# Barrow entropy and stochastic gravitational wave background spectrum
generated from cosmological QCD phase transition

Zhong-Wen Feng1 Email<EMAIL_ADDRESS>1 Physics and Space Science College,
China West Normal University, Nanchong, 637009, China

###### Abstract

In recent years, it is argued that Barrow entropy (a theory based on quantum
gravity effects, which suggests that the area law of the horizon entropy can
receive a fractal correction) can have significant effects on the properties
of the universe. In this manuscript, we investigate the stochastic
gravitational wave background generated during the first-order cosmological
QCD phase transition of the early universe in the framework of the Barrow
entropy. We first derived the Barrow corrections to the expression of
stochastic gravitational wave background spectrum in presence of trace
anomaly. Then, by taking account of Bubble wall collisions, sound waves and
magnetohydrodynamic turbulence as the sources of stochastic gravitational
wave, an analysis of the influence of Barrow entropy on the total energy
density and the peak signal of stochastic gravitational wave signal is carried
out. Finally, we discuss the possibility of detectors for the detection of
these stochastic gravitational wave signals. Our results show that effect of
Barrow entropy plays an important role in the temporal evolution of
temperature of the universe as a function of the Hubble parameter, which leads
to the signal of stochastic gravitational wave to shift towards the lower
frequency regime, making the signal of stochastic gravitational wave possible
to be probed by relevant ongoing and future detectors.

Barrow entropy; Stochastic gravitational wave background; QCD phase transition

##  I Introduction

The gravitational wave (GW), as ripples in spacetimes, was first predicted by
Albert Einstein in 1916, based on the theory of General Relativity (GR). Over
the past one hundred years, physicists have done everything they can to detect
the signal of gravitational waves, and hope it would could improve people’
understanding of the nature in the same way that electromagnetic waves have
done. Fortunately, in 2015, the LIGO Scientific and Virgo Collaborations used
sophisticated gravitational wave detectors to receive the signal of GWs form a
binary black hole merger event for the first time cha1 . This GW events not
only reaffirms the validity of GR, but also gives powerful evidence for the
existence of black holes. Subsequently, more GW signals from compact binary
(such as black hole-neutron star cha2  and neutron star-neutron star cha3 )
inspiral events have been detected. These results have made it possible to
probe the mysteries of the universe from different perspectives, and have
opened up a new field of physical and astronomical research-the multi-
messenger astronomy. Now many more sensitive GW detectors are being built and
will be operational in the near future (see Refs. cha4 ; cha5 ; cha6 ; cha7 ;
cha8 ; cha9  for example).

Despite the fact that GW from compact binary excitation events are the main
focus today, it is also worth noting that GW generated by other sources also
contain a wealth of unique physical information. In particular, many works
have been argued that the early universe have undergone several phase
transitions during its evolution, if the phase transitions are strongly first
order, they would in turn have generated low-frequency GW cha10 ; cha11 ;
cha12 . These GW hardly interact with any matters, at all, they would
superimpose and populate the entire universe, and are therefore called
stochastic gravitational wave background (SGWB). According to the standard
model of particle physics, it is believed that there are at least two types of
first-order cosmological phase transitions, viz. electroweak phase transition
(at $T\sim 100$ GeV) with accompanying electroweak symmetry breaking cha13 ;
cha14 ; cha15 , and the quantum chromodynamics (QCD) phase transition (at
$T\sim 0.1-0.2$ GeV) that breaks the chiral symmetry cha16 ; cha17 . In recent
years, there has been an increasing interest in SGWB generated by QCD phase
transitions, which is mainly due to two reasons: one is that the frequency
range of SGWB generated by QCD phase transitions is expected to be in the
magnitude of $10^{-9}$ Hz $\sim 10^{-6}$ Hz, which reaches the sensitivity of
the future-planned GW detectors, such as the European Pulsar Timing Array
(EPTA), International Pulsar Timing Array (IPTA), Square Kilometer Array
(SKA), and North American Nanohertz Observatory for Gravitational Wave
(NANOGrav). Therefore, it is believed that SGWB is likely to be detected in
the near future. The other one is that the different materials and fields is
able to influence the first-order phase transition in the QCD epoch and in
turn change the features of the SGWB cha19 ; cha20 ; chc1 ; chc2 ; chc3 ; chc4
. Since the signals of SGWB may contain traces of early universe, it is
possible to use them to analyze the properties of the early universe and
explore new physics within therein cha18 .

On the other hand, it has been argued that quantum gravity (QG) effect play a
significant role in the generation of SGWB chc6+ ; cha21 ; cha22 ; cha23 ;
cha24 ; chc6 . For instance, the authors in Refs. cha21 ; cha22 ; cha23
showed that the different kinds of generalized uncertainty principle (one of
the most important QG model) can correct the thermal properties of the early
universe, resulting in changes to the energy density and frequency of SGWB.
Calcagni and Kuroyanagi cha24  analyzed the possibility of five cosmological
quantum-gravity scenarios generating SGWB, and discussed whether these signals
could be detected by existing or future detectors. Subsequently, according to
a very general class of ultraviolet complete theories of QG enjoying Weyl
conformal invariance, Calcagni and Modesto chc6  constructed a new
cosmological model and analyzed the properties of the GW generated therein.
Their results predict that these SGWB could be detected in the next 5-10
years. Considering that faint QG effects are amplified as GW propagate
throughout the universe, the above works may open a new window for us to study
the effects of QG (see Ref. chc7  and references therein).

