# Gravitational wave probes of Barrow cosmology with LISA standard sirens

Mahnaz Asghari  Alireza Allahyari  and David F. Mota

###### Abstract

We study the Barrow cosmological model, which proposes that quantum gravity
effects create a complex, fractal structure for the universe’s apparent
horizon. We leverage the thermodynamics-gravity conjecture. By applying the
Clausius relation to the apparent horizon of the Friedmann-Lemaître-Robertson-
Walker universe within this framework, we derive modified field equations
where the Barrow entropy is linked to the horizon. We assess the Barrow
cosmology against current observations – cosmic microwave background ,
supernovae , and baryon acoustic oscillations data – and include projections
for future Laser Interferometer Space Antenna (LISA) standard sirens (SS). Our
numerical results suggest a modest improvement in the Hubble tension for
Barrow cosmology with phantom dark energy behavior, compared to the standard
cosmological model. Furthermore, incorporating simulated LISA SS data
alongside existing observational constraints tightens the limitations on
cosmological parameters, particularly the deformation exponent.

##  1 Introduction

It is known that there is a remarkable connection between thermodynamics and
gravity [1, 2, 3], where accordingly, the Einstein field equations can be
considered as the equation of state for the spacetime [4]. From a cosmological
perspective, the thermodynamics-gravity conjecture yields the Friedmann
equations in the form of the first law of thermodynamics and vice versa [5, 6,
7, 8, 9, 10, 11, 12]. So, contemplating the spacetime as a thermodynamic
system, we may write the field equations of Friedmann-Lemaître-Robertson-
Walker (FLRW) universe by making use of the Clausius relation $\delta
Q=T\delta S$ at the apparent horizon of universe. Correspondingly, applying
Bekenstein-Hawking area law entropy [13, 3] in Clausius relation yields field
equations in concordance $\Lambda$CDM model, while modified theories of
gravity are reproduced by making corrections to the standard area law [14].

Regarding modifications on area law entropy relation, there are logarithmic as
well as power-law corrections to Bekenstein-Hawking entropy due to quantum
effects, which have been extensively explored in the literature [15, 16, 17,
18, 19, 20, 21, 22, 23, 24, 25]. Also, one can consider non-extensive Tsallis
entropy [26] proposed by the fact that Boltzmann-Gibbs additive entropy should
be generalized to non-additive one in large scale gravitational systems [27,
28, 29, 30, 31, 32, 33]. Related investigations on Tsallis entropy are
developed in [34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45].

Furthermore, J. D. Barrow in 2020, introduced a fractal structure for the
black hole horizon geometry due to quantum-gravitational effects [46]. The
Barrow formulation of black hole entropy is given by [46]

$S=\Big{(}\frac{A}{A_{0}}\Big{)}^{1+\Delta/2}\;,$ (1.1)

where $A$ stands for the black hole horizon area, $A_{0}$ indicates the Planck
area, and the deformation exponent $\Delta$ ($0\leq\Delta\leq 1$ [46, 47, 48])
specifies deviations from the area law formula. Choosing $\Delta=0$
corresponds to the standard Bekenstein-Hawking entropy, and consequently shows
the simplest horizon construction. On the other hand, $\Delta=1$ describes the
most intricate and fractal structure. For some theoretical explorations on
Barrow entropy in the cosmological context, see [49, 50, 51, 52, 53, 54, 55,
56, 57, 58, 59, 60, 61, 62, 63]. Moreover, it is important to investigate
Barrow cosmology from an observational viewpoint [64, 65, 66, 67]. In this
direction, one powerful probe to inspect alternative cosmological theories to
standard $\Lambda$CDM model is the gravitational wave (GW) astronomy.

The direct detection of GW signal from the binary black hole coalescence
GW150914 by the LIGO111Laser Interferometer Gravitational-Wave Observatory
collaboration in 2015 [68], provided significant advancements in investigating
the gravitational physics. Thereafter in 2017, the LIGO-Virgo detector network
identified a GW signal emitted by the coalescing binary neutron star GW170817
[69], which was associated with a short $\gamma$-ray burst GRB170817A [70],
observed by the Fermi Gamma-Ray Burst Monitor [71, 72, 73, 74] and INTEGRAL
[75, 76]. The intriguing coincident detection of the GW event and its
electromagnetic (EM) counterpart inaugurated the era of multi-messenger
astrophysics [77].

Considering cosmological applications, one can regard coalescing binary
systems as standard sirens (SS) (i.e. the gravitational analogue of standard
candles) which are utilized to estimate the Hubble constant [78, 79]. Actually
it is possible to determine the luminosity distance to the compact binary
coalescence by measuring its generated GW signal, and then the SS measurement
of $d_{L}$ results in estimating cosmological parameters [80, 81, 82]. In
2017, the first SS determination of $H_{0}$, which was independent of the
cosmic distance ladder, reported the constraint of $H_{0}=70.0^{+12.0}_{-8.0}$
$\mathrm{km\,s^{-1}\,Mpc^{-1}}$ [83]. Afterwards, another measurement of
$H_{0}$ based on multiple GW observations resulted in
$H_{0}=68.7^{+17.0}_{-7.8}$ $\mathrm{km\,s^{-1}\,Mpc^{-1}}$, in which
according to a statistical method, informations from GW170817 along with its
EM counterpart was combined with GW signals from binary black hole events with
no expected EM counterparts [84]. Regarding unsatisfactory accuracy on Hubble
constant from SS measurements, it is important to exploit next generation GW
observatories such as the ground-based detectors Einstein Telescope [85, 86]
and Cosmic Explorer [87], as well as the space-based detector LISA222Laser
Interferometer Space Antenna [88, 89] to enhance constraints on cosmological
parameters.

The space-borne interferometer LISA, expected to be launched in 2034, has the
potentiality to detect GWs in a band from below $10^{-4}$ Hz to above
$10^{-1}$ Hz. The low frequency range of the LISA mission provides the
opportunity to probe SS signals at much higher redshifts as well as exploring
more massive black hole mergers [88]. There are several studies on applying SS
in investigating the observational verification of cosmological models. For
some recent related works see e.g. [90, 91, 92, 93, 94, 95, 96, 97, 98, 99,
100, 101, 102, 103].

The present study is devoted to exploiting the LISA SS to put constraints on
parameters of Barrow cosmological model. In particular, we apply GW SS
together with cosmic microwave background (CMB), supernovae (SN), and baryon
acoustic oscillations (BAO) measurements to examine the effectiveness of
Barrow cosmology in resolving the Hubble tension [104, 105]. The methodology
of our investigation is similar to the paper [67], in which we detected a
slight reduction in $H_{0}$ tension.

This paper is organized as follows. Section 2 considers modified field
equations based on Barrow corrections to the area law entropy, as well as
introducing modifications to GW propagation in Barrow model. Section 3 is
dedicated to observational probes employed in our study, which consists of
current observations along with future LISA SS data. We also present the
applied method for generating the GW mock catalog in section 3. Numerical
results as well as observational constraints on Barrow model are reported in
Section 4. We conclude in section 5. Throughout the paper we consider the
units where $k_{\mathrm{B}}=c=\hbar=1$.

##  2 Barrow cosmology

This section is dedicated to describing modified field equations according to
Barrow entropy corrections, as well as introducing modifications to GW
propagation in Barrow cosmology.

###  2.1 Modified field equations

In pursuance of extracting field equations from Barrow entropy, one should
apply the Clausius relation $\delta Q=T\delta S$ at the apparent horizon of
FLRW universe, which is considered as a whole thermodynamic system. For this
purpose, we regard a spatially flat universe, described by the FLRW metric

$\mathrm{d}s^{2}=a^{2}(\tau)\Big{(}-\mathrm{d}\tau^{2}+\mathrm{d}\vec{x}^{2}\Big{)}\;,$
(2.1)

in background level. Then, it is also important to contemplate the area $A$ in
Barrow entropy relation (1.1) as the apparent horizon of the universe area
$A=4\pi\tilde{r}^{2}_{\mathrm{A}}$, with the apparent horizon radius
$\tilde{r}_{\mathrm{A}}=\Big{(}H^{2}+\frac{K}{a^{2}}\Big{)}^{-1/2}$ [106]
(where $H$ is the Hubble parameter, and $K=-1,0,1$ is the curvature constant
describing open, flat and closed universe, respectively).

