Barrow entropy at
apparent horizon of FLRW universe, one should replace the entropy in Clausius
relation $\delta Q=T\delta S$ by the entropy expression (1), as well as
considering $A$ as the apparent horizon area given by
$A=4\pi\tilde{r}^{2}_{\mathrm{A}}$. Our approach in the present study is
similar to the one in paper [76] which considers non-additive Tsallis entropy
correction to area law relation. However, it should be mentioned that the
physical motivation and principles are completely different in two
investigations. Particularly, Tsallis non-additive entropy correction is
motivated by generalizing standard thermodynamics to a non-extensive one,
while Barrow correction to entropy originates from intricate, fractal
structure on horizon induced by quantum-gravitational effects.

It is worth mentioning that applying the Bekenstein-Hawking entropy in
Clausius relation leads to the Einstein field equations in standard
$\mathrm{\Lambda}$CDM model [7]. It is known that the standard model of
cosmology is in excellent agreement with the majority of observational
constraints, however, it suffers from some observational tensions which
inspire investigations beyond the standard model. Specifically, low redshift
measurements of the Hubble constant [77, 78, 79, 80], and also local
determinations of the growth of structure [81] are inconsistent with the
Planck cosmic microwave background (CMB) observations [82]. Accordingly, in
this paper we study whether the discrepancies between local and global
measurements can be relieved in Barrow cosmology. Throughout this paper we set
$k_{\mathrm{B}}=c=\hbar=1$, for simplicity.

This paper is structured as follows. In section 2 we derive modified field
equations describing the evolutions of universe when the horizon entropy is
given by Eq. (1). Numerical solutions based on Barrow entropy corrections to
the field equations are presented in Section 3. Section 4 is dedicated to
constraining Barrow cosmology with observational data. We summarize our
conclusions in section 5.

##  2 Modified gravitational field equations from Barrow corrections

In this part, we are going to derive the corresponding field equations in the
cosmological setup when the entropy associated with the apparent horizon is in
the form of Barrow entropy. In this respect, we consider a spatially flat FLRW
universe, with the background metric given by

$\mathrm{d}s^{2}=a^{2}(\tau)\big{(}-\mathrm{d}\tau^{2}+\mathrm{d}\boldsymbol{x}^{2}\big{)}\;,$
(2)

where $\tau$ is the conformal time. Also the perturbed line element in linear
perturbation theory in conformal Newtonian gauge reads

$\mathrm{d}s^{2}=a^{2}(\tau)\Big{(}-\big{(}1+2\mathrm{\Psi}\big{)}\mathrm{d}\tau^{2}+\big{(}1-2\mathrm{\Phi}\big{)}\mathrm{d}\boldsymbol{x}^{2}\Big{)}\;,$
(3)

with gravitational potentials $\mathrm{\Psi}$ and $\mathrm{\Phi}$. Similarly
in synchronous gauge we have

$\mathrm{d}s^{2}=a^{2}(\tau)\Big{(}-\mathrm{d}\tau^{2}+\big{(}\delta_{ij}+h_{ij}\big{)}\mathrm{d}x^{i}\mathrm{d}x^{j}\Big{)}\;,$
(4)

where $h_{ij}=\mathrm{diag}(-2\eta,-2\eta,h+4\eta)$, with scalar perturbations
$h$ and $\eta$.

Furthermore, we consider the universe consists of radiation (R), matter (M)
(dark matter (DM) and baryons (B)) and dark energy (DE), which are assumed to
be perfect fluids with the energy-momentum tensor defined as

$T_{\mu\nu(i)}=\big{(}\rho_{i}+p_{i}\big{)}u_{\mu(i)}u_{\nu(i)}+g_{\mu\nu}p_{i}\;,$
(5)

where $\rho_{i}=\bar{\rho}_{i}+\delta\rho_{i}$ is the energy density,
$p_{i}=\bar{p}_{i}+\delta p_{i}$ is the pressure, and $u_{\mu(i)}$ is the
four-velocity of $i^{th}$ component in the universe (and a bar indicates the
background level).

Employing the Clausius relation, $\delta Q=T\delta S$, which is satisfied on a
local causal horizon $\mathcal{H}$, we derive the gravitational field
equations. Considering the universe as a thermodynamic system, we apply the
Clausius relation on the apparent horizon of the universe with radius
$\tilde{r}_{\mathrm{A}}$, defined as [83]

$\tilde{r}_{\mathrm{A}}=\Big{(}H^{2}+\frac{K}{a^{2}}\Big{)}^{-1/2}\;,$ (6)

where $H$ is the Hubble parameter and $K=-1,0,1$ is the curvature constant
corresponding to open, flat and closed universe, respectively. Also the
associated temperature with the apparent horizon is given by

$T=\frac{\kappa}{2\pi}\;,$ (7)

where $\kappa$ is the surface gravity at apparent horizon. As mentioned
before, in the interest of deriving field equations in Barrow cosmology, we
should apply the Borrow entropy defined in equation (1) in Clausius relation.
So $\delta S$ takes the form

$\delta
S=\Big{(}1+\frac{\mathrm{\Delta}}{2}\Big{)}A_{0}^{-1-\mathrm{\Delta}/2}A^{\mathrm{\Delta}/2}\delta
A\;.$ (8)

According to Ref. [4, 7], $\delta Q$ can be written as

$\delta Q=-\kappa\int_{\mathcal{H}}\lambda
T_{\mu\nu}k^{\mu}k^{\nu}\mathrm{d}\lambda\mathrm{d}A\;,$ (9)

and $\delta A$ is given by

$\displaystyle\delta
A=\int_{\mathcal{H}}\theta\mathrm{d}\lambda\mathrm{d}A\;,$ (10)

with expansion $\theta$ defined as

$\displaystyle\theta=-\lambda R_{\mu\nu}k^{\mu}k^{\nu}\;.$ (11)