In 2020, based on the QG effects, Barrow cha28  proposed a possible fractal
correction to the the area law of the horizon entropy (hereafter called the
“Barrow entropy”), which is expressed as follows:

${S_{B}}={\left({\frac{A}{{4G}}}\right)^{1+\frac{\delta}{2}}}={\left({{S_{0}}}\right)^{1+\frac{\delta}{2}}},$
(1)

where $A$ is the standard horizon area, ${S_{0}}$ is the holographic entropy
since its proportional to the boundary area of the system chb4 , the Barrow
exponent $\delta\in\left[{0,1}\right]$ is a quantity indicating the effect of
quantum gravitational deformation. For $\delta=0$, the original expression of
area-law entropy is recovered, which corresponds to the simplest horizon
structure, whereas $\delta=1$ indicates the most sophisticated surface
structure. This interesting model has attracted a lot of attention since its
inception, and its application quickly expanded from the thermodynamics of
black holes to the evolution of the universe cha29 ; cha30 ; cha31 ; cha32 ;
cha33 ; cha34 ; cha35 ; cha36 ; cha37 ; cha38 . According to the above works,
it is reason to believe that the Barrow entropy could correct the
thermodynamic properties and evolution of the universe. Then some questions
arises as whether the SGWB generated during the QCD phase transition epoch of
the modified universe were affected by the Barrow entropy? Are their signals
being picked up by GW detectors? To this end, we attempt to answer those
questions in the present manuscript. By utilizing the Barrow entropy (1), we
derive the modified SGWB spectrum, the characteristic frequency, and the
fractional energy density are analyzed. It turns out that the Barrow
corrections to the peak of SGWB signals are lower than the original case, so
that different detectors can detect SGWB signals with different $\delta$.

The paper is organized as follows. In the next section, according to the
Barrow entropy, we derive the general expression for the SGWB spectrum
measured today. In section III, by comparing the equation of state phase of
the ideal gas with the QCD equation of state provided by recent lattice
results, we obtain the expression for the effective equation of state used in
this study. In Section IV, by discussing the role of Barrow entropy on the
amplitude and frequency of SGWB, we predict the response of the modified SGWB
signs in ongoing and proposed detectors for future observations. Finally, the
conclusion is given in section V.

##  II Modified SGWB spectrum from Barrow corrections

A consensus was reached that SGWB is generated during the QCD and electroweak
phase transition periods, and propagated to the current epoch. Therefore, for
investigating the SGWB today, we consider the GW from the epoch of phase
transition to the present time. By assuming that the universe is expanding
adiabatically due to the phase transition, the entropy density can be
estimated to be $s\sim{a^{3}}{S_{0}}\sim{a^{3}}\xi{g_{s}}{T^{3}}$, where $a$,
${g_{s}}$ and $\xi$ represent the scale factor, the effective number of
degrees of freedom involved in entropy density and an undetermined parameter,
respectively cha39 . Now, applying the Barrow entropy (1), the relevant
entropy density becomes

$\displaystyle{s_{B}}={a^{3}}{\left({\frac{{2{\pi^{2}}}}{{45}}{g_{\pi}}{T^{3}}}\right)^{1+\frac{\delta}{2}}},$
(2)

where we take
$\xi={{2{\pi^{2}}}\mathord{\left/{\vphantom{{2{\pi^{2}}}{45}}}\right.\kern-1.2pt}{45}}$
in accordance with the results in Ref. cha22 . Since the entropy density in
the adiabatic expanding universe remains constant even beyond equilibrium
(i.e.,
${{\dot{s}}\mathord{\left/{\vphantom{{\dot{s}}s}}\right.\kern-1.2pt}s}=0$),
the time variation of universe temperature can be expressed as

$\displaystyle\frac{{{\text{d}}T}}{{{\text{d}}t}}=-HT\Xi{\left({T,{g_{s}},\delta}\right)^{-1}},$
(3)

where $H$ is the Hubble parameter, and
$\Xi\left({T,{g_{s}},\delta}\right)=1+\frac{T}{{3{g_{s}}}}\frac{{{\text{d}}{g_{s}}}}{{{\text{d}}T}}+\frac{\delta}{2}\left({1+\frac{T}{{3{g_{s}}}}\frac{{{\text{d}}{g_{s}}}}{{{\text{d}}T}}}\right)$.
It is clear that the original case
$\frac{{{\text{d}}T}}{{{\text{d}}t}}=-HT{\left({1+\frac{T}{{3{g_{s}}}}\frac{{{\text{d}}{g_{s}}}}{{{\text{d}}T}}}\right)^{-1}}$
is recovered when $\delta=0$. Next, due to the Hubble parameter, one can
rewrite Eq. (3) in terms of scale factor

$\displaystyle\frac{{{a_{*}}}}{{{a_{0}}}}=\exp\left[{\int_{{T_{*}}}^{{T_{0}}}{\frac{1}{T}\Xi\left({T,{g_{s}},\delta}\right)}{\text{d}}T}\right].$
(4)

Here and also in what follows, the subscripts “$*$” and “$0$” denote the
quantities at the epochs of phase transition and today, respectively.
Furthermore, based on the relation between the scale factor and red-shifted
${{{\nu_{0{\text{peak}}}}}\mathord{\left/{\vphantom{{{\nu_{0{\text{peak}}}}}{{\nu_{*}}}}}\right.\kern-1.2pt}{{\nu_{*}}}}={{{a_{*}}}\mathord{\left/{\vphantom{{{a_{*}}}{{a_{0}}}}}\right.\kern-1.2pt}{{a_{0}}}}$,
the peak frequency of SGWB red-shifted to current epoch is given by

$\displaystyle{{\frac{{{\nu_{0{\text{peak}}}}}}{{{\nu_{*}}}}}}={{\frac{{{a_{*}}}}{{{a_{0}}}}}}=\frac{{{T_{0}}}}{{{T_{*}}}}{\left[{\frac{{{g_{s}}\left({{T_{0}}}\right)}}{{{g_{s}}\left({{T_{*}}}\right)}}}\right]^{\frac{{2+\delta}}{6}}}{\left({\frac{{{T_{0}}}}{{{T_{*}}}}}\right)^{\frac{\delta}{2}}}.$
(5)