Taking the differential of Barrow entropy relation (1.1), results in

$\delta
S=\Big{(}1+\frac{\Delta}{2}\Big{)}A_{0}^{-1-\Delta/2}A^{\Delta/2}\delta A\;.$
(2.2)

And additionally, following the Refs. [4, 14], $\delta Q$ is defined as

$\delta Q=-\kappa\int_{\mathcal{H}}\lambda
T_{\mu\nu}k^{\mu}k^{\nu}\mathrm{d}\lambda\mathrm{d}A\;,$ (2.3)

in which $\mathcal{H}$ stands for the apparent horizon, $\delta
A=\int_{\mathcal{H}}\theta\mathrm{d}\lambda\mathrm{d}A$, and $\theta=-\lambda
R_{\mu\nu}k^{\mu}k^{\nu}$. Also the content of universe including radiation
(R), matter (M) (dark matter (DM) and baryons (B)) and dark energy (DE), is
considered as a perfect fluid with the energy-momentum tensor
$T_{\mu\nu}=\big{(}\rho+p\big{)}u_{\mu}u_{\nu}+g_{\mu\nu}p$. Furthermore, the
temperature corresponds to the apparent horizon with the surface gravity
$\kappa$ is

$T=\frac{\kappa}{2\pi}\;.$ (2.4)

Replacing equations (2.2), (2.3) and (2.4) in Clausius relation, yields

$\int_{\mathcal{H}}(-\lambda)\Bigg{(}-2\pi
T_{\mu\nu}+\Big{(}1+\frac{\Delta}{2}\Big{)}A_{0}^{-1-\Delta/2}R_{\mu\nu}A^{\Delta/2}\Bigg{)}k^{\mu}k^{\nu}\mathrm{d}\lambda\mathrm{d}A=0\;,$
(2.5)

where by introducing the scalar $f$, for all null vectors $k^{\mu}$ we obtain

$-2\pi
T_{\mu\nu}+\Big{(}1+\frac{\Delta}{2}\Big{)}A_{0}^{-1-\Delta/2}R_{\mu\nu}A^{\Delta/2}=fg_{\mu\nu}\;.$
(2.6)

Then, imposing the energy-momentum conservation results in

$\Big{(}1+\frac{\Delta}{2}\Big{)}A_{0}^{-1-\Delta/2}\Bigg{(}\frac{1}{2}\big{(}\partial_{\nu}R\big{)}A^{\Delta/2}+R_{\mu\nu}\partial^{\mu}A^{\Delta/2}\Bigg{)}=\partial_{\nu}f\,.$
(2.7)

So, since the LHS of (2.7) is not the gradient of a scalar, one can conclude
that on the account of non-equilibrium thermodynamics, the Clausius relation
is not satisfied. Accordingly, we apply the entropy balance relation [14]

$\delta S=\frac{\delta Q}{T}+\mathrm{d}_{i}S\;,$ (2.8)

in which $\mathrm{d}_{i}S$ is the produced entropy owing to irreversible
processes inside the system [107]. Then, contemplating the energy-momentum
conservation, $\mathrm{d}_{i}S$ can be written as

$\mathrm{d}_{i}S=\Big{(}1+\frac{\Delta}{2}\Big{)}A_{0}^{-1-\Delta/2}\int_{\mathcal{H}}(-\lambda)\nabla_{\mu}\nabla_{\nu}A^{\Delta/2}k^{\mu}k^{\nu}\mathrm{d}\lambda\mathrm{d}A\;.$
(2.9)

Thereupon, by substituting $\mathrm{d}_{i}S$ in the entropy balance relation
(2.8) and repeating the above calculations (refer to the Ref. [67] for more
details), finally we find

$\Big{(}1+\frac{\Delta}{2}\Big{)}A_{0}^{-1-\Delta/2}\Bigg{(}\frac{1}{2}\big{(}\partial_{\nu}R\big{)}A^{\Delta/2}-\partial_{\nu}\Box
A^{\Delta/2}\Bigg{)}=\partial_{\nu}f\;.$ (2.10)

Now, considering the scalar $\mathcal{L}$ as $\mathcal{L}=RA^{\Delta/2}$,
yields

$\displaystyle\frac{\partial\mathcal{L}}{\partial R}=A^{\Delta/2}\;,$
$\displaystyle\partial_{\nu}R=\frac{\partial R}{\partial
x^{\nu}}=\frac{\partial
R}{\partial\mathcal{L}}\frac{\partial\mathcal{L}}{\partial
x^{\nu}}=A^{-\Delta/2}\partial_{\nu}\mathcal{L}\;,$
$\displaystyle\to\big{(}\partial_{\nu}R\big{)}A^{\Delta/2}=\partial_{\nu}\mathcal{L}\;,$
(2.11)

and accordingly we find the scalar $f$ as

$f=\Big{(}1+\frac{\Delta}{2}\Big{)}A_{0}^{-1-\Delta/2}\Big{(}\frac{1}{2}RA^{\Delta/2}-\Box
A^{\Delta/2}\Big{)}\;.$ (2.12)

Thereupon, modified field equations based on Barrow entropy take the form

$R_{\mu\nu}A^{\Delta/2}-\nabla_{\mu}\nabla_{\nu}A^{\Delta/2}-\frac{1}{2}RA^{\Delta/2}g_{\mu\nu}+\Box
A^{\Delta/2}g_{\mu\nu}=\frac{4\pi}{2+\Delta}A_{0}^{1+\Delta/2}T_{\mu\nu}\;.$
(2.13)

Consequently, by applying the Clausius relation at the apparent horizon of
FLRW universe along with regarding quantum-gravitational effects on the
apparent horizon, modified gravitational field equations in Barrow cosmology
have been extracted. It can be easily seen that choosing $\Delta=0$ reduces to
the standard field equations in general relativity. Contemplating
$A^{\Delta/2}=(4\pi)^{\Delta/2}H^{-\Delta}$ for a spatially flat universe,
modified field equations in background level can be derived as

$H^{2-\Delta}-\Delta H^{-\Delta}H^{\prime}\frac{1}{a}=X^{\Delta}\,\frac{8\pi
G}{3}\sum_{i}\bar{\rho}_{i}\;,$ (2.14)
$H^{-\Delta}\Bigg{\\{}\big{(}\Delta-2\big{)}H^{\prime}\frac{1}{a}-3H^{2}+\Delta\bigg{(}\frac{H^{\prime\prime}}{H}-\big{(}1+\Delta\big{)}\Big{(}\frac{H^{\prime}}{H}\Big{)}^{2}\bigg{)}\frac{1}{a^{2}}\Bigg{\\}}=X^{\Delta}\,8\pi
G\sum_{i}\bar{p}_{i}\;,$ (2.15)

where a prime indicates a deviation with respect to the conformal time, and
also $A_{0}$ is considered as

$A_{0}=\big{(}4\pi\big{)}^{\Delta/(2+\Delta)}\Big{(}2\big{(}2+\Delta\big{)}G\Big{)}^{2/(2+\Delta)}X^{2\Delta/(2+\Delta)}\;,$

with the constant $X$ of the dimension of length. Considering equations (2.14)
and (2.15), the modified Friedmann equation in regard to Barrow entropy
corrections takes the form

$H^{2-\Delta}=\frac{1}{1+2\Delta}X^{\Delta}\,\frac{8\pi
G}{3}\sum_{i}\bar{\rho}_{i}\;.$ (2.16)

Furthermore, modified Friedmann equation in term of total density parameter
$\Omega_{\mathrm{tot}}=\bar{\rho}_{\mathrm{tot}}/\rho_{\mathrm{cr}}$ (with
$\rho_{\mathrm{cr}}={3H^{2}}/{(8\pi G)}$ and
$\bar{\rho}_{\mathrm{tot}}=\sum_{i}\bar{\rho}_{i}$) becomes

$\Omega_{\mathrm{tot}}=\big{(}1+2\Delta\big{)}X^{-\Delta}H^{-\Delta}\;.$
(2.17)

Then, as expected, Friedmann equation in standard cosmology will be recovered
in case of $\Delta=0$.

It is also interesting to notice that the cosmic accelerated expansion in
Barrow cosmology is achieved when $w_{\mathrm{tot}}<-(1+\Delta)/3$. So,
contemplating the allowed range of $\Delta$ ($0\leq\Delta\leq 1$), the most
intricate structure of apparent horizon imposes the condition
$w_{\mathrm{tot}}<-2/3$, compared to the standard gravity in which we have
$w_{\mathrm{tot}}<-1/3$. It implies that Barrow cosmology late time
acceleration will occur for more negative values of the equation of state
parameter.