Thus, considering equations (7),(8) and (9), Clausius relation takes the form

$\displaystyle\kappa\int_{\mathcal{H}}(-\lambda)T_{\mu\nu}k^{\mu}k^{\nu}\mathrm{d}\lambda\mathrm{d}A$
$\displaystyle=\frac{\kappa}{2\pi}\Big{(}1+\frac{\mathrm{\Delta}}{2}\Big{)}A_{0}^{-1-\mathrm{\Delta}/2}\int_{\mathcal{H}}(-\lambda)R_{\mu\nu}k^{\mu}k^{\nu}A^{\mathrm{\Delta}/2}\mathrm{d}\lambda\mathrm{d}A\;,$
(12) $\displaystyle\to\int_{\mathcal{H}}$
$\displaystyle(-\lambda)\Bigg{(}-2\pi
T_{\mu\nu}+\Big{(}1+\frac{\mathrm{\Delta}}{2}\Big{)}A_{0}^{-1-\mathrm{\Delta}/2}R_{\mu\nu}A^{\mathrm{\Delta}/2}\Bigg{)}$
$\displaystyle\times k^{\mu}k^{\nu}\mathrm{d}\lambda\mathrm{d}A=0\;.$ (13)

Then, for all null vectors $k^{\mu}$ we find

$-2\pi
T_{\mu\nu}+\Big{(}1+\frac{\mathrm{\Delta}}{2}\Big{)}A_{0}^{-1-\mathrm{\Delta}/2}R_{\mu\nu}A^{\mathrm{\Delta}/2}=fg_{\mu\nu}\;,$
(14)

where $f$ is a scalar. So according to energy-momentum conservation
($\nabla^{\mu}T_{\mu\nu}=0$) we must have

$\nabla^{\mu}\Bigg{(}\Big{(}1+\frac{\mathrm{\Delta}}{2}\Big{)}A_{0}^{-1-\mathrm{\Delta}/2}R_{\mu\nu}A^{\mathrm{\Delta}/2}-fg_{\mu\nu}\Bigg{)}=0\;,$
(15)
$\to\Big{(}1+\frac{\mathrm{\Delta}}{2}\Big{)}A_{0}^{-1-\mathrm{\Delta}/2}\Bigg{(}\frac{1}{2}\big{(}\partial_{\nu}R\big{)}A^{\mathrm{\Delta}/2}+R_{\mu\nu}\partial^{\mu}A^{\mathrm{\Delta}/2}\Bigg{)}=\partial_{\nu}f\,.$
(16)

Considering Eq. (16), the LHS is not the gradient of a scalar, which means
that corresponds to non-equilibrium behaviour of thermodynamics, the Clausius
relation is not satisfied. Thus, the Clausius relation should be replaced by
the entropy balance relation [7]

$\delta S=\frac{\delta Q}{T}+\mathrm{d}_{i}S\;,$ (17)

where $\mathrm{d}_{i}S$ is the entropy produced inside the system caused by
irreversible transformations of the system [84]. Then in order to resolve the
contradiction with energy-momentum conservation, we assume $\mathrm{d}_{i}S$
as the following form

$\mathrm{d}_{i}S=\Big{(}1+\frac{\mathrm{\Delta}}{2}\Big{)}A_{0}^{-1-\mathrm{\Delta}/2}\int_{\mathcal{H}}(-\lambda)\nabla_{\mu}\nabla_{\nu}A^{\mathrm{\Delta}/2}k^{\mu}k^{\nu}\mathrm{d}\lambda\mathrm{d}A\;.$
(18)

Substituting $\mathrm{d}_{i}S$ in relation (17) results in

$\displaystyle\int_{\mathcal{H}}$
$\displaystyle(-\lambda)\Bigg{(}\Big{(}1+\frac{\mathrm{\Delta}}{2}\Big{)}A_{0}^{-1-\mathrm{\Delta}/2}R_{\mu\nu}A^{\mathrm{\Delta}/2}-2\pi
T_{\mu\nu}$
$\displaystyle-\Big{(}1+\frac{\mathrm{\Delta}}{2}\Big{)}A_{0}^{-1-\mathrm{\Delta}/2}\nabla_{\mu}\nabla_{\nu}A^{\mathrm{\Delta}/2}\Bigg{)}k^{\mu}k^{\nu}\mathrm{d}\lambda\mathrm{d}A=0\;.$
(19)

Again, for all null vectors $k^{\mu}$ we should have

$\displaystyle\Big{(}1+\frac{\mathrm{\Delta}}{2}\Big{)}A_{0}^{-1-\mathrm{\Delta}/2}R_{\mu\nu}A^{\mathrm{\Delta}/2}-2\pi
T_{\mu\nu}$
$\displaystyle-\Big{(}1+\frac{\mathrm{\Delta}}{2}\Big{)}A_{0}^{-1-\mathrm{\Delta}/2}\nabla_{\mu}\nabla_{\nu}A^{\mathrm{\Delta}/2}=fg_{\mu\nu}\;.$
(20)

Then, considering energy-momentum conservation and after doing some
calculations we obtain

$\Big{(}1+\frac{\mathrm{\Delta}}{2}\Big{)}A_{0}^{-1-\mathrm{\Delta}/2}\Bigg{(}\frac{1}{2}\big{(}\partial_{\nu}R\big{)}A^{\mathrm{\Delta}/2}-\partial_{\nu}\Box
A^{\mathrm{\Delta}/2}\Bigg{)}=\partial_{\nu}f\;.$ (21)