In order to gain insight into the effect of Barrow entropy on
${{{\nu_{0{\text{peak}}}}}\mathord{\left/{\vphantom{{{\nu_{0{\text{peak}}}}}{{\nu_{*}}}}}\right.\kern-1.2pt}{{\nu_{*}}}}$,
we depicted Fig. 1. It is found that different values of Barrow exponent
$\delta$ give a similar behavior for the ratio
${{{\nu_{0{\text{peak}}}}}\mathord{\left/{\vphantom{{{\nu_{0{\text{peak}}}}}{{\nu_{*}}}}}\right.\kern-1.2pt}{{\nu_{*}}}}$,
that is, the
${{{\nu_{0{\text{peak}}}}}\mathord{\left/{\vphantom{{{\nu_{0{\text{peak}}}}}{{\nu_{*}}}}}\right.\kern-1.2pt}{{\nu_{*}}}}$
increases as the temperature $T_{*}$ decreases. Meanwhile, when the $T_{*}$ is
fixed, the ratio
${{{\nu_{0{\text{peak}}}}}\mathord{\left/{\vphantom{{{\nu_{0{\text{peak}}}}}{{\nu_{*}}}}}\right.\kern-1.2pt}{{\nu_{*}}}}$
decreases significantly with increasing $\delta$.

Figure 1:  ${\nu_{0{\text{peak}}}}/{\nu_{*}}$ for different values of Barrow
exponent $\delta$ with ${T_{0}}=2.725K=2.348\times{10^{-13}}{\text{GeV}}$,
${g_{s}}\left({{T_{0}}}\right)=3.4$ and ${g_{s}}\left({{T_{*}}}\right)=35$.

Next, considering the SGWB is ultimately decoupled to the rest of the
universe, one can derive the energy density of the SGWB at today epoch by
employing the Boltzmann equation
${{{\text{d}}\left({{\rho_{{\text{gw}}}}{a^{4}}}\right)}\mathord{\left/{\vphantom{{{\text{d}}\left({{\rho_{{\text{gw}}}}{a^{4}}}\right)}{{\text{d}}t}}}\right.\kern-1.2pt}{{\text{d}}t}}=0$,
which yields

$\displaystyle{\rho_{{\text{gw}}}}\left({{T_{0}}}\right)$
$\displaystyle={\rho_{{\text{gw}}}}\left({{T_{*}}}\right){\left({\frac{{{a_{*}}}}{{{a_{0}}}}}\right)^{4}}$
$\displaystyle={\rho_{{\text{gw}}}}\left({{T_{*}}}\right)\exp\left[{\int_{{T_{*}}}^{{T_{0}}}{\frac{4}{T}\Xi\left({T,{g_{s}},\delta}\right){\text{d}}T}}\right].$
(6)

Then, by using the definitions of the density parameter of SGWB in the today’s
epoch
${\Omega_{{\text{gw}}}}={{{\rho_{{\text{gw}}}}\left({{T_{0}}}\right)}\mathord{\left/{\vphantom{{{\rho_{{\text{gw}}}}\left({{T_{0}}}\right)}{{\rho_{{\text{cr}}}}\left({{T_{0}}}\right)}}}\right.\kern-1.2pt}{{\rho_{{\text{cr}}}}\left({{T_{0}}}\right)}}$
as well as its counterpart in phase transition era
${\Omega_{{\text{gw}}*}}={{{\rho_{{\text{gw}}}}\left({{T_{*}}}\right)}\mathord{\left/{\vphantom{{{\rho_{{\text{gw}}}}\left({{T_{*}}}\right)}{{\rho_{{\text{cr}}}}\left({{T_{*}}}\right)}}}\right.\kern-1.2pt}{{\rho_{{\text{cr}}}}\left({{T_{*}}}\right)}}$,
the expression of density parameter of SGWB in today’s epoch is given by

$\displaystyle{\Omega_{{\text{gw}}}}={\Omega_{{\text{gw}}*}}{\left({\frac{{{H_{*}}}}{{{H_{0}}}}}\right)^{2}}\exp\left[{\int_{{T_{*}}}^{{T_{0}}}{\frac{4}{T}\Xi\left({T,{g_{s}},\delta}\right){\text{d}}T}}\right],$
(7)

where

$\displaystyle{\left({\frac{{{H_{*}}}}{{{H_{0}}}}}\right)^{2}}=\frac{{{\rho_{{\text{cr}}}}\left({{T_{*}}}\right)}}{{{\rho_{{\text{cr}}}}\left({{T_{0}}}\right)}}.$
(8)

In order to obtain the ratio of the Hubble parameters during the phase
transition to its current value, it is necessary to apply the continuity
equation
${\dot{\rho}_{t}}=-3H{\rho_{t}}\left({1+{{{P_{t}}}\mathord{\left/{\vphantom{{{P_{t}}}{{\rho_{t}}}}}\right.\kern-1.2pt}{{\rho_{t}}}}}\right)$
with the total pressure density of the universe $P_{t}$ and total energy
density of the universe $\rho_{t}$. Inserting Eq. (3) into the continuity
equation, one has

$\displaystyle\frac{{{\text{d}}{\rho_{t}}}}{{{\rho_{t}}}}=\frac{3}{T}\left({1+{\omega_{{\text{ef\/f}}}}}\right)\Xi\left({T,{g_{s}},\delta}\right){\text{d}}T.$
(9)

where
${\omega_{{\text{ef\/f}}}}={{{P_{t}}}\mathord{\left/{\vphantom{{{P_{t}}}{{\rho_{t}}}}}\right.\kern-1.2pt}{{\rho_{t}}}}$
is the effective equation of state parameter. By integrating Eq. (9) from the
early time in the radiation dominated ${T_{r}}={10^{4}}$ GeV to the time of
phase transition ${T_{*}}$, the critical energy density of radiation in the
phase transition period is given by