Considering cosmological perturbations in FLRW universe, for the scalar mode
of metric perturbations we have

$\mathrm{d}s^{2}=a^{2}(\tau)\Big{(}-\mathrm{d}\tau^{2}+\big{(}\delta_{ij}+h_{ij}\big{)}\mathrm{d}x^{i}\mathrm{d}x^{j}\Big{)}\;,$
(2.18)

where
$h_{ij}(\vec{x},\tau)=\int\mathrm{d}^{3}k\,e^{i\vec{k}.\vec{x}}\bigg{(}\hat{k}_{i}\hat{k}_{j}h(\vec{k},\tau)+\Big{(}\hat{k}_{i}\hat{k}_{j}-\frac{1}{3}\delta_{ij}\Big{)}6\eta(\vec{k},\tau)\bigg{)}$,
with scalar perturbations $h$ and $\eta$, and $\vec{k}=k\hat{k}$ [108]. So
modified field equations (2.13) to linear order of perturbations take the form

$\Big{(}\frac{a^{\prime}}{a^{2}}\Big{)}^{-\Delta}\Bigg{\\{}\frac{a^{\prime}}{a}h^{\prime}-2k^{2}\eta-\frac{1}{2}\Delta
h^{\prime}\bigg{(}\frac{a^{\prime\prime}}{a^{\prime}}-2\frac{a^{\prime}}{a}\bigg{)}\Bigg{\\}}=X^{\Delta}\,a^{2}\,8\pi
G\sum_{i}\delta\rho_{i}\;,$ (2.19)
$\Big{(}\frac{a^{\prime}}{a^{2}}\Big{)}^{-\Delta}\,k^{2}\eta^{\prime}=X^{\Delta}\,a^{2}\,4\pi
G\sum_{i}\big{(}\bar{\rho}_{i}+\bar{p}_{i}\big{)}\theta_{i}\;,$ (2.20)
$\displaystyle\Big{(}\frac{a^{\prime}}{a^{2}}\Big{)}^{-\Delta}\Bigg{\\{}\frac{1}{2}h^{\prime\prime}+3\eta^{\prime\prime}+\frac{a^{\prime}}{a}h^{\prime}+6\frac{a^{\prime}}{a}\eta^{\prime}-k^{2}\eta-\frac{1}{2}\Delta\bigg{(}\frac{a^{\prime\prime}}{a^{\prime}}-2\frac{a^{\prime}}{a}\bigg{)}\bigg{(}h^{\prime}+6\eta^{\prime}\bigg{)}\Bigg{\\}}=0\;,$
(2.21)
$\Big{(}\frac{a^{\prime}}{a^{2}}\Big{)}^{-\Delta}\Bigg{\\{}-2\frac{a^{\prime}}{a}h^{\prime}-h^{\prime\prime}+2k^{2}\eta+\Delta\bigg{(}\frac{a^{\prime\prime}}{a^{\prime}}-2\frac{a^{\prime}}{a}\bigg{)}h^{\prime}\Bigg{\\}}=X^{\Delta}\,a^{2}\,24\pi
G\sum_{i}\delta p_{i}\;.$ (2.22)

Moreover, the energy-momentum conservation equations are irrelevant to Barrow
entropy corrections, so continuity and Euler equations for matter and dark
energy components are given by

$\displaystyle\delta^{\prime}_{\mathrm{M}}=-\theta_{\mathrm{M}}-\frac{1}{2}h^{\prime}\;,$
(2.23)
$\displaystyle\theta^{\prime}_{\mathrm{M}}=-\frac{a^{\prime}}{a}\theta_{\mathrm{M}}\;,$
(2.24) $\displaystyle\delta^{\prime}_{\mathrm{DE}}$
$\displaystyle=-3\frac{a^{\prime}}{a}\big{(}c^{2}_{s,\mathrm{DE}}-w_{\mathrm{DE}}\big{)}\delta_{\mathrm{DE}}-\frac{1}{2}h^{\prime}\big{(}1+w_{\mathrm{DE}}\big{)}$
$\displaystyle-\big{(}1+w_{\mathrm{DE}}\big{)}\bigg{(}1+9\Big{(}\frac{a^{\prime}}{a}\Big{)}^{2}\big{(}c^{2}_{s,\mathrm{DE}}-c^{2}_{a,\mathrm{DE}}\big{)}\frac{1}{k^{2}}\bigg{)}\theta_{\mathrm{DE}}\;,$
(2.25)
$\displaystyle\theta^{\prime}_{\mathrm{DE}}=\frac{a^{\prime}}{a}\big{(}3c^{2}_{s,\mathrm{DE}}-1\big{)}\theta_{\mathrm{DE}}+\frac{k^{2}c^{2}_{s,\mathrm{DE}}}{1+w_{\mathrm{DE}}}\delta_{\mathrm{DE}}\;.$
(2.26)

###  2.2 Modified gravitational wave propagation

In this part, we turn our attention to the modified propagation of GWs in
accordance with Barrow corrections to the area law entropy. To this end, we
contemplate the tensorial part of perturbed field equations (2.13) in absence
of anisotropic stress, which reads

$h^{\prime\prime}_{(+,\times)}+\Big{(}2\frac{a^{\prime}}{a}-a^{2}\Delta\big{(}\frac{a^{\prime\prime}}{a^{\prime}}-2\frac{a^{\prime}}{a}\big{)}\Big{)}h^{\prime}_{(+,\times)}+k^{2}h_{(+,\times)}=0\;,$
(2.27)

with the two plus and cross polarizations. From equation (2.27) it can be
understood that the friction term is modified, while the propagation speed of
GWs remains unchanged which is compatible with observations [70]. Also it is
obvious that for $\Delta=0$, the free propagation of tensor perturbations in
general relativity will be recovered. In order to solve equation (2.27), it is
convenient to consider the function $\chi_{(+,\times)}(\tau,k)$ defined as

$\chi_{(+,\times)}(\tau,k)=\tilde{a}(\tau)h_{(+,\times)}(\tau,k)\;,$ (2.28)

where in our modified cosmological model, $\tilde{a}(\tau)$ is given by

$\frac{\tilde{a}^{\prime}}{\tilde{a}}=\frac{a^{\prime}}{a}-a^{2}\Delta\bigg{(}\frac{1}{2}\frac{a^{\prime\prime}}{a^{\prime}}-\frac{a^{\prime}}{a}\bigg{)}=Ha-\frac{1}{2}a^{2}\Delta\frac{H^{\prime}}{H}\;.$
(2.29)

Thus, it can be inferred that in modified theories of gravity, the GW
amplitude decreases as $1/{\tilde{a}}$ during propagation through cosmological
distances. Hence, it is suggested to consider the GW luminosity distance
$d_{L}^{\mathrm{gw}}(z)$ in modified gravity which is defined as [109, 90]

$d_{L}^{\mathrm{gw}}(z)=\frac{1}{(1+z)\tilde{a}(z)}d_{L}^{\mathrm{em}}(z)\;,$
(2.30)

where $d_{L}^{\mathrm{em}}(z)=(1+z)\int_{0}^{z}{\frac{\mathrm{d}z}{H(z)}}$ is
the standard luminosity distance. Then, from equation (2.29) in Barrow
cosmology we obtain

$d_{L}^{\mathrm{gw}}(z)=d_{L}^{\mathrm{em}}(z)\exp{\bigg{(}\frac{1}{2}\Delta\int_{0}^{z}{\mathrm{d}z\,\frac{1}{(1+z)^{2}H(z)}\frac{\mathrm{d}H(z)}{\mathrm{d}z}}\bigg{)}}\;,$
(2.31)

with the Hubble parameter $H(z)$ defined in equation (2.16).

In the next section, we introduce the observational tests employed in this
investigation to put constraints on the parameters of Barrow cosmological
model.

##  3 Observational probes

In order to confront Barrow model with observations, we apply the following
current and future data in our numerical analysis.

  * •

CMB: We contemplate CMB measurements form Planck 2018 data release, which
consists of high-$l$ TT,TE,EE, low-$l$ EE, low-$l$ TT, and lensing data [110].

  * •

SN: Considering the type Ia supernovae (SN Ia) distance measurements, we take
into account the Pantheon sample containing 1048 data points in the range of
$0.01<z<2.3$ [111].

  * •

BAO: We make use of the 6dF Galaxy Survey (6dFGS) BAO detection which provides
constraining the distance-redshift relation at $z_{\mathrm{eff}}=0.106$ [112],
and also we consider the SDSS333Sloan Digital Sky Survey-III BOSS444Baryon
Oscillation Spectroscopic Survey data which include LOWZ
($z_{\mathrm{eff}}=0.32$) and CMASS ($z_{\mathrm{eff}}=0.57$) galaxy samples
[113].

  * •

LISA standard sirens: The space-based detector LISA has the potential to probe
signals from SS of massive black hole binaries (MBHBs) in the range $10^{3}$
to $10^{7}$ solar masses [88, 114]. MBHBs are powerful sources of GWs in the
universe, as they are supposed to merge at the center of galaxies, which are
gas-rich nuclear environments, and so these events are expected to produce a
detectable EM counterpart. Moreover, detecting MBHBs in the center of galaxies
is so useful to improve our understanding of galaxy evolution and cosmic
structure formation.

In order to generate our catalog of SS events, we apply the methodology
outlined in [100], which considers the redshift distribution of SS according
to the explanations in Ref. [115]. Based on competing scenarios for the
initial conditions for the massive black hole formation and evolution at high
redshifts, there are three distinct populations of MBHBs, namely Pop III,
Delay, and No Delay population models [116]. Pop III model is a light-seed
scenario in which massive black holes are considered to form from the remnants
of population III stars. On the other hand, the heavy-seed scenario assumes
that massive black holes evolve from the collapse of protogalactic disks,
leading to Delay (where a finite time delay between the merger of host
galaxies and black hole coalescence is assumed), and No Delay (with
considering no delays between the galaxy and black hole mergers) models [116].