Here, we can choose the scalar $\mathcal{L}$ as
$\mathcal{L}=RA^{\mathrm{\Delta}/2}$, which reads

$\displaystyle\frac{\partial\mathcal{L}}{\partial R}=A^{\mathrm{\Delta}/2}\;,$
$\displaystyle\partial_{\nu}R=\frac{\partial R}{\partial
x^{\nu}}=\frac{\partial
R}{\partial\mathcal{L}}\frac{\partial\mathcal{L}}{\partial
x^{\nu}}=A^{-\mathrm{\Delta}/2}\partial_{\nu}\mathcal{L}\;,$
$\displaystyle\to\big{(}\partial_{\nu}R\big{)}A^{\mathrm{\Delta}/2}=\partial_{\nu}\mathcal{L}\;.$
(22)

Therefor, the scalar $f$ becomes

$\displaystyle f$
$\displaystyle=\Big{(}1+\frac{\mathrm{\Delta}}{2}\Big{)}A_{0}^{-1-\mathrm{\Delta}/2}\Big{(}\frac{1}{2}\mathcal{L}-\Box
A^{\mathrm{\Delta}/2}\Big{)}$
$\displaystyle=\Big{(}1+\frac{\mathrm{\Delta}}{2}\Big{)}A_{0}^{-1-\mathrm{\Delta}/2}\Big{(}\frac{1}{2}RA^{\mathrm{\Delta}/2}-\Box
A^{\mathrm{\Delta}/2}\Big{)}\;.$ (23)

Thus, the modified field equations in Barrow cosmology can be written as

$\displaystyle
R_{\mu\nu}A^{\mathrm{\Delta}/2}-\nabla_{\mu}\nabla_{\nu}A^{\mathrm{\Delta}/2}-\frac{1}{2}RA^{\mathrm{\Delta}/2}g_{\mu\nu}+\Box
A^{\mathrm{\Delta}/2}g_{\mu\nu}$
$\displaystyle=\frac{4\pi}{2+\mathrm{\Delta}}A_{0}^{1+\mathrm{\Delta}/2}T_{\mu\nu}\;,$
(24)

in which for a flat spacetime we have

$A^{\mathrm{\Delta}/2}=(4\pi)^{\mathrm{\Delta}/2}\Big{(}\frac{a^{\prime}}{a^{2}}\Big{)}^{-\mathrm{\Delta}}=(4\pi)^{\mathrm{\mathrm{\Delta}}/2}H^{-\mathrm{\mathrm{\Delta}}}\;,$
(25)

where a prime indicates a deviation with respect to the conformal time. In
this way, we derive the modified Einstein field equations based on Barrow
corrections to area law entropy, caused by quantum-gravitational effects on
apparent horizon of FLRW universe. Considering $\mathrm{\Delta}=0$, the
standard field equations in Einstein gravity will be recovered. The $(00)$ and
$(ii)$ components of gravitational field equations (2) at background level
take the form

$\Big{(}\frac{a^{\prime}}{a^{2}}\Big{)}^{-\mathrm{\Delta}}\Bigg{\\{}\Big{(}\frac{a^{\prime}}{a}\Big{)}^{2}-\mathrm{\Delta}\bigg{(}\frac{a^{\prime\prime}}{a}-2\Big{(}\frac{a^{\prime}}{a}\Big{)}^{2}\bigg{)}\Bigg{\\}}=\frac{8\pi
G}{3}a^{2}\sum_{i}\bar{\rho}_{i}\;,$ (26)
$\displaystyle\Big{(}\frac{a^{\prime}}{a^{2}}\Big{)}^{-\mathrm{\Delta}}\Bigg{\\{}-2\frac{a^{\prime\prime}}{a}+\Big{(}\frac{a^{\prime}}{a}\Big{)}^{2}+\mathrm{\Delta}\Bigg{[}-\frac{a^{\prime\prime}}{a}+\frac{a^{\prime\prime\prime}}{a^{\prime}}-\Big{(}\frac{a^{\prime\prime}}{a^{\prime}}\Big{)}^{2}$
$\displaystyle-\mathrm{\Delta}\bigg{(}\frac{a^{\prime\prime}}{a^{\prime}}-2\frac{a^{\prime}}{a}\bigg{)}^{2}\Bigg{]}\Bigg{\\}}=8\pi
Ga^{2}\sum_{i}\bar{p}_{i}\;,$ (27)

where we have defined $A_{0}$ as

$A_{0}=\big{(}4\pi\big{)}^{(\mathrm{\Delta}-2)/(2+\mathrm{\Delta})}\Big{(}\big{(}2+\mathrm{\Delta}\big{)}8\pi
G\Big{)}^{2/(2+\mathrm{\Delta})}\;.$

Also background level equations in term of Hubble parameter are given by

$H^{2-\mathrm{\Delta}}-\mathrm{\Delta}H^{-\mathrm{\Delta}}H^{\prime}\frac{1}{a}=\frac{8\pi
G}{3}\sum_{i}\bar{\rho}_{i}\;,$ (28) $\displaystyle
H^{-\mathrm{\Delta}}\Bigg{\\{}\big{(}\mathrm{\Delta}-2\big{)}H^{\prime}\frac{1}{a}-3H^{2}$
$\displaystyle+\mathrm{\Delta}\bigg{(}\frac{H^{\prime\prime}}{H}-\big{(}1+\mathrm{\Delta}\big{)}\Big{(}\frac{H^{\prime}}{H}\Big{)}^{2}\bigg{)}\frac{1}{a^{2}}\Bigg{\\}}=8\pi
G\sum_{i}\bar{p}_{i}\;.$ (29)