$\displaystyle{\rho_{cr}}\left({{T_{*}}}\right)={\rho_{r}}\left({{T_{r}}}\right)\exp\left[{\int_{{T_{r}}}^{{T_{*}}}{\frac{3}{T}\left({1+{\omega_{{\text{ef\/f}}}}}\right)\Xi\left({T,{g_{s}},\delta}\right){\text{d}}T}}\right].$
(10)

Then, substituting Eq. (10) into Eq. (8), and using the relation
$H_{*}^{2}={\rho_{*}}$ chd1 , one gets

$\displaystyle{\left({\frac{{{H_{*}}}}{{{H_{0}}}}}\right)^{2}}$
$\displaystyle={\Omega_{r0}}\frac{{{\rho_{r}}\left({{T_{r}}}\right)}}{{{\rho_{r}}\left({{T_{0}}}\right)}}$
$\displaystyle\times\exp\left[{\int_{{T_{r}}}^{{T_{*}}}{\frac{3}{T}\left({1+{\omega_{{\text{ef\/f}}}}}\right)\Xi\left({T,{g_{s}},\delta}\right){\text{d}}T}}\right],$
(11)

where
${\Omega_{r0}}={{{\rho_{r}}\left({{T_{0}}}\right)}\mathord{\left/{\vphantom{{{\rho_{r}}\left({{T_{0}}}\right)}{{\rho_{cr}}\left({{T_{0}}}\right)}}}\right.\kern-1.2pt}{{\rho_{cr}}\left({{T_{0}}}\right)}}\simeq
8.5\times{10^{-5}}$ represents the current value of fractional energy density
of radiation. Moreover, according to the viewpoint in Ref. cha23 , it is
proved that
${{{\rho_{r}}\left({{T_{r}}}\right)}\mathord{\left/{\vphantom{{{\rho_{r}}\left({{T_{r}}}\right)}{{\rho_{r}}\left({{T_{0}}}\right)}}}\right.\kern-1.2pt}{{\rho_{r}}\left({{T_{0}}}\right)}}\simeq{\left({{{{a_{0}}}\mathord{\left/{\vphantom{{{a_{0}}}{{a_{r}}}}}\right.\kern-1.2pt}{{a_{r}}}}}\right)^{4}}$,
hence, Eq. (II) can be rewritten as

$\displaystyle{\left({\frac{{{H_{*}}}}{{{H_{0}}}}}\right)^{2}}$
$\displaystyle={\Omega_{r0}}\exp\left[{\int_{{T_{0}}}^{{T_{r}}}{\frac{4}{T}\Xi\left({T,{g_{s}},\delta}\right){\text{d}}T}}\right]$
$\displaystyle\times\exp\left[{\int_{{T_{r}}}^{{T_{*}}}{\frac{3}{T}\left({1+{\omega_{{\text{ef\/f}}}}}\right)\Xi\left({T,{g_{s}},\delta}\right)){\text{d}}T}}\right].$
(12)

As a result, we find the GW spectrum measured today, from Eq. (7), as

$\displaystyle{\Omega_{{\text{gw}}}}$
$\displaystyle={\Omega_{r0}}{\Omega_{{\text{gw}}*}}\exp\left[{\int_{{T_{*}}}^{{T_{r}}}{\frac{4}{T}\Xi\left({T,{g_{s}},\delta}\right){\text{d}}T}}\right]$
$\displaystyle\times\exp\left[{\int_{{T_{r}}}^{{T_{*}}}{\frac{3}{T}\left({1+{\omega_{{\text{ef\/f}}}}}\right)\Xi\left({T,{g_{s}},\delta}\right){\text{d}}T}}\right].$
(13)

From Eq. (II) and Eq. (II), it is found that the ratio
${{{H_{*}}}\mathord{\left/{\vphantom{{{H_{*}}}{{H_{0}}}}}\right.\kern-1.2pt}{{H_{0}}}}$
and ${\Omega_{{\text{gw}}}}$ are related to the effective equation of state
parameter ${\omega_{\text{ef\/f}}}$ and the Barrow exponent $\delta$.
Therefore, in the following content, we will discuss the influence of these
two parameters on the generation of SGWB arising from the QCD phase
transition.

##  III The equation of state of SGWB

In this section, it is necessary to analyze the influence of the equation of
state on SGWB. For the ultra-relativistic ideal gas with non-interacting
particle, the effect equation of state is equal to
${\omega_{{\text{ef\/f}}}}={1\mathord{\left/{\vphantom{13}}\right.\kern-1.2pt}3}$,
which leads to Eq. (II) and Eq. (II) turning into the following forms

$\displaystyle{\left({\frac{{{H_{*}}}}{{{H_{0}}}}}\right)^{2}}={\Omega_{r0}}{\left({\frac{{{T_{*}}}}{{{T_{0}}}}}\right)^{2\left({2+\delta}\right)}}{\left[{\frac{{{g_{s}}\left({{T_{*}}}\right)}}{{{g_{s}}\left({{T_{0}}}\right)}}}\right]^{\frac{{2\left({2+\delta}\right)}}{3}}},$
(14)

and

$\displaystyle{\Omega_{{\text{gw}}}}={\Omega_{r0}}{\Omega_{{\text{gw}}*}}.$
(15)

Obviously, in the ultra-relativistic gas with non-interacting particle case,
Eq. (II) and Eq. (II) become very concise. In particular, the SGWB spectrum
measured today (15) reduces to the original scenario, which means the Barrow
exponent $\delta$ is no longer affecting SGWB spectrum measured today.
However, QCD interactions can lead to deviation from
${\omega_{{\text{ef\/f}}}}={1\mathord{\left/{\vphantom{13}}\right.\kern-1.2pt}3}$
cha39 , resulting in SGWB that are significantly different from those in the
ideal gas case. Therefore, we explore the equation of state around QCD epoch.