It is also important to compute the realistic $1\sigma$ luminosity distance
error to MBHBs, according to the LISA sensitivity, which can be estimated as
[117, 100]

$\sigma_{\mathrm{LISA}}^{2}=\sigma_{\mathrm{delens}}^{2}+\sigma_{\mathrm{v}}^{2}+\sigma_{\mathrm{inst}}^{2}+\bigg{(}\frac{\mathrm{d}}{\mathrm{d}z}(d_{L})\sigma_{\mathrm{photo}}\bigg{)}^{2}\;,$
(3.1)

which consists of error contributions from weak lensing, peculiar velocity,
LISA instrument, and redshift measurements. The weak lensing contribution is
given by

$\sigma_{\mathrm{delens}}(z)=F_{\mathrm{delens}}(z)\sigma_{\mathrm{lens}}(z)\;,$
(3.2)

where

$\displaystyle
F_{\mathrm{delens}}(z)=1-\frac{0.3}{\pi/2}\arctan\bigg{(}\frac{z}{0.073}\bigg{)}\;,$
(3.3)
$\displaystyle\frac{\sigma_{\mathrm{lens}}(z)}{d_{L}(z)}=0.066\bigg{(}\frac{1-(1+z)^{-0.25}}{0.25}\bigg{)}^{1.8}\;.$
(3.4)

The peculiar velocity uncertainty contribution takes the form

$\frac{\sigma_{\mathrm{v}}(z)}{d_{L}(z)}=\bigg{(}1+\frac{c(1+z)^{2}}{H(z)d_{L}(z)}\bigg{)}\frac{500\,\mathrm{km/s}}{c}\;.$
(3.5)

The LISA instrumental error is

$\frac{\sigma_{\mathrm{inst}}(z)}{d_{L}(z)}=0.05\bigg{(}\frac{d_{L}(z)}{36.6\,\mathrm{Gpc}}\bigg{)}\;.$
(3.6)

And finally, for redshift measurement error we have

$\sigma_{\mathrm{photo}}(z)=0.03(1+z)\;,\quad\text{if}\;\;z>2\;.$ (3.7)

Throughout this study, we generate an SS mock catalog for each MBHB
population, i.e. Pop III, Delay, and No Delay models, considering a ten-year
LISA mission. The fiducial model is assumed to be the Barrow cosmology
described in section 2, where for the cosmological parameters we have
considered the obtained best fit values based on the ”CMB+SN+BAO” dataset
displayed in table 1. Accordingly, figure 1 depicts the evolution of GW
luminosity distance with redshift for the three LISA mock catalogs.

Figure 1: Mock LISA SS for ten years of operation from three MBHB populations.

Next section is devoted to numerical results as well as observational
constraints on Barrow cosmology based on the introduced data sets in this
part.

##  4 Results

In the interest of a numerical investigation on Barrow cosmology, we make use
of our modified version of the publicly available Boltzmann code
CLASS555Cosmic Linear Anisotropy Solving System [118], based on field
equations of Barrow cosmological model. To this end, the value of cosmological
parameters are set to the Planck 2018 data release [110], while considering
$w_{\mathrm{DE}}=-0.98$ and $c_{s,\mathrm{DE}}^{2}=1$ for the dark energy
component.

The CMB temperature power spectra diagrams in Barrow cosmology are illustrated
in figure 2. Upper panels represent the power spectra for different values of
$\Delta$, where $X$ fixed to be $1$, while lower panels depict the same
diagrams for different values of $X$, with $\Delta$ considered to be $0.015$.
(Notice that the case $X=1$ recovers the results reported in Ref. [67].)

Figure 2: Upper panels: The CMB temperature power spectra (left) and their
relative ratio with respect to standard cosmological model (right) for
different values of $\Delta$, regarding $X=1$. Lower panels: The analogous
diagrams for different values of $X$, where $\Delta=0.015$.

Regarding the importance of the evolution of Hubble parameter in Barrow
cosmology, figure 3 displays the expansion history of the universe as
described in the modified Friedmann equation (2.16). Considering figure 3, it
is evident that there is an enhancement in the current value of Hubble
parameter for non zero values of $\Delta$, which reveals more consistency with
low-redshift estimations of Hubble constant.

Figure 3: Evolution of Hubble parameter with regard to conformal time in
Barrow cosmology compared to $\Lambda$CDM model, for different values of
$\Delta$, while $X=1$ (left panel), and analogous diagrams for different
values of $X$, considering $\Delta=0.015$ (right panel).

Accordingly, for more precise investigations on the capability of Barrow model
in reducing the Hubble tension, we fit the model to current and future
observational data by performing an MCMC666Markov Chain Monte Carlo method
through the Monte Python code [119, 120]. In order to use the mock LISA SS
data in our MCMC analysis, we add three created likelihoods based on Pop III,
Delay, and No Delay population models to the Monte Python code. The
cosmological parameter set contemplated in MCMC approach includes {
$100\,\Omega_{\mathrm{B},0}h^{2}$, $\Omega_{\mathrm{DM},0}h^{2}$,
$100\,\theta_{s}$, $\ln(10^{10}A_{s})$, $n_{s}$, $\tau_{\mathrm{reio}}$,
$w_{\mathrm{DE}}$, $\Delta$, $X$ }, which are the six parameters of standard
model of cosmology, and also the dark energy equation of state
$w_{\mathrm{DE}}$, the deformation exponent $\Delta$, and the constant $X$.
There are also some derived parameters in MCMC analysis which consist of the
reionization redshift $z_{\mathrm{reio}}$, the matter density parameter
$\Omega_{\mathrm{M},0}$, the Hubble constant $H_{0}$, and the structure growth
parameter $\sigma_{8}$. Moreover, preliminary numerical investigations suggest
the prior ranges [$0$, $0.015$] and [$0.1$, $100$] for $\Delta$ and $X$,
respectively, while we consider no prior range for $w_{\mathrm{DE}}$.

In pursuance of deriving constraints on cosmological parameters of Barrow
model, we first apply the ”CMB+SN+BAO” dataset, where the results for Barrow
cosmology compared to the $\Lambda$CDM model are reported in table 1.

Table 1: Best fit values along with $1\sigma$ and $2\sigma$ confidence level
intervals from the ”CMB+SN+BAO” dataset for $\Lambda$CDM and Barrow cosmology.

|  $\Lambda$CDM | Barrow cosmology
---|---|---
parameter | best fit | 68% & 95% limits | best fit | 68% & 95% limits
$100\,\Omega_{\mathrm{B},0}h^{2}$ | $2.246$ | $2.245^{+0.013+0.027}_{-0.014-0.028}$ | $2.236$ | $2.240^{+0.014+0.030}_{-0.015-0.028}$
$\Omega_{\mathrm{DM},0}h^{2}$ | $0.1191$ | $0.1189^{+0.00096+0.0018}_{-0.00086-0.0020}$ | $0.1196$ | $0.1198^{+0.0011+0.0023}_{-0.0011-0.0023}$
$100\,\theta_{s}$ | $1.042$ | $1.042^{+0.00029+0.00057}_{-0.00030-0.00059}$ | $1.042$ | $1.042^{+0.00029+0.00059}_{-0.00030-0.00060}$
$\ln(10^{10}A_{s})$ | $3.048$ | $3.047^{+0.013+0.028}_{-0.014-0.028}$ | $3.041$ | $3.047^{+0.013+0.027}_{-0.015-0.027}$
$n_{s}$ | $0.9675$ | $0.9673^{+0.0036+0.0076}_{-0.0038-0.0074}$ | $0.9682$ | $0.9670^{+0.0042+0.0079}_{-0.0039-0.0084}$
$\tau_{\mathrm{reio}}$ | $0.05699$ | $0.05662^{+0.0063+0.014}_{-0.0073-0.013}$ | $0.05364$ | $0.05519^{+0.0063+0.013}_{-0.0076-0.014}$
$w_{\mathrm{DE}}$ | — | — | $-1.042$ | $-1.012^{+0.036+0.074}_{-0.036-0.071}$
$\Delta$ | — | — | $0.0002865$ | $0.0007894^{+0.00020+0.0012}_{-0.00079-0.00079}$
$X$ [Mpc] | — | — | $72.61$ | unconstrained
$z_{\mathrm{reio}}$ | $7.921$ | $7.870^{+0.64+1.4}_{-0.72-1.3}$ | $7.606$ | $7.745^{+0.67+1.3}_{-0.73-1.4}$
$\Omega_{\mathrm{M},0}$ | $0.3038$ | $0.3022^{+0.0057+0.011}_{-0.0050-0.011}$ | $0.2910$ | $0.2939^{+0.0078+0.016}_{-0.0081-0.016}$
$H_{0}\;[\mathrm{km\,s^{-1}\,Mpc^{-1}}]$ | $68.28$ | $68.41^{+0.37+0.90}_{-0.46-0.82}$ | $69.84$ | $69.58^{+0.91+1.9}_{-0.94-1.8}$
$\sigma_{8}$ | $0.8225$ | $0.8217^{+0.0056+0.012}_{-0.0061-0.011}$ | $0.8372$ | $0.8359^{+0.012+0.022}_{-0.012-0.023}$

Also, the $1\sigma$ and $2\sigma$ confidence contours based on ”CMB+SN+BAO”
dataset are demonstrated in figure 4.