Then, from (28) and (2), the first modified Friedmann equation can be derived
as

$H^{2-\mathrm{\Delta}}=\frac{1}{1+2\mathrm{\Delta}}\frac{8\pi
G}{3}\sum_{i}\bar{\rho}_{i}\;.$ (30)

It is also convenient to rewrite Friedmann equation in term of total density
parameter defined as
$\mathrm{\Omega}_{\mathrm{tot}}=\bar{\rho}_{\mathrm{tot}}/\rho_{\mathrm{cr}}$,
where $\rho_{\mathrm{cr}}={3H^{2}}/{(8\pi G)}$ and
$\bar{\rho}_{\mathrm{tot}}=\sum_{i}\bar{\rho}_{i}$. Hence, equation (30) can
be written as

$\mathrm{\Omega}_{\mathrm{tot}}=\big{(}1+2\mathrm{\Delta}\big{)}H^{-\mathrm{\Delta}}\;.$
(31)

It should be noted that, in the limit $\mathrm{\Delta}\to 0$, the Friedmann
equation takes the standard form in general relativity.

Defining the total equation of state parameter $w_{\mathrm{tot}}$ as
$w_{\mathrm{tot}}=\bar{p}_{\mathrm{tot}}/\bar{\rho}_{\mathrm{tot}}$, the
accelerated expansion of universe in Barrow cosmology is satisfied when
$w_{\mathrm{tot}}<-(1+\mathrm{\Delta})/3$. Taking into account the fact that
$0\leq\mathrm{\Delta}\leq 1$, Barrow corrections predict a more negative
equation of state parameter in an accelerating universe (for example for
$\mathrm{\Delta}=1$ we obtain $w_{\mathrm{tot}}<-2/3$), while in the limit
$\mathrm{\Delta}\to 0$ the accelerated expansion condition in standard
cosmology will be restored.

Taking into account the modified field equations (2) to linear order of
perturbations, in conformal Newtonian gauge (con) we have

$\displaystyle\Big{(}\frac{a^{\prime}}{a^{2}}\Big{)}^{-\mathrm{\Delta}}\Bigg{\\{}3\frac{a^{\prime}}{a}\mathrm{\Phi}^{\prime}+k^{2}\mathrm{\Phi}+3\Big{(}\frac{a^{\prime}}{a}\Big{)}^{2}\mathrm{\Psi}$
$\displaystyle-\frac{3}{2}\mathrm{\Delta}\bigg{(}\frac{a^{\prime\prime}}{a^{\prime}}-2\frac{a^{\prime}}{a}\bigg{)}\bigg{(}\mathrm{\Phi}^{\prime}+2\frac{a^{\prime}}{a}\mathrm{\Psi}\bigg{)}\Bigg{\\}}=-4\pi
Ga^{2}\sum_{i}\delta\rho_{i(\mathrm{con})}\;,$ (32)
$\displaystyle\Big{(}\frac{a^{\prime}}{a^{2}}\Big{)}^{-\mathrm{\Delta}}\Bigg{\\{}k^{2}\mathrm{\Phi}^{\prime}+\frac{a^{\prime}}{a}k^{2}\mathrm{\Psi}-\frac{1}{2}\mathrm{\Delta}\bigg{(}\frac{a^{\prime\prime}}{a^{\prime}}-2\frac{a^{\prime}}{a}\bigg{)}k^{2}\mathrm{\Psi}\Bigg{\\}}$
$\displaystyle=4\pi
Ga^{2}\sum_{i}\big{(}\bar{\rho}_{i}+\bar{p}_{i}\big{)}\theta_{i(\mathrm{con})}\;,$
(33) $\mathrm{\Phi}=\mathrm{\Psi}\;,$ (34)
$\displaystyle\Big{(}\frac{a^{\prime}}{a^{2}}\Big{)}^{-\mathrm{\Delta}}\Bigg{\\{}2\frac{a^{\prime\prime}}{a}\mathrm{\Psi}-\Big{(}\frac{a^{\prime}}{a}\Big{)}^{2}\mathrm{\Psi}+\frac{a^{\prime}}{a}\mathrm{\Psi}^{\prime}+2\frac{a^{\prime}}{a}\mathrm{\Phi}^{\prime}+\mathrm{\Phi}^{\prime\prime}$
$\displaystyle+\frac{k^{2}}{3}\big{(}\mathrm{\Phi}-\mathrm{\Psi}\big{)}+\mathrm{\Delta}\Bigg{\\{}\mathrm{\Psi}\Bigg{[}\mathrm{\Delta}\bigg{(}\frac{a^{\prime\prime}}{a^{\prime}}-2\frac{a^{\prime}}{a}\bigg{)}^{2}-\frac{a^{\prime\prime\prime}}{a^{\prime}}+\frac{a^{\prime\prime}}{a}$
$\displaystyle+\Big{(}\frac{a^{\prime\prime}}{a^{\prime}}\Big{)}^{2}\Bigg{]}+\mathrm{\Psi}^{\prime}\bigg{(}\frac{a^{\prime}}{a}-\frac{1}{2}\frac{a^{\prime\prime}}{a^{\prime}}\bigg{)}+\mathrm{\Phi}^{\prime}\bigg{(}2\frac{a^{\prime}}{a}-\frac{a^{\prime\prime}}{a^{\prime}}\bigg{)}\Bigg{\\}}\Bigg{\\}}$
$\displaystyle=4\pi Ga^{2}\sum_{i}\delta p_{i(\mathrm{con})}\;,$ (35)