In Ref. cha40 , the authors pointed out that the effect of QCD interaction can
be introduced by using the results of modern lattice calculation, which
employs the ${N_{f}}=2+1$ flavours and covers the temperature $0.1-0.4$ GeV.
The QCD equation of state resulting from the pressure parameterization due to
the strong interactions between $u$, $d$, $s$ quarks and gluons, as

$\displaystyle F\left(T\right)=\frac{P}{{{T^{4}}}}$
$\displaystyle=\frac{1}{2}\left\\{{1+\tanh\left[{{c_{\tau}}\left({\tau-{\tau_{0}}}\right)}\right]}\right\\}\frac{{{p_{i}}+\frac{{{a_{n}}}}{\tau}+\frac{{{b_{n}}}}{{{\tau^{2}}}}+\frac{{{c_{n}}}}{{{\tau^{4}}}}}}{{1+\frac{{{a_{d}}}}{\tau}+\frac{{{b_{d}}}}{{{\tau^{2}}}}+\frac{{{c_{d}}}}{{{\tau^{4}}}}}},$
(16)

where
$\tau={T\mathord{\left/{\vphantom{T{{T_{c}}}}}\right.\kern-1.2pt}{{T_{c}}}}$
with the phase transition temperature ${T_{c}}=0.145$ GeV,
${p_{i}}={{19{\pi^{2}}}\mathord{\left/{\vphantom{{19{\pi^{2}}}{36}}}\right.\kern-1.2pt}{36}}$
represents the ideal gas value of
${P\mathord{\left/{\vphantom{P{{T^{4}}}}}\right.\kern-1.2pt}{{T^{4}}}}$ for
QCD with three massless quarks. The other coefficients, at temperatures above
100 MeV, can be expressed as ${c_{\tau}}=3.6706$, ${\tau_{0}}=0.9761$,
${a_{n}}=-8.7704$, ${b_{n}}=3.9200$, ${c_{n}}=0.3419$, ${a_{d}}=-1.2600$,
${b_{d}}=0.8425$, ${c_{d}}=-0.0475$. Moreover, according to Eq. (III), the
trace anomaly can be expressed in terms of energy density and pressure as
cha41

$\displaystyle\frac{{\rho-3P}}{{{T^{4}}}}=T\frac{{{\text{d}}F\left(T\right)}}{{{\text{d}}T}}=T\frac{{\text{d}}}{{{\text{d}}T}}\left({\frac{P}{{{T^{4}}}}}\right),$
(17)

and the the effective equation of state with trace anomaly effect is given by

$\displaystyle{\omega_{{\text{ef\/f}}}}={\left[{\frac{T}{{F\left(T\right)}}\frac{{{\text{d}}F\left(T\right)}}{{{\text{d}}T}}+3}\right]^{-1}}.$
(18)

In order to probe the role of trace anomaly effect on the effective equation
of state, we display Fig. 2.

Figure 2:  The effective equation of state ${\omega_{{\text{ef\/f}}}}$ as a
function of transition temperature ${T_{*}}$.

From Fig. 2, one can see there is a very large gap between the blue dashed
curve for the effective equation of state with trace anomaly effect and black
solid curve for the ideal gas ${\omega_{{\text{ef\/f}}}}$ around the critical
temperature region ($0.1-0.2$ GeV), which indicates the effect of trace
anomaly around the QCD phase transition epoch cannot be neglected. However, at
the high temperatures (${T_{*}}\geq 5$ GeV), the two curves converge to each
other and hence are asymptotically equal.

By putting Eq. (18) into Eq. (II), the ratio between Hubble parameter at the
epoch of transition to its counterpart at current epoch
${\left({{{{H_{*}}}\mathord{\left/{\vphantom{{{H_{*}}}{{H_{0}}}}}\right.\kern-1.2pt}{{H_{0}}}}}\right)^{2}}$
takes the form

$\displaystyle{\left({\frac{{{H_{*}}}}{{{H_{0}}}}}\right)^{2}}$
$\displaystyle={\Omega_{r0}}{\left({\frac{{{T_{r}}}}{{{T_{0}}}}}\right)^{2\left({2+\delta}\right)}}{\left[{\frac{{{g_{s}}\left({{T_{r}}}\right)}}{{{g_{s}}\left({{T_{0}}}\right)}}}\right]^{\frac{{2\left({2+\delta}\right)}}{3}}}{\left[{\frac{{{g_{s}}{{\left({{T_{*}}}\right)}^{1+\omega\left({{T_{*}}}\right)}}}}{{{g_{s}}{{\left({{T_{r}}}\right)}^{1+\omega\left({{T_{*}}}\right)}}}}}\right]^{\frac{{2+\delta}}{2}}}$
$\displaystyle\times\exp\left[{\int_{{T_{r}}}^{{T_{*}}}{\frac{3}{T}\left({1+{\omega_{{\text{ef\/f}}}}}\right)\left({1+\frac{\delta}{2}}\right){\text{d}}T}}\right].$
(19)

According to the above equation, we plot the
${\left({{{{H_{*}}}\mathord{\left/{\vphantom{{{H_{*}}}{{H_{0}}}}}\right.\kern-1.2pt}{{H_{0}}}}}\right)^{2}}$
versus transition temperature ${T_{*}}$ in Fig. 3.

Figure 3: (a) The ${H_{*}}/{H_{0}}$ as a function of transition temperature
${T_{*}}$ without Barrow exponent. (b) The ${H_{*}}/{H_{0}}$ as a function of
transition temperature ${T_{*}}$ for different Barrow exponent. We set
${T_{r}}={10^{4}}{\text{GeV}}$.