Figure 4: One-dimensional marginalized posterior and two-dimensional
marginalized contours with $68\%$ and $95\%$ confidence levels from
”CMB+SN+BAO” dataset, for some selected parameters of Barrow model (blue)
compared to $\Lambda$CDM (purple).

Numerical results exhibit an increase in the value of Hubble constant in
Barrow model compared to $\Lambda$CDM, which is positively influenced by the
phantom nature of dark energy. Thus, we observe a slight alleviation in Hubble
tension for Barrow cosmology with a phantom dark energy, while the tension can
not be completely released. On the other hand, the constant $X$ is degenerate
with other model parameters, and remains unconstrained for this dataset.

Considering the derived best fit values for cosmological parameters of Barrow
model in table 1, it is possible to generate three LISA SS mock catalogs,
namely, Pop III, Delay, and No Delay, based on the fiducial Barrow cosmology,
as displayed in figure 1. Thereupon, we are able to confront Barrow model with
SS data, by employing three different datasets combinations: ”CMB+SN+BAO+Pop
III”, ”CMB+SN+BAO+Delay”, and ”CMB+SN+BAO+No Delay”. Correspondingly,
observational constraints on cosmological parameters of Barrow model are
summarized in table 2.

Table 2: Best fit values along with $1\sigma$ and $2\sigma$ confidence level
intervals from ”CMB+SN+BAO+Pop III”, ”CMB+SN+BAO+Delay”, and ”CMB+SN+BAO+No
Delay” datasets, regarding Barrow cosmology.

| CMB+SN+BAO+Pop III | CMB+SN+BAO+Delay | CMB+SN+BAO+No Delay
---|---|---|---
parameter | best fit | 68% & 95% limits | best fit | 68% & 95% limits | best fit | 68% & 95% limits
$100\,\Omega_{\mathrm{B},0}h^{2}$ | $2.238$ | $2.241^{+0.015+0.030}_{-0.015-0.032}$ | $2.230$ | $2.238^{+0.013+0.028}_{-0.014-0.027}$ | $2.247$ | $2.241^{+0.014+0.026}_{-0.014-0.027}$
$\Omega_{\mathrm{DM},0}h^{2}$ | $0.1200$ | $0.1197^{+0.0010+0.0023}_{-0.0012-0.0021}$ | $0.1201$ | $0.1198^{+0.0011+0.0023}_{-0.0011-0.0022}$ | $0.1194$ | $0.1197^{+0.00096+0.0021}_{-0.0012-0.0021}$
$100\,\theta_{s}$ | $1.042$ | $1.042^{+0.00030+0.00056}_{-0.00027-0.00062}$ | $1.042$ | $1.042^{+0.00028+0.00057}_{-0.00030-0.00057}$ | $1.042$ | $1.042^{+0.00029+0.00056}_{-0.00028-0.00056}$
$\ln(10^{10}A_{s})$ | $3.048$ | $3.047^{+0.013+0.028}_{-0.016-0.029}$ | $3.046$ | $3.045^{+0.012+0.029}_{-0.017-0.027}$ | $3.049$ | $3.045^{+0.013+0.027}_{-0.014-0.028}$
$n_{s}$ | $0.9661$ | $0.9673^{+0.0044+0.0079}_{-0.0039-0.0082}$ | $0.9659$ | $0.9664^{+0.0040+0.0083}_{-0.0042-0.0082}$ | $0.9647$ | $0.9673^{+0.0038+0.0080}_{-0.0038-0.0077}$
$\tau_{\mathrm{reio}}$ | $0.05578$ | $0.05575^{+0.0065+0.014}_{-0.0081-0.014}$ | $0.05699$ | $0.05476^{+0.0061+0.014}_{-0.0082-0.014}$ | $0.05648$ | $0.05486^{+0.0062+0.014}_{-0.0075-0.014}$
$w_{\mathrm{DE}}$ | $-1.012$ | $-1.010^{+0.036+0.071}_{-0.035-0.072}$ | $-1.032$ | $-1.009^{+0.030+0.065}_{-0.033-0.062}$ | $-1.057$ | $-1.042^{+0.029+0.060}_{-0.030-0.060}$
$\Delta$ | $0.0005820$ | $0.0007850^{+0.00032+0.00089}_{-0.00065-0.00078}$ | $0.0002594$ | $0.0004571^{+0.00011+0.00071}_{-0.00046-0.00046}$ | $0.0005151$ | $0.0006632^{+0.00021+0.00079}_{-0.00063-0.00066}$
$X$ [Mpc] | $38.51$ | unconstrained | $41.08$ | unconstrained | $1.712$ | unconstrained
$z_{\mathrm{reio}}$ | $7.829$ | $7.796^{+0.68+1.4}_{-0.79-1.4}$ | $7.968$ | $7.706^{+0.65+1.4}_{-0.79-1.4}$ | $7.858$ | $7.702^{+0.64+1.4}_{-0.74-1.4}$
$\Omega_{\mathrm{M},0}$ | $0.2971$ | $0.2936^{+0.0074+0.015}_{-0.0075-0.015}$ | $0.2972$ | $0.2989^{+0.0056+0.011}_{-0.0057-0.011}$ | $0.2841$ | $0.2867^{+0.0041+0.0084}_{-0.0045-0.0085}$
$H_{0}\;[\mathrm{km\,s^{-1}\,Mpc^{-1}}]$ | $69.23$ | $69.58^{+0.89+1.8}_{-0.86-1.8}$ | $69.21$ | $68.98^{+0.64+1.3}_{-0.63-1.3}$ | $70.68$ | $70.40^{+0.45+0.97}_{-0.52-0.95}$
$\sigma_{8}$ | $0.8355$ | $0.8352^{+0.012+0.024}_{-0.012-0.024}$ | $0.8376$ | $0.8309^{+0.012+0.021}_{-0.011-0.022}$ | $0.8452$ | $0.8428^{+0.0096+0.019}_{-0.0092-0.019}$

Furthermore, two-dimensional marginalized posterior distributions for selected
cosmological parameters of Barrow model are represented in figure 5.

Figure 5: One-dimensional marginalized posterior and two-dimensional
marginalized contours with $68\%$ and $95\%$ confidence levels from
”CMB+SN+BAO” (red), ”CMB+SN+BAO+Pop III” (blue), ”CMB+SN+BAO+Delay” (green),
and ”CMB+SN+BAO+No Delay” (orange) datasets, for some selected parameters of
Barrow model.

Regarding numerical investigations on Barrow cosmology, we find that adding
the mock GW SS data to current cosmological probes can provide better
constraints on the model parameters. In particular, we notice that the
exponent $\Delta$ is partially better constrained after the addition of Pop
III data, while including Delay data would result in tighter constraints on
the most of cosmological parameters. On the other hand, when we add No Delay
data, much tighter constraints on the model parameters would be obtained.
Moreover, the addition of No Delay data is also more effective in ameliorating
the Hubble tension by predicting higher values for $H_{0}$, which is certainly
affected by the phantom behavior of the dark energy equation of state.

It is also interesting to consider the measurement precision of the Hubble
constant defined as $\sigma(H_{0})/H_{0}$, where $\sigma(H_{0})$ is the
$1\sigma$ error of Hubble constant. According to MCMC analysis, the obtained
measurement precision of $H_{0}$ based on ”CMB+SN+BAO” dataset is $2.66\%$,
where adding the Pop III data would not improve it significantly by providing
the constraint precision of $2.52\%$. However, in case of ”CMB+SN+BAO+Delay”
dataset the constraint precision of $H_{0}$ would improve to $1.84\%$, while
for the ”CMB+SN+BAO+No Delay” dataset we measure a considerable precision of
$1.38\%$. Thus, it can be concluded that the mock LISA SS data is capable of
providing noticeable precision to constraint Barrow cosmological model.

##  5 Conclusions

Barrow cosmology is based on Barrow corrections to the area law entropy of the
apparent horizon of the universe, resulted from quantum-gravitational
deformation effects. Regarding the prominent analogy between thermodynamics
and gravity, one can drive modified field equations of Barrow model by
associating the Barrow entropy to the apparent horizon of FLRW universe, as
described in section 2, where the propagation of GWs in Barrow cosmology is
also investigated. The purpose of the present paper is to study the Barrow
cosmological model with observational probes, including current data along
with the mock LISA SS data.