while in synchronous gauge (syn) we can write

$\displaystyle\Big{(}\frac{a^{\prime}}{a^{2}}\Big{)}^{-\mathrm{\Delta}}\Bigg{\\{}\frac{a^{\prime}}{a}h^{\prime}-2k^{2}\eta-\frac{1}{2}\mathrm{\Delta}h^{\prime}\bigg{(}\frac{a^{\prime\prime}}{a^{\prime}}-2\frac{a^{\prime}}{a}\bigg{)}\Bigg{\\}}$
$\displaystyle=8\pi Ga^{2}\sum_{i}\delta\rho_{i(\mathrm{syn})}\;,$ (36)
$\Big{(}\frac{a^{\prime}}{a^{2}}\Big{)}^{-\mathrm{\Delta}}\,k^{2}\eta^{\prime}=4\pi
Ga^{2}\sum_{i}\big{(}\bar{\rho}_{i}+\bar{p}_{i}\big{)}\theta_{i(\mathrm{syn})}\;,$
(37)
$\displaystyle\Big{(}\frac{a^{\prime}}{a^{2}}\Big{)}^{-\mathrm{\Delta}}\Bigg{\\{}\frac{1}{2}h^{\prime\prime}+3\eta^{\prime\prime}+\frac{a^{\prime}}{a}h^{\prime}+6\frac{a^{\prime}}{a}\eta^{\prime}-k^{2}\eta$
$\displaystyle-\frac{1}{2}\mathrm{\Delta}\bigg{(}\frac{a^{\prime\prime}}{a^{\prime}}-2\frac{a^{\prime}}{a}\bigg{)}\bigg{(}h^{\prime}+6\eta^{\prime}\bigg{)}\Bigg{\\}}=0\;,$
(38)
$\displaystyle\Big{(}\frac{a^{\prime}}{a^{2}}\Big{)}^{-\mathrm{\Delta}}\Bigg{\\{}-2\frac{a^{\prime}}{a}h^{\prime}-h^{\prime\prime}+2k^{2}\eta+\mathrm{\Delta}\bigg{(}\frac{a^{\prime\prime}}{a^{\prime}}-2\frac{a^{\prime}}{a}\bigg{)}h^{\prime}\Bigg{\\}}$
$\displaystyle=24\pi Ga^{2}\sum_{i}\delta p_{i(\mathrm{syn})}\;.$ (39)

Choosing $\mathrm{\Delta}=0$ would result in standard field equations at
perturbation level in Einstein gravity.

Furthermore, considering energy-momentum conservation, continuity and Euler
equations would not be affected by Barrow corrections, and so conservation
equations are identical to those in general relativity. In the rest of the
paper we explore Barrow cosmology in the synchronous gauge, hence,
conservation equations for matter and dark energy components take the form

$\displaystyle\delta^{\prime}_{\mathrm{M(syn)}}=-\theta_{\mathrm{M(syn)}}-\frac{1}{2}h^{\prime}\;,$
(40)
$\displaystyle\theta^{\prime}_{\mathrm{M(syn)}}=-\frac{a^{\prime}}{a}\theta_{\mathrm{M(syn)}}\;,$
(41)
$\displaystyle\delta^{\prime}_{\mathrm{DE(syn)}}=-3\frac{a^{\prime}}{a}\big{(}c^{2}_{s,\mathrm{DE}}-w_{\mathrm{DE}}\big{)}\delta_{\mathrm{DE(syn)}}-\frac{1}{2}h^{\prime}\big{(}1+w_{\mathrm{DE}}\big{)}$
$\displaystyle-\big{(}1+w_{\mathrm{DE}}\big{)}\bigg{(}1+9\Big{(}\frac{a^{\prime}}{a}\Big{)}^{2}\big{(}c^{2}_{s,\mathrm{DE}}-c^{2}_{a,\mathrm{DE}}\big{)}\frac{1}{k^{2}}\bigg{)}\theta_{\mathrm{DE(syn)}}\;,$
(42)
$\displaystyle\theta^{\prime}_{\mathrm{DE(syn)}}=\frac{a^{\prime}}{a}\big{(}3c^{2}_{s,\mathrm{DE}}-1\big{)}\theta_{\mathrm{DE(syn)}}+\frac{k^{2}c^{2}_{s,\mathrm{DE}}}{1+w_{\mathrm{DE}}}\delta_{\mathrm{DE(syn)}}\;.$
(43)

In the next section, we study Barrow cosmology using a modified version of the
Boltzmann code CLASS111Cosmic Linear Anisotropy Solving System [85], in which
we have included the Barrow parameter $\mathrm{\Delta}$ that quantifies
deviations from standard cosmology. Moreover, in order to constrain
cosmological parameters from current observations, we employ the Markov Chain
Monte Carlo (MCMC) method by making use of the Monte Python code [86, 87].

##  3 Numerical analysis

In order to study, numerically, the Barrow cosmology model, we modify the
CLASS code according to the Barrow cosmology field equations described in
section 2. In this direction, we consider Planck $2018$ results [82] for the
cosmological parameters, read $\mathrm{\Omega}_{\mathrm{B},0}h^{2}=0.02242$,
$\mathrm{\Omega}_{\mathrm{DM},0}h^{2}=0.11933$, $H_{0}=67.66$
$\frac{km}{sMpc}$, $A_{s}=2.105\times 10^{-9}$, and
$\tau_{\mathrm{reio}}=0.0561$, and also we choose $w_{\mathrm{DE}}=-0.98$ and
$c_{s,\mathrm{DE}}^{2}=1$ for the dark energy fluid.