In Fig. 3, we show the
${{{H_{*}}}\mathord{\left/{\vphantom{{{H_{*}}}{{H_{0}}}}}\right.\kern-1.2pt}{{H_{0}}}}$
as a function of transition temperature ${T_{*}}$ with $\delta=0$. The black
solid curve represents
${{{H_{*}}}\mathord{\left/{\vphantom{{{H_{*}}}{{H_{0}}}}}\right.\kern-1.2pt}{{H_{0}}}}$
without trace anomaly, whereas the trace anomaly case is illustrated by the
red dashed curve. It can be seen that the slope of the black solid curve
remains constant and its rises uniformly with the increase in transition
temperature ${T_{*}}$. However, when considering the effect of the trace
anomaly, the slope of red dashed curve is significantly lower than that of
black solid curve in the region ${T_{*}}\leqslant 0.13$ GeV. For
${T_{*}}>0.13$ GeV, the slope of the red dashed curve gets raised, which it
catches up with black solid curve at ${T_{*}}\approx 5$ GeV. It is again shown
that in order to study the SGWB generated from cosmological QCD phase
transition epoch, the effect of trace anomaly must be taken into account.
Therefore, we will consider the effect of trace anomaly in the followings.
From Fig. 3, by taking account the effect of trace anomaly, one can see the
relationship between
${{{H_{*}}}\mathord{\left/{\vphantom{{{H_{*}}}{{H_{0}}}}}\right.\kern-1.2pt}{{H_{0}}}}$
and transition temperature ${T_{*}}$ for different Barrow exponent. The value
of Barrow exponent decreases from top (red dashed curve) to bottom (blue solid
curve). It is clear that the ratio
${{{H_{*}}}\mathord{\left/{\vphantom{{{H_{*}}}{{H_{0}}}}}\right.\kern-1.2pt}{{H_{0}}}}$
with different Barrow exponent $\delta$ have a similar behavior with changing
transition temperature ${T_{*}}$. If fixing the transition temperature, the
ratio increases with increasing Barrow exponent.

Furthermore, applying the effect of trace anomaly and Eq. (II), the ratio
${{{\Omega_{{\text{gw}}}}}\mathord{\left/{\vphantom{{{\Omega_{{\text{gw}}}}}{{\Omega_{{\text{gw}}*}}}}}\right.\kern-1.2pt}{{\Omega_{{\text{gw}}*}}}}$
can be rewritten as

$\displaystyle\frac{{{\Omega_{{\text{gw}}}}}}{{{\Omega_{{\text{gw}}*}}}}$
$\displaystyle={\Omega_{r0}}{\left({\frac{{{T_{r}}}}{{{T_{*}}}}}\right)^{2\left({2+\delta}\right)}}{\left[{\frac{{{g_{s}}\left({{T_{r}}}\right)}}{{{g_{s}}\left({{T_{*}}}\right)}}}\right]^{\frac{2}{3}\left({2+\delta}\right)}}{\left[{\frac{{{g_{s}}{{\left({{T_{*}}}\right)}^{1+\omega\left({{T_{*}}}\right)}}}}{{{g_{s}}{{\left({{T_{r}}}\right)}^{1+\omega\left({{T_{r}}}\right)}}}}}\right]^{\frac{{2+\delta}}{2}}}$
$\displaystyle\times\exp\left[{\int_{{T_{r}}}^{{T_{*}}}{\frac{3}{T}\left({1+{\omega_{{\text{eff}}}}}\right)\left({1+\frac{\delta}{2}}\right){\text{d}}T}}\right].$
(20)

By setting ${T_{r}}={10^{4}}$ GeV and ${g_{s}}\left({{T_{r}}}\right)=106$, the
relationship between
${{{\Omega_{{\text{gw}}}}}\mathord{\left/{\vphantom{{{\Omega_{{\text{gw}}}}}{{\Omega_{{\text{gw}}*}}}}}\right.\kern-1.2pt}{{\Omega_{{\text{gw}}*}}}}$
as a function of transition temperature ${T_{*}}$ for different $\delta$ is
displayed in Fig. 4.

Figure 4:  The relationship between
${\Omega_{{\text{gw}}}}/{\Omega_{{\text{gw}}*}}$ as a function of transition
temperature ${T_{*}}$ for different $\delta$.

In Fig. 4, the black horizontal line represents the equation of state for
ultra-relativistic gas with non-interacting particles, i. e.,
${\omega_{{\text{ef\/f}}}}={1\mathord{\left/{\vphantom{13}}\right.\kern-1.2pt}3}$.
The black solid curve, red dashed curve, and blue dotted curve illustrate the
${{{\Omega_{{\text{gw}}}}}\mathord{\left/{\vphantom{{{\Omega_{{\text{gw}}}}}{{\Omega_{{\text{gw}}*}}}}}\right.\kern-1.2pt}{{\Omega_{{\text{gw}}*}}}}$
with $\delta=0$, $0.5$ and $1$, respectively. Before ${T_{*}}\leqslant
0.13{\text{GeV}}$, the
${{{\Omega_{{\text{gw}}}}}\mathord{\left/{\vphantom{{{\Omega_{{\text{gw}}}}}{{\Omega_{{\text{gw}}*}}}}}\right.\kern-1.2pt}{{\Omega_{{\text{gw}}*}}}}$
influenced by Barrow exponent is higher than the original case, whereas it is
below the original case when ${T_{*}}>0.13{\text{GeV}}$. In the high
temperature region, the QCD equation of state becomes the case of ultra-
relativistic gas with non-interacting particles, which leads to all the curves
intersect the black dotted line at $2$ GeV .