LISA is a future space-borne GW observatory, designed to detect GW signals
from MBHBs in the milli-Hz frequency band. We generate three SS catalogs,
namely Pop III, Delay, and No Delay, according to ten years of LISA mission
duration, where the Barrow cosmology constrained by the ”CMB+SN+BAO” dataset
is contemplated as the fiducial model.

Numerical investigations based on dataset ”CMB+SN+BAO” indicate that Barrow
model with a phantom dark energy can slightly relieve the Hubble tension,
while the problem cannot be solved completely. Furthermore, when we add LISA
SS data to current observations, constraints on the model parameters would be
improved. Specifically, in case of ”CMB+SN+BAO+Pop III” dataset, better
constraints on the deformation exponents $\Delta$ is resulted. Moreover, the
model parameters are constrained more tightly for the dataset
”CMB+SN+BAO+Delay”. In addition, we observe significantly tighter constraints
on cosmological parameters in case of ”CMB+SN+BAO+No Delay” dataset, and also,
the Hubble tension is reduced more impressively in this case.

##  6 Acknowledgements

DFM thanks the Research Council of Norway for their support and the resources
provided by UNINETT Sigma2 – the National Infrastructure for High Performance
Computing and Data Storage in Norway. AA acknowledges SarAmadan and school of
physics of IPM.