Figure (1) displays the CMB temperature anisotropy and matter power spectra
diagrams in Barrow cosmology compared to $\mathrm{\Lambda}$CDM.

Figure 1: The CMB power spectra (upper left) and their relative ratio with
respect to $\mathrm{\Lambda}$CDM (upper right) for different values of
$\mathrm{\Delta}$. Lower panels show the analogous diagrams for the matter
power spectrum.

Matter power spectra diagrams show an enhancement in structure growth in the
Barrow cosmology model, which is inconsistent with low redshift measurements
of galaxy clustering. The evolution of matter density contrast illustrated in
figure (2) would also reflect the increase in the growth of structures in
Barrow cosmology.

Figure 2: Matter density contrast diagrams in term of conformal time in Barrow
cosmology compared to $\mathrm{\Lambda}$CDM.

On the other hand, considering the Friedmann equation (30), it is possible to
explore the expansion history of the universe in Barrow cosmology as
demonstrated in figure (3). According to this figure it can be understood that
the current value of Hubble parameter in Barrow cosmology is more compatible
with its local determinations in comparison with the $\mathrm{\Lambda}$CDM
model.

Figure 3: Hubble parameter in term of conformal time in Barrow cosmology
compared to $\mathrm{\Lambda}$CDM.

##  4 Comparison with observational data

In this section, we put constraints on the parameters of Barrow cosmology, by
applying an MCMC approach through the Monte Python code [86, 87]. The set of
cosmological parameters used in MCMC analysis consists of:
{$100\,\mathrm{\Omega}_{\mathrm{B},0}h^{2}$,
$\mathrm{\Omega}_{\mathrm{DM},0}h^{2}$, $100\,\theta_{s}$,
$\ln(10^{10}A_{s})$, $n_{s}$, $\tau_{\mathrm{reio}}$, $w_{\mathrm{DE}}$,
$\mathrm{\Delta}$},
where $\mathrm{\Omega}_{\mathrm{B},0}h^{2}$ and
$\mathrm{\Omega}_{\mathrm{DM},0}h^{2}$ represent the baryon and cold dark
matter densities relative to the critical density respectively, $\theta_{s}$
is the ratio of the sound horizon to the angular diameter distance at
decoupling, $A_{s}$ stands for the amplitude of the primordial scalar
perturbation spectrum, $n_{s}$ is the scalar spectral index,
$\tau_{\mathrm{reio}}$ is the optical depth to reionization, $w_{\mathrm{DE}}$
is the dark energy equation of state parameter, and $\mathrm{\Delta}$
indicates the Barrow parameter. Furthermore, we have four derived parameters
including the reionization redshift ($z_{\mathrm{reio}}$), the matter density
parameter ($\mathrm{\Omega}_{\mathrm{M},0}$), the Hubble constant ($H_{0}$),
and the root-mean-square mass fluctuations on scales of 8 $h^{-1}$ Mpc
($\sigma_{8}$). Considering preliminary numerical studies, we choose the prior
range [$0$, $0.015$] for the Barrow parameter, and additionally, we set no
prior range on $w_{\mathrm{DE}}$.

The following likelihoods are utilized in MCMC method: The Planck likelihood
with Planck 2018 data (containing high-$l$ TT,TE,EE, low-$l$ EE, low-$l$ TT,
and lensing) [82], the Planck-SZ likelihood for the Sunyaev-Zeldovich effect
measured by Planck [88, 89], the CFHTLenS likelihood with the weak lensing
data [90, 91], the Pantheon likelihood with the supernovae data [92], the BAO
likelihood with the baryon acoustic oscillations data [93, 94], and the BAORSD
likelihood for BAO and redshift-space distortions (RSD) measurements [95, 96].

In order to constrain the cosmological model under consideration, we use two
different dataset combinations: "Planck + Pantheon + BAO" (hereafter dataset
(I)) and "Planck + Planck-SZ + CFHTLenS + Pantheon + BAO + BAORSD" (hereafter
dataset (II)). Table (1) represents the observational constraints from two
different datasets (I) and (II).

Table 1: Best fit values as well as 68% and 95% confidence level intervals for
cosmological parameters of $\mathrm{\Lambda}$CDM, Barrow cosmology (with
Barrow parameter $\mathrm{\Delta}$), and also TMG model (with Tsallis
parameter $\beta$) from Ref. [76].