##  IV Modified QCD sources of SGWB

During the first-order phase transitions, the true vacuum bubbles nucleate and
expand in the false vacuum. Collisions of these bubbles create a quadrupole
moment in the cosmic fluid when they occur and lead to the generation of SGWB.
Generally, the dominant component of the SGWB signal are: (i) the collisions
of bubble wall collisions che1 , (ii) the sound waves che2 , and (iii) the
magnetohydrodynamic turbulence which produced after bubble collision in the
plasma che3 ; che4 . Therefore, the contributions from the three sources to
${\Omega_{gw*}}={\Omega_{gw}}\left(T_{*}\right)$, respectively, can be given
by:
(a) Bubble wall collisions (BC)

$\displaystyle\Omega_{{\text{gw}}*}^{{\text{BC}}}(\nu)={\left({\frac{{{H_{*}}}}{\beta}}\right)^{2}}{\chi_{{\text{BC}}}}\left({\frac{{0.11{\mu^{3}}}}{{0.42+{\mu^{2}}}}}\right){S_{{\text{BC}}}},{\chi_{{\text{BC}}}}={\left({\frac{{{\kappa_{{\text{BC}}}}}}{{1+\epsilon}}}\right)^{2}}.$
(21)

(b) Sound waves (SW)

$\displaystyle\Omega_{{\text{gw}}*}^{{\text{SW}}}(\nu)={\left({\frac{{{H_{*}}}}{\beta}}\right)^{2}}{\chi_{{\text{SW}}}}\mu{S_{{\text{SW}}}},{\chi_{{\text{SW}}}}={\left({\frac{{{\kappa_{{\text{SW}}}}}}{{1+\epsilon}}}\right)^{2}}.$
(22)

(c) Magnetohydrodynamic (MHD) turbulence

$\displaystyle\Omega_{{\text{gw}}*}^{{\text{MHD}}}(\nu)=\left({\frac{{{H_{*}}}}{\beta}}\right){\chi_{{\text{MHD}}}}{S_{{\text{MHD}}}},{\chi_{{\text{MHD}}}}={\left({\frac{{{\kappa_{{\text{MHD}}}}}}{{1+\epsilon}}}\right)^{\frac{3}{2}}}.$
(23)

In Eq. (21)-Eq. (23), the $\beta$ is the inverse time duration of the phase
transition, ${H_{*}}$ is the Hubble parameter at the time of production of
GWs, $\mu$ is the velocity of wall, $\epsilon$ denotes the ratio of the vacuum
energy density released in the phase transition to that of the radiation,
${\kappa_{{\text{BC}}}}$, ${\kappa_{{\text{SW}}}}$ and
${\kappa_{{\text{MHD}}}}$ are represent the fraction of the latent heat of the
phase transition deposited on the BC, SW, and MHD turbulence, respectively.
The spectral functions of the SGWB which are characterized from numerical fits
as

$\displaystyle{S_{\text{BC}}}=\frac{{3.8{{\left({{\nu\mathord{\left/{\vphantom{\nu{{\nu_{\text{BC}}}}}}\right.\kern-1.2pt}{{\nu_{\text{BC}}}}}}\right)}^{2.8}}}}{{1+2.8{{\left({{\nu\mathord{\left/{\vphantom{\nu{{\nu_{\text{BC}}}}}}\right.\kern-1.2pt}{{\nu_{\text{BC}}}}}}\right)}^{3.8}}}},$
(24a)
$\displaystyle{S_{\text{SW}}}={\left({\frac{\nu}{{{\nu_{\text{SW}}}}}}\right)^{3}}{\left[{\frac{7}{{4+3{{\left({{\nu\mathord{\left/{\vphantom{\nu{{\nu_{\text{SW}}}}}}\right.\kern-1.2pt}{{\nu_{\text{SW}}}}}}\right)}^{2}}}}}\right]^{3.5}},$
(24b)
$\displaystyle{S_{{\text{MHD}}}}=\frac{{\mu{{\left({\frac{\nu}{{{\nu_{{\text{MHD}}}}}}}\right)}^{3}}}}{{{{\left({1+\frac{\nu}{{{\nu_{{\text{MHD}}}}}}}\right)}^{\frac{{11}}{3}}}\left\\{{1+\frac{{8\pi\nu}}{{{H_{*}}}}{{\left[{\frac{{{T_{0}}}}{{{T_{*}}}}\frac{{{g_{s}}{{\left({{T_{0}}}\right)}^{\frac{{2+\delta}}{6}}}}}{{{g_{s}}{{\left({{T_{*}}}\right)}^{\frac{{2+\delta}}{6}}}}}{{\left({\frac{{{T_{0}}}}{{{T_{*}}}}}\right)}^{\frac{\delta}{2}}}}\right]}^{-1}}}\right\\}}},$
(24c)

with the peak frequencies as

$\displaystyle{\nu_{\text{BC}}}$
$\displaystyle=\frac{{0.62\beta}}{{1.8-0.1\mu+{\mu^{2}}}}\left({\frac{{{a_{*}}}}{{{a_{0}}}}}\right)$
$\displaystyle=\frac{{0.62\beta}}{{1.8-0.1\mu+{\mu^{2}}}}\frac{{{T_{0}}}}{{{T_{*}}}}\frac{{{g_{s}}{{\left({{T_{0}}}\right)}^{\frac{{2+\delta}}{6}}}}}{{{g_{s}}{{\left({{T_{*}}}\right)}^{\frac{{2+\delta}}{6}}}}}{\left({\frac{{{T_{0}}}}{{{T_{*}}}}}\right)^{\frac{\delta}{2}}},$
(25a)
$\displaystyle{\nu_{\text{SW}}}=\frac{{38\beta}}{{31\mu}}\left({\frac{{{a_{*}}}}{{{a_{0}}}}}\right)=\frac{{38\beta}}{{31\mu}}\frac{{{T_{0}}}}{{{T_{*}}}}\frac{{{g_{s}}{{\left({{T_{0}}}\right)}^{\frac{{2+\delta}}{6}}}}}{{{g_{s}}{{\left({{T_{*}}}\right)}^{\frac{{2+\delta}}{6}}}}}{\left({\frac{{{T_{0}}}}{{{T_{*}}}}}\right)^{\frac{\delta}{2}}},$
(25b) $\displaystyle{\nu_{{\text{MHD}}}}$
$\displaystyle=\frac{{7\beta}}{{4\mu}}\left({\frac{{{a_{*}}}}{{{a_{0}}}}}\right)$
$\displaystyle=\frac{{7\beta}}{{4\mu}}\left[{\frac{{{T_{0}}}}{{{T_{*}}}}\frac{{{g_{s}}{{\left({{T_{0}}}\right)}^{\frac{{2+\delta}}{6}}}}}{{{g_{s}}{{\left({{T_{*}}}\right)}^{\frac{{2+\delta}}{6}}}}}{{\left({\frac{{{T_{0}}}}{{{T_{*}}}}}\right)}^{\frac{\delta}{2}}}}\right].$
(25c)