## References

  * [1] J. M. Bardeen, B. Carter and S. W. Hawking, _The four laws of black hole mechanics_ , _Comm. Math. Phys._ 31 (1973) 161–170.
  * [2] J. D. Bekenstein, _Black holes and entropy_ , _Phys. Rev. D_ 7 (Apr, 1973) 2333–2346.
  * [3] S. W. Hawking, _Particle creation by black holes_ , _Communications in Mathematical Physics_ 43 (Aug, 1975) 199–220.
  * [4] T. Jacobson, _Thermodynamics of spacetime: The Einstein equation of state_, _Phys. Rev. Lett._ 75 (Aug, 1995) 1260–1263.
  * [5] E. Verlinde, “On the holographic principle in a radiation dominated universe.” arXiv:hep-th/0008140, Aug, 2000.
  * [6] B. Wang, E. Abdalla and R.-K. Su, _Relating friedmann equation to Cardy formula in universes with cosmological constant_, _Physics Letters B_ 503 (2001) 394 – 398.
  * [7] A. V. Frolov and L. Kofman, _Inflation and de Sitter thermodynamics_, _Journal of Cosmology and Astroparticle Physics_ 2003 (may, 2003) 009–009.
  * [8] U. H. Danielsson, _Transplanckian energy production and slow roll inflation_ , _Phys. Rev. D_ 71 (Jan, 2005) 023516.
  * [9] R.-G. Cai and S. P. Kim, _First law of thermodynamics and friedmann equations of Friedmann–Robertson–Walker universe_ , _Journal of High Energy Physics_ 2005 (feb, 2005) 050–050.
  * [10] R. Bousso, _Cosmology and the $\mathrm{S}$ matrix_, _Phys. Rev. D_ 71 (Mar, 2005) 064024.
  * [11] M. Akbar and R.-G. Cai, _Thermodynamic behavior of the Friedmann equation at the apparent horizon of the FRW universe_, _Phys. Rev. D_ 75 (Apr, 2007) 084003.
  * [12] A. Sheykhi, _Thermodynamics of apparent horizon and modified Friedmann equations_ , _The European Physical Journal C_ 69 (Sep, 2010) 265.
  * [13] S. W. Hawking, _Black hole explosions?_ , _Nature_ 248 (Mar., 1974) 30–31.
  * [14] C. Eling, R. Guedens and T. Jacobson, _Nonequilibrium thermodynamics of spacetime_ , _Phys. Rev. Lett._ 96 (Mar, 2006) 121301.
  * [15] C. Rovelli, _Black hole entropy from loop quantum gravity_ , _Phys. Rev. Lett._ 77 (Oct, 1996) 3288–3291.
  * [16] R. B. Mann and S. N. Solodukhin, _Quantum scalar field on a three-dimensional (BTZ) black hole instanton: Heat kernel, effective action, and thermodynamics_ , _Phys. Rev. D_ 55 (Mar, 1997) 3622–3632.
  * [17] A. Ashtekar, J. Baez, A. Corichi and K. Krasnov, _Quantum geometry and black hole entropy_ , _Phys. Rev. Lett._ 80 (Feb, 1998) 904–907.
  * [18] R. K. Kaul and P. Majumdar, _Logarithmic correction to the Bekenstein-Hawking entropy_ , _Phys. Rev. Lett._ 84 (Jun, 2000) 5255–5257.
  * [19] S. Das, P. Majumdar and R. K. Bhaduri, _General logarithmic corrections to black-hole entropy_ , _Classical and Quantum Gravity_ 19 (apr, 2002) 2355–2367.
  * [20] R. Banerjee and B. R. Majhi, _Quantum tunneling and back reaction_ , _Physics Letters B_ 662 (2008) 62 – 65.
  * [21] R. Banerjee and B. R. Majhi, _Quantum tunneling beyond semiclassical approximation_ , _Journal of High Energy Physics_ 2008 (jun, 2008) 095–095.
  * [22] J. Zhang, _Black hole quantum tunnelling and black hole entropy correction_ , _Physics Letters B_ 668 (2008) 353 – 356.
  * [23] S. Das, S. Shankaranarayanan and S. Sur, _Power-law corrections to entanglement entropy of horizons_ , _Phys. Rev. D_ 77 (Mar, 2008) 064013.
  * [24] S. Das, S. Shankaranarayanan and S. Sur, “Black hole entropy from entanglement: A review.” arXiv:0806.0402, 2008.
  * [25] N. Radicella and D. Pavón, _The generalized second law in universes with quantum corrected entropy relations_ , _Physics Letters B_ 691 (2010) 121 – 126.
  * [26] C. Tsallis and L. J. L. Cirto, _Black hole thermodynamical entropy_ , _The European Physical Journal C_ 73 (Jul, 2013) 2487.
  * [27] J. W. Gibbs, _Elementary Principles in Statistical Mechanics: Developed with Especial Reference to the Rational Foundation of Thermodynamics_.  Cambridge Library Collection - Mathematics. Cambridge University Press, 2010, 10.1017/CBO9780511686948.
  * [28] C. Tsallis, _Possible generalization of Boltzmann-Gibbs statistics_ , _Journal of Statistical Physics_ 52 (1988) 479.
  * [29] M. L. Lyra and C. Tsallis, _Nonextensivity and multifractality in low-dimensional dissipative systems_ , _Phys. Rev. Lett._ 80 (Jan, 1998) 53–56.
  * [30] C. Tsallis, R. Mendes and A. Plastino, _The role of constraints within generalized nonextensive statistics_ , _Physica A: Statistical Mechanics and its Applications_ 261 (1998) 534 – 554.
  * [31] C. Tsallis, _The nonadditive entropy $s_{q}$ and its applications in physics and elsewhere: Some remarks_, _Entropy_ 13 (Sep, 2011) 1765–1804.
  * [32] C. Tsallis, “From nolinear statistical mechanics to nonlinear quantum mechanics - concepts and applications.” arXiv:1202.3178, 2012.
  * [33] E. M. Barboza Jr, R. d. C. Nunes, E. M. Abreu and J. A. Neto, _Dark energy models through nonextensive tsallis’ statistics_ , _Physica A: Statistical Mechanics and its Applications_ 436 (2015) 301–310.
  * [34] A. Sheykhi, _Modified friedmann equations from Tsallis entropy_ , _Physics Letters B_ 785 (2018) 118 – 126.
  * [35] A. Sayahian Jahromi, S. A. Moosavi, H. Moradpour, J. P. Morais Graça, I. P. Lobo, I. G. Salako et al., _Generalized entropy formalism and a new holographic dark energy model_ , _Physics Letters B_ 780 (May, 2018) 21–24.
  * [36] M. Abdollahi Zadeh, A. Sheykhi, H. Moradpour and K. Bamba, _Note on tsallis holographic dark energy_ , _The European Physical Journal C_ 78 (Nov, 2018) 940.
  * [37] E. N. Saridakis, K. Bamba, R. Myrzakulov and F. K. Anagnostopoulos, _Holographic dark energy through tsallis entropy_ , _Journal of Cosmology and Astroparticle Physics_ 2018 (Dec, 2018) 012–012.
  * [38] E. Sadri, _Observational constraints on interacting tsallis holographic dark energy model_ , _The European Physical Journal C_ 79 (Sep, 2019) 762.
  * [39] A. Sheykhi, _New explanation for accelerated expansion and flat galactic rotation curves_ , _The European Physical Journal C_ 80 (2020) 25.
  * [40] A. A. Mamon, A. H. Ziaie and K. Bamba, _A generalized interacting tsallis holographic dark energy model and its thermodynamic implications_ , _The European Physical Journal C_ 80 (Oct, 2020) 974.
  * [41] A. Ghoshal and G. Lambiase, “Constraints on Tsallis cosmology from big bang nucleosynthesis and dark matter freeze-out.” arXiv:2104.11296, 2021.
  * [42] W. J. C. da Silva and R. Silva, _Cosmological perturbations in the tsallis holographic dark energy scenarios_ , _The European Physical Journal Plus_ 136 (May, 2021) 543.
  * [43] S. Nojiri, S. D. Odintsov and T. Paul, _Different Faces of Generalized Holographic Dark Energy_ , _Symmetry_ 13 (May, 2021) 928, [2105.08438].
  * [44] M. Asghari and A. Sheykhi, _Observational constraints on Tsallis modified gravity_ , _Monthly Notices of the Royal Astronomical Society_ (09, 2021) , [https://academic.oup.com/mnras/advance-article-pdf/doi/10.1093/mnras/stab2671/40417595/stab2671.pdf].
  * [45] P. Jizba and G. Lambiase, _Tsallis cosmology and its applications in dark matter physics with focus on IceCube high-energy neutrino data_ , _Eur. Phys. J. C_ 82 (Dec, 2022) 1123, [2206.12910].
  * [46] J. D. Barrow, _The area of a rough black hole_ , _Physics Letters B_ 808 (2020) 135643.
  * [47] S. Hawking, _Spacetime foam_ , _Nuclear Physics B_ 144 (1978) 349–362.
  * [48] G. ’t Hooft, _On the quantum structure of a black hole_ , _Nuclear Physics B_ 256 (1985) 727–745.
  * [49] E. N. Saridakis, _Modified cosmology through spacetime thermodynamics and Barrow horizon entropy_ , _Journal of Cosmology and Astroparticle Physics_ 2020 (Jul, 2020) 031–031.
  * [50] B. Das and B. Pandey, “A study of holographic dark energy models using configuration entropy.” arXiv:2011.07337, 2020.
  * [51] E. N. Saridakis, _Barrow holographic dark energy_ , _Phys. Rev. D_ 102 (Dec, 2020) 123525.
  * [52] S. Srivastava and U. K. Sharma, _Barrow holographic dark energy with hubble horizon as IR cutoff_ , _International Journal of Geometric Methods in Modern Physics_ 18 (2021) 2150014.
  * [53] U. K. Sharma, G. Varshney and V. C. Dubey, _Barrow agegraphic dark energy_ , _International Journal of Modern Physics D_ 30 (Jan, 2021) 2150021.
  * [54] A. Mamon, A. Paliathanasis and S. Saha, _Dynamics of an interacting Barrow holographic dark energy model and its thermodynamic implications_ , _Eur. Phys. J. Plus_ 136 (Jan, 2021) 134.
  * [55] A. Pradhan, A. Dixit and V. K. Bhardwaj, _Barrow HDE model for statefinder diagnostic in FLRW universe_ , _International Journal of Modern Physics A_ 36 (Feb, 2021) 2150030.
  * [56] A. Dixit, V. K. Bharadwaj and A. Pradhan, “Barrow HDE model for statefinder diagnostic in non-flat FRW universe.” arXiv:2103.08339, Mar, 2021.
  * [57] P. Adhikary, S. Das, S. Basilakos and E. N. Saridakis, “Barrow holographic dark energy in non-flat universe.” arXiv:2104.13118, Apr, 2021.
  * [58] V. K. Bhardwaj, A. Dixit and A. Pradhan, _Statefinder hierarchy model for the barrow holographic dark energy_ , _New Astronomy_ 88 (2021) 101623.
  * [59] A. Sheykhi, _Barrow entropy corrections to friedmann equations_ , _Phys. Rev. D_ 103 (Jun, 2021) 123503.
  * [60] E. N. Saridakis and S. Basilakos, _The generalized second law of thermodynamics with barrow entropy_ , _The European Physical Journal C_ 81 (2021) 644.
  * [61] A. Sheykhi and B. Farsi, _Growth of perturbations in Tsallis and Barrow cosmology_ , _Eur. Phys. J. C_ 82 (2022) 1111, [2205.04138].
  * [62] A. Sheykhi, _Modified cosmology through Barrow entropy_ , _Phys. Rev. D_ 107 (2023) 023505, [2210.12525].
  * [63] A. Sheykhi and M. S. Hamedan, _Holographic dark energy in modified Barrow cosmology_ , _Entropy_ 25 (2023) 569, [2211.00088].
  * [64] F. K. Anagnostopoulos, S. Basilakos and E. N. Saridakis, _Observational constraints on Barrow holographic dark energy_ , _The European Physical Journal C_ 80 (Sep, 2020) 826.
  * [65] J. D. Barrow, S. Basilakos and E. N. Saridakis, _Big bang nucleosynthesis constraints on Barrow entropy_ , _Physics Letters B_ 815 (2021) 136134.
  * [66] G. Leon, J. Magaña, A. Hernández-Almada, M. A. García-Aspeitia, T. Verdugo and V. Motta, “Barrow entropy cosmology: an observational approach with a hint of stability analysis.” arXiv:2108.10998, Aug., 2021.
  * [67] M. Asghari and A. Sheykhi, _Observational constraints of the modified cosmology through Barrow entropy_ , _Eur. Phys. J. C_ 82 (2022) 388, [2110.00059].