| Planck + Pantheon + BAO | Planck + Planck-SZ + CFHTLenS + Pantheon + BAO + BAORSD
---|---|---
|  $\mathrm{\Lambda}$CDM | Barrow cosmology |  $\mathrm{\Lambda}$CDM | Barrow cosmology | TMG model
parameter | best fit | 68% & 95% limits | best fit | 68% & 95% limits | best fit | 68% & 95% limits | best fit | 68% & 95% limits | best fit | 68% & 95% limits
$100\,\mathrm{\Omega}_{\mathrm{B},0}h^{2}$ | $2.246$ | $2.245^{+0.013+0.027}_{-0.014-0.028}$ | $2.235$ | $2.239^{+0.015+0.030}_{-0.015-0.028}$ | $2.261$ | $2.263^{+0.012+0.026}_{-0.013-0.025}$ | $2.270$ | $2.264^{+0.013+0.027}_{-0.013-0.027}$ | $2.272$ | $2.268^{+0.014+0.027}_{-0.015-0.028}$
$\mathrm{\Omega}_{\mathrm{DM},0}h^{2}$ | $0.1191$ | $0.1189^{+0.00096+0.0018}_{-0.00086-0.0020}$ | $0.1203$ | $0.1197^{+0.0012+0.0021}_{-0.00095-0.0023}$ | $0.1163$ | $0.1164^{+0.00078+0.0015}_{-0.00079-0.0015}$ | $0.1161$ | $0.1164^{+0.00092+0.0016}_{-0.00076-0.0017}$ | $0.1166$ | $0.1160^{+0.00097+0.0016}_{-0.00077-0.0017}$
$100\,\theta_{s}$ | $1.042$ | $1.042^{+0.00029+0.00057}_{-0.00030-0.00059}$ | $1.042$ | $1.042^{+0.00029+0.00055}_{-0.00029-0.00057}$ | $1.042$ | $1.042^{+0.00029+0.00055}_{-0.00026-0.00053}$ | $1.042$ | $1.042^{+0.00029+0.00059}_{-0.00029-0.00056}$ | $1.042$ | $1.042^{+0.00026+0.00054}_{-0.00028-0.00052}$
$\ln(10^{10}A_{s})$ | $3.048$ | $3.047^{+0.013+0.028}_{-0.014-0.028}$ | $3.040$ | $3.046^{+0.012+0.029}_{-0.015-0.027}$ | $3.034$ | $3.024^{+0.010+0.023}_{-0.014-0.021}$ | $3.020$ | $3.028^{+0.012+0.025}_{-0.014-0.022}$ | $3.026$ | $3.028^{+0.011+0.024}_{-0.014-0.025}$
$n_{s}$ | $0.9675$ | $0.9673^{+0.0036+0.0076}_{-0.0038-0.0074}$ | $0.9685$ | $0.9665^{+0.0042+0.0079}_{-0.0040-0.0081}$ | $0.9712$ | $0.9719^{+0.0036+0.0072}_{-0.0039-0.0074}$ | $0.9706$ | $0.9728^{+0.0038+0.0078}_{-0.0041-0.0077}$ | $0.9706$ | $0.9720^{+0.0038+0.0074}_{-0.0040-0.0076}$
$\tau_{\mathrm{reio}}$ | $0.05699$ | $0.05662^{+0.0063+0.014}_{-0.0073-0.013}$ | $0.05135$ | $0.05483^{+0.0061+0.014}_{-0.0080-0.014}$ | $0.05358$ | $0.04963^{+0.0041+0.010}_{-0.0074-0.0096}$ | $0.04839$ | $0.05074^{+0.0054+0.012}_{-0.0077-0.011}$ | $0.05102$ | $0.05158^{+0.0059+0.011}_{-0.0072-0.012}$
$w_{\mathrm{DE}}$ ($w_{\mathrm{f}}$) | — | — | $-1.026$ | $-1.009^{+0.036+0.070}_{-0.039-0.072}$ | — | — | $-0.9762$ | $-0.9569^{+0.033+0.065}_{-0.032-0.066}$ | $-0.9677$ | $-0.9944^{+0.041+0.089}_{-0.046-0.084}$
$\mathrm{\Delta}$ ($\beta$) | — | — | $0.0008549$ | $0.0007799^{+0.00021+0.0011}_{-0.00078-0.00078}$ | — | — | $0.0002247$ | $0.0005340^{+0.00013+0.00092}_{-0.00053-0.00053}$ | $0.9999$ | $0.9997^{+0.00047+0.00098}_{-0.00048-0.00090}$
$z_{\mathrm{reio}}$ | $7.921$ | $7.870^{+0.64+1.4}_{-0.72-1.3}$ | $7.376$ | $7.702^{+0.62+1.4}_{-0.78-1.4}$ | $7.502$ | $7.084^{+0.50+1.0}_{-0.69-1.0}$ | $6.951$ | $7.196^{+0.55+1.1}_{-0.78-1.2}$ | $7.231$ | $7.275^{+0.59+1.2}_{-0.72-1.2}$
$\mathrm{\Omega}_{\mathrm{M},0}$ | $0.3038$ | $0.3022^{+0.0057+0.011}_{-0.0050-0.011}$ | $0.2922$ | $0.2953^{+0.0077+0.016}_{-0.0084-0.016}$ | $0.2871$ | $0.2876^{+0.0043+0.0086}_{-0.0044-0.0086}$ | $0.2897$ | $0.2938^{+0.0078+0.017}_{-0.0084-0.017}$ | $0.2994$ | $0.2945^{+0.0076+0.016}_{-0.0074-0.015}$
$H_{0}\;[\mathrm{\frac{km}{s\,Mpc}}]$ | $68.28$ | $68.41^{+0.37+0.90}_{-0.46-0.82}$ | $69.86$ | $69.39^{+0.96+1.8}_{-0.90-1.9}$ | $69.56$ | $69.54^{+0.37+0.73}_{-0.36-0.71}$ | $69.23$ | $68.82^{+0.94+1.9}_{-0.91-1.9}$ | $68.20$ | $68.62^{+0.79+1.7}_{-0.89-1.8}$
$\sigma_{8}$ | $0.8225$ | $0.8217^{+0.0056+0.012}_{-0.0061-0.011}$ | $0.8408$ | $0.8343^{+0.012+0.024}_{-0.012-0.024}$ | $0.8079$ | $0.8044^{+0.0045+0.0096}_{-0.0051-0.0091}$ | $0.7958$ | $0.7978^{+0.0096+0.019}_{-0.0092-0.019}$ | $0.7927$ | $0.7956^{+0.0097+0.019}_{-0.0092-0.019}$

Marginalized $1\sigma$ and $2\sigma$ confidence level contour plots from
datasets (I) and (II) for selected cosmological parameters of Barrow cosmology
are also depicted in figure (4). Furthermore, in order to compare Barrow
cosmology with Tsallis modified gravity (TMG), the results based on TMG model
according to Ref. [76] are also displayed.