It is should be noted that the parameters $\epsilon$ and $\kappa$ in $\chi$
play essential roles in the definition of the peak position and amplitude of
the SGWB signal, nevertheless, since they are model-dependent, there is still
no solid method to find a specific expression for $\kappa$. According to the
viewpoints in Refs. cha19 ; cha39 , we set ${\kern 1.0pt}{\kern
1.0pt}{\chi_{\text{BC}}}={\kern 1.0pt}{\kern 1.0pt}{\chi_{\text{SW}}}={\kern
1.0pt}{\kern 1.0pt}{\chi_{\text{MHD}}}=0.05$, $\beta=n{H_{*}}$, $n=5$,
$\mu=0.7$, and the Hubble parameter at phase transition epoch can be expressed
as

$\displaystyle{H_{*}}=\sqrt{\frac{{8\pi}}{{3m_{p}^{2}}}\rho\left({{T_{*}}}\right)},$
(26)

where $m_{p}$ is the Planck mass,
$\rho\left({{T_{*}}}\right)=T_{*}^{5}\left[{{{{\text{d}}F\left({{T_{*}}}\right)}\mathord{\left/{\vphantom{{{\text{d}}F\left({{T_{*}}}\right)}{{\text{d}}{T_{*}}}}}\right.\kern-1.2pt}{{\text{d}}{T_{*}}}}}\right]+3T_{*}^{4}F\left({{T_{*}}}\right)$
is the energy density at transition temperature ${T_{*}}$. It is well known
that the QCD phase transitions occurs at temperatures ranges from about a few
hundred MeV, whose exact value depends on the type of QCD matters. For the
sake of simplicity, here we assume that GW is generated immediately after the
the phase transition occurrence, hence, the temperature ${T_{*}}$ is estimated
to be the critical phase change temperature ${T_{c}}$, that is,
${T_{*}}={T_{c}}=0.145$ GeV. As a result, according to the expressions of
today’s peak frequency of the SGWB generated BC, SW, MHD turbulence, the total
peak frequency of SGWB in the Barrow entropy background is given by

$\displaystyle{\nu_{{\text{total}}}}$
$\displaystyle=\left({\frac{{0.62\beta}}{{1.8-0.1\mu+{\mu^{2}}}}+\frac{{38\beta}}{{31\mu}}+\frac{{7\beta}}{{4\mu}}}\right)\left({\frac{{{a_{*}}}}{{{a_{0}}}}}\right)$
$\displaystyle=\left({\frac{{0.62\beta}}{{1.8-0.1\mu+{\mu^{2}}}}+\frac{{38\beta}}{{31\mu}}+\frac{{7\beta}}{{4\mu}}}\right){\left[{\frac{{{g_{s}}\left({{T_{0}}}\right)}}{{{g_{s}}\left({{T_{*}}}\right)}}}\right]^{\frac{{2+\delta}}{6}}}$
$\displaystyle\times{\left({\frac{{{T_{0}}}}{{{T_{*}}}}}\right)^{1+\frac{\delta}{2}}}.$
(27)

Finally, based on Eq. (III)-Eq. (26), the total energy density of the SGWB can
be expressed as follows

$\displaystyle{\Omega_{{\text{gw}}}}{h^{2}}$
$\displaystyle=\left[{\Omega_{{\text{gw}}}^{BC}\left(\nu\right)+\Omega_{{\text{gw}}}^{SW}\left(\nu\right)+\Omega_{{\text{gw}}}^{{\text{MHD}}}\left(\nu\right)}\right]{h^{2}}$
$\displaystyle=\left[{\Omega_{gw*}^{BC}\left(\nu\right)+\Omega_{gw*}^{SW}\left(\nu\right)+\Omega_{{\text{gw}}*}^{{\text{MHD}}}\left(\nu\right)}\right]{\Omega_{r0}}{h^{2}}$
$\displaystyle\times{\left({\frac{{{T_{r}}}}{{{T_{*}}}}}\right)^{2\left({2+\delta}\right)}}{\left[{\frac{{{g_{s}}\left({{T_{r}}}\right)}}{{{g_{s}}\left({{T_{*}}}\right)}}}\right]^{\frac{2}{3}\left({2+\delta}\right)}}{\left[{\frac{{{g_{s}}{{\left({{T_{*}}}\right)}^{1+\omega\left({{T_{*}}}\right)}}}}{{{g_{s}}{{\left({{T_{r}}}\right)}^{1+\omega\left({{T_{r}}}\right)}}}}}\right]^{\frac{{2+\delta}}{2}}}$
$\displaystyle\times\exp\left[{\int_{{T_{r}}}^{{T_{*}}}{\frac{3}{T}\left({1+{\omega_{{\text{eff}}}}}\right)\left({1+\frac{\delta}{2}}\right){\text{d}}T}}\right],$
(28)

where
$h={{{H_{0}}}\mathord{\left/{\vphantom{{{H_{0}}}{100{\text{k}}{{\text{m}}^{-1}}}}}\right.\kern-1.2pt}{100{\text{k}}{{\text{m}}^{-1}}}}{\text{s
Mpc}}$. Substituting the parameters mentioned above into Eq. (IV), we show the
SGWB spectrum from QCD phase transitions due to BC, SW and MHD turbulence for
different parameter $\delta$ in Fig. 5.

Figure 5: The SGWB spectrum from QCD phase transition as