  * [68] LIGO Scientific, Virgo collaboration, B. P. Abbott et al., _Observation of Gravitational Waves from a Binary Black Hole Merger_ , _Phys. Rev. Lett._ 116 (2016) 061102, [1602.03837].
  * [69] LIGO Scientific, Virgo collaboration, B. P. Abbott et al., _GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral_ , _Phys. Rev. Lett._ 119 (2017) 161101, [1710.05832].
  * [70] LIGO Scientific, Virgo, Fermi-GBM, INTEGRAL collaboration, B. P. Abbott et al., _Gravitational Waves and Gamma-rays from a Binary Neutron Star Merger: GW170817 and GRB 170817A_ , _Astrophys. J. Lett._ 848 (2017) L13, [1710.05834].
  * [71] A. von Kienlin, C. Meegan and A. Goldstein, _GRB 170817A: Fermi GBM detection._ , _GRB Coordinates Network_ 21520 (Jan., 2017) 1\.
  * [72] V. Connaughton et al., _LIGO/Virgo G298048: Fermi GBM trigger 170817.529 and LIGO single IFO trigger_ , _GRB Coordinates Network_ 21506 (Aug., 2017) 1.
  * [73] A. Goldstein et al., _LIGO/Virgo G298048 - Update on Fermi/GBM GRB 170817A Analysis_ , _GRB Coordinates Network_ 21528 (Aug., 2017) 1.
  * [74] A. Goldstein et al., _An Ordinary Short Gamma-Ray Burst with Extraordinary Implications: Fermi-GBM Detection of GRB 170817A_ , _Astrophys. J. Lett._ 848 (2017) L14, [1710.05446].
  * [75] V. Savchenko et al., _LIGO/Virgo G298048: INTEGRAL detection of a prompt gamma-ray counterpart_ , _GRB Coordinates Network_ 21507 (Aug., 2017) 1.
  * [76] V. Savchenko et al., _INTEGRAL Detection of the First Prompt Gamma-Ray Signal Coincident with the Gravitational-wave Event GW170817_ , _Astrophys. J. Lett._ 848 (2017) L15, [1710.05449].
  * [77] P. Mészáros, D. B. Fox, C. Hanna and K. Murase, _Multi-Messenger Astrophysics_ , _Nature Rev. Phys._ 1 (2019) 585–599, [1906.10212].
  * [78] B. F. Schutz, _Determining the Hubble constant from gravitational wave observations_ , _Nature_ 323 (Sept., 1986) 310–311.
  * [79] B. F. Schutz, _Lighthouses of Gravitational Wave Astronomy_ , in _Lighthouses of the Universe: The Most Luminous Celestial Objects and Their Use for Cosmology_ (M. Gilfanov, R. Sunyeav and E. Churazov, eds.), p. 207, Jan., 2002, gr-qc/0111095, DOI.
  * [80] D. E. Holz and S. A. Hughes, _Using gravitational-wave standard sirens_ , _Astrophys. J._ 629 (2005) 15–22, [astro-ph/0504616].
  * [81] C. Cutler and D. E. Holz, _Ultrahigh precision cosmology from gravitational waves_ , _Phys. Rev. D_ 80 (Nov, 2009) 104009.
  * [82] A. Palmese et al., _Gravitational Wave Cosmology and Astrophysics with Large Spectroscopic Galaxy Surveys_ , _BAAS_ 51 (May, 2019) 310, [1903.04730].
  * [83] LIGO Scientific, Virgo, 1M2H, Dark Energy Camera GW-E, DES, DLT40, Las Cumbres Observatory, VINROUGE, MASTER collaboration, B. P. Abbott et al., _A gravitational-wave standard siren measurement of the Hubble constant_ , _Nature_ 551 (2017) 85–88, [1710.05835].
  * [84] LIGO Scientific, Virgo, VIRGO collaboration, B. P. Abbott et al., _A Gravitational-wave Measurement of the Hubble Constant Following the Second Observing Run of Advanced LIGO and Virgo_ , _Astrophys. J._ 909 (2021) 218, [1908.06060].
  * [85] M. Punturo et al., _The Einstein Telescope: A third-generation gravitational wave observatory_ , _Class. Quant. Grav._ 27 (2010) 194002.
  * [86] M. Maggiore et al., _Science Case for the Einstein Telescope_ , _JCAP_ 03 (2020) 050, [1912.02622].
  * [87] D. Reitze et al., _Cosmic Explorer: The U.S. Contribution to Gravitational-Wave Astronomy beyond LIGO_ , _Bull. Am. Astron. Soc._ 51 (2019) 035, [1907.04833].
  * [88] P. Amaro-Seoane et al., _Laser Interferometer Space Antenna_ , _arXiv e-prints_ (Feb., 2017) arXiv:1702.00786, [1702.00786].
  * [89] J. I. Thorpe et al., _The Laser Interferometer Space Antenna: Unveiling the Millihertz Gravitational Wave Sky_ , in _Bulletin of the American Astronomical Society_ , vol. 51, p. 77, Sept., 2019, 1907.06482, DOI.
  * [90] E. Belgacem, Y. Dirian, S. Foffa and M. Maggiore, _Modified gravitational-wave propagation and standard sirens_ , _Phys. Rev. D_ 98 (Jul, 2018) 023510.
  * [91] A. Nishizawa and S. Arai, _Generalized framework for testing gravity with gravitational-wave propagation. iii. future prospect_ , _Phys. Rev. D_ 99 (May, 2019) 104038.
  * [92] LISA Cosmology Working Group collaboration, E. Belgacem et al., _Testing modified gravity at cosmological distances with LISA standard sirens_ , _JCAP_ 07 (Jul, 2019) 024, [1906.01593].
  * [93] W. Yang, S. Vagnozzi, E. Di Valentino, R. C. Nunes, S. Pan and D. F. Mota, _Listening to the sound of dark sector interactions with gravitational wave standard sirens_ , _JCAP_ 07 (Jul, 2019) 037, [1905.08286].
  * [94] R. C. Nunes, M. E. S. Alves and J. C. N. de Araujo, _Forecast constraints on $f({T})$ gravity with gravitational waves from compact binary coalescences_, _Phys. Rev. D_ 100 (Sep, 2019) 064012.
  * [95] S. Mastrogiovanni, D. A. Steer and M. Barsuglia, _Probing modified gravity theories and cosmology using gravitational-waves and associated electromagnetic counterparts_ , _Phys. Rev. D_ 102 (Aug, 2020) 044009.
  * [96] P. Baral, S. K. Roy and S. Pal, _Prospects of probing dark energy with eLISA: Standard versus null diagnostics_ , _Monthly Notices of the Royal Astronomical Society_ 500 (10, 2020) 2896–2907, [https://academic.oup.com/mnras/article-pdf/500/3/2896/34572597/staa3346.pdf].
  * [97] A. Finke, S. Foffa, F. Iacovelli, M. Maggiore and M. Mancarella, _Cosmology with LIGO/Virgo dark sirens: Hubble parameter and modified gravitational wave propagation_ , _JCAP_ 08 (Aug, 2021) 026, [2101.12660].
  * [98] A. Bonilla, S. Kumar, R. C. Nunes and S. Pan, _Reconstruction of the dark sectors’ interaction: A model-independent inference and forecast from GW standard sirens_ , _Monthly Notices of the Royal Astronomical Society_ 512 (03, 2022) 4231–4238, [https://academic.oup.com/mnras/article-pdf/512/3/4231/43291045/stac687.pdf].
  * [99] A. Allahyari, R. C. Nunes and D. F. Mota, _No slip gravity in light of LISA standard sirens_ , _Monthly Notices of the Royal Astronomical Society_ 514 (05, 2022) 1274–1281, [https://academic.oup.com/mnras/article-pdf/514/1/1274/43986124/stac1445.pdf].
  * [100] J. Ferreira, T. Barreiro, J. Mimoso and N. J. Nunes, _Forecasting $f({Q})$ cosmology with $\mathrm{\Lambda}\mathrm{CDM}$ background using standard sirens_, _Phys. Rev. D_ 105 (Jun, 2022) 123531.
  * [101] R. D’Agostino and R. C. Nunes, _Forecasting constraints on deviations from general relativity in $f({Q})$ gravity with standard sirens_, _Phys. Rev. D_ 106 (Dec, 2022) 124053.
  * [102] I. S. Matos, E. Bellini, M. O. Calvão and M. Kunz, “Testing gravity with gravitational wave friction and gravitational slip.” arXiv:2210.12174, 10, 2022\.
  * [103] W. Yang, S. Pan, E. Di Valentino, C. Escamilla-Rivera and A. Paliathanasis, _Exploring bulk viscous unified scenarios with gravitational waves standard sirens_ , _Mon. Not. Roy. Astron. Soc._ 520 (2023) 1146–1154, [2301.03969].
  * [104] E. Di Valentino, O. Mena, S. Pan, L. Visinelli, W. Yang, A. Melchiorri et al., _In the realm of the Hubble tension—a review of solutions_ , _Class. Quant. Grav._ 38 (2021) 153001, [2103.01183].
  * [105] E. Abdalla et al., _Cosmology intertwined: A review of the particle physics, astrophysics, and cosmology associated with the cosmological tensions and anomalies_ , _JHEAp_ 34 (2022) 49–211, [2203.06142].
  * [106] D. Bak and S.-J. Rey, _Cosmic holography_ , _Classical and Quantum Gravity_ 17 (jul, 2000) L83–L89.
  * [107] S. De Groot and P. Mazur, _Non-equilibrium Thermodynamics_.  Dover books on physics and chemistry. North-Holland Publishing Company, 1962.
  * [108] C.-P. Ma and E. Bertschinger, _Cosmological perturbation theory in the synchronous and conformal Newtonian gauges_ , _The Astrophysical Journal_ 455 (1995) 7.
  * [109] E. Belgacem, Y. Dirian, S. Foffa and M. Maggiore, _Gravitational-wave luminosity distance in modified gravity theories_ , _Phys. Rev. D_ 97 (2018) 104066, [1712.08108].
  * [110] N. Aghanim et al., _Planck 2018 results. VI. Cosmological parameters_, _Astronomy & Astrophysics_ 641 (Sep, 2020) A6.
  * [111] D. M. Scolnic et al., _The complete light-curve sample of spectroscopically confirmed SNe Ia from pan-STARRS1 and cosmological constraints from the combined pantheon sample_ , _The Astrophysical Journal_ 859 (May, 2018) 101.
  * [112] F. Beutler, C. Blake, M. Colless, D. H. Jones, L. Staveley-Smith, L. Campbell et al., _The 6 DF Galaxy Survey: Baryon Acoustic Oscillations and the Local Hubble Constant_, _Monthly Notices of the Royal Astronomical Society_ 416 (2011) 3017–3032.
  * [113] L. Anderson et al., _The clustering of galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey: Baryon Acoustic Oscillations in the Data Releases 10 and 11 Galaxy Samples_, _Monthly Notices of the Royal Astronomical Society_ 441 (2014) 24–62.
  * [114] E. Barausse et al., _Massive Black Hole Science with eLISA_ , _J. Phys. Conf. Ser._ 610 (2015) 012001, [1410.2907].
  * [115] C. Caprini and N. Tamanini, _Constraining early and interacting dark energy with gravitational wave standard sirens: the potential of the eLISA mission_ , _JCAP_ 10 (2016) 006, [1607.08755].
  * [116] N. Tamanini, C. Caprini, E. Barausse, A. Sesana, A. Klein and A. Petiteau, _Science with the space-based interferometer eLISA. III: probing the expansion of the universe using gravitational wave standard sirens_ , _Journal of Cosmology and Astroparticle Physics_ 2016 (2016) 002.
  * [117] L. Speri, N. Tamanini, R. R. Caldwell, J. R. Gair and B. Wang, _Testing the quasar hubble diagram with LISA standard sirens_ , _Phys. Rev. D_ 103 (Apr, 2021) 083526.
  * [118] D. Blas, J. Lesgourgues and T. Tram, _The Cosmic Linear Anisotropy Solving System (CLASS). Part II: Approximation schemes_, _Journal of Cosmology and Astroparticle Physics_ 2011 (2011) 034.
  * [119] B. Audren, J. Lesgourgues, K. Benabed and S. Prunet, _Conservative Constraints on Early Cosmology: an illustration of the Monte Python cosmological parameter inference code_, _JCAP_ 1302 (2013) 001, [1210.7183].
  * [120] T. Brinckmann and J. Lesgourgues, _MontePython 3: boosted MCMC sampler and other features_ , _Phys. Dark Univ._ 24 (2019) 100260, [1804.07261].