Figure 4: The one-dimensional posterior distribution and two-dimensional
posterior contours with 68% and 95% confidence limits for the selected
cosmological parameters of Barrow cosmology from dataset (I) (blue) and
dataset (II) (orange) (left pannel), and also for TMG model from dataset (II)
according to Ref. [76] (right pannel).

Considering data set (I), we can see an enhancement in the Hubble constant in
Barrow cosmology compared to standard model, which indicates a slight
amelioration in $H_{0}$ tension. Moreover, dark energy has a phantom behaviour
according to the best fit and mean value of $w_{\mathrm{DE}}$, which is
influential in alleviating the $H_{0}$ tension. However, a quintessential
character of dark energy is also allowed within the $1\sigma$ confidence
level, where $-1.048<w_{\mathrm{DE}}<-0.9735$. On the other hand, dataset (II)
results show a suppression in the growth of structure in Barrow cosmology with
respect to $\mathrm{\Lambda}$CDM model. Actually, one anticipates higher
values of $\sigma_{8}$ in Barrow cosmology according to numerical results
described in section 3. However, the quintessential behaviour of dark energy
can reduce the structure growth and consequently provide more compatible
results with local galaxy surveys. It should be noted that, considering
datasets (I) and (II), simultaneous alleviation of existing discrepancies
between local observations and CMB measurements is not possible in Barrow
cosmology due to the correlation between $\sigma_{8}$ and $H_{0}$.

Moreover, it is worth to compare Barrow cosmology with TMG model based on
numerical results from dataset (II). Considering the fact that Barrow
cosmology and TMG model are established on absolutely different physical
principles, the growth of structure is slightly reduced in both Barrow and
Tsallis scenarios, related to the quintessential behaviour of dark energy. The
structure growth suppression is more important in TMG model compared to Barrow
cosmology, according to the behaviour of $\beta$ and $\mathrm{\Delta}$.
Specifically, the derived best fit and mean value of $\mathrm{\Delta}$
correspond to higher values of $\sigma_{8}$, in conflict with the
quintessential character of dark energy which results in a decrease in the
growth of structure.

Also, the Akaike information criterion (AIC) defined as [97, 98]

$\mathrm{AIC}=-2\ln{\mathcal{L}_{\mathrm{max}}}+2K,$ (44)

with $\mathcal{L}_{\mathrm{max}}$ the maximum likelihood function and $K$ the
number of free parameters, is helpful to evaluate which model is more
supported by observational data. considering the two different datasets, we
obtain the following results

$\displaystyle\mathrm{dataset(I):}$ $\displaystyle\;\mathrm{AIC_{(\Lambda
CDM)}}=3817.12\;,\;\mathrm{AIC_{(Barrow)}}=3820.20$
$\displaystyle\to\;\mathrm{\Delta AIC}=3.08\;,$
$\displaystyle\mathrm{dataset(II):}$ $\displaystyle\;\mathrm{AIC_{(\Lambda
CDM)}}=3847.12\;,\;\mathrm{AIC_{(Barrow)}}=3850.72$
$\displaystyle\to\;\mathrm{\Delta AIC}=3.60\;.$

Therefor, we can understand that the $\mathrm{\Lambda}$CDM model is better
fitted to both datasets (I) and (II) compared to Barrow cosmology, however,
one can not rule out the Barrow cosmology model.

##  5 Conclusions

It is argued by Barrow [56] that black hole horizon might have an intricate,
fractal structure caused by quantum gravitational corrections to the area law
of entropy. In this respect, Barrow entropy relation (1) is associated to
black hole horizon, with the new exponent $\mathrm{\Delta}$ measures deviation
from standard cosmology. Furthermore, Einstein field equations can be derived
from the first law of thermodynamics at apparent horizon of FLRW universe and
vice versa, inspired by the remarkable analogy between thermodynamics and
gravity. In this direction, it is possible to associate an entropy to the
apparent horizon as the same expression of black hole entropy, just by
replacing the black hole horizon radius by the apparent horizon one.
Accordingly, we derived modified field equations in Barrow cosmology by
applying Barrow entropy in Clausius relation. Then, by performing the MCMC
analysis, we put constraints on the cosmological parameters, specifically, the
Barrow parameter $\mathrm{\Delta}$ which quantifies deviations from
$\mathrm{\Lambda}$CDM, based on two different combinations of datasets:
"Planck + Pantheon + BAO" (dataset (I)) and "Planck + Planck-SZ + CFHTLenS +
Pantheon + BAO + BAORSD" (dataset (II)).

Numerical results from dataset (I) indicate that the $H_{0}$ tension can be
slightly relieved in Barrow cosmology with a phantom behaviour of dark energy.
On the other hand, numerical results based on dataset (II) indicate that the
quintessential nature of dark energy can slightly alleviate the $\sigma_{8}$
tension. However, the correlation between $\sigma_{8}$ and $H_{0}$ shows that
a full reconciliation between local and global observations is not possible.

For the last point, it is interesting to compare the obtained constraints on
Barrow cosmology with the results based on Tsallis cosmology reported in [76].
Considering both Barrow and Tsallis scenarios, there is a slight alleviation
in $\sigma_{8}$ tension with a quintessential behaviour of dark energy
according to dataset (II), which is more significant in TMG model. However,
considering physical principles and the motivation of correction, Barrow
cosmology and TMG model describe two different corrections to the area law of
entropy.

## acknowledgements

We thank Shiraz University Research Council.

## Data Availability Statement

This manuscript has no associated data or the data will not be deposited.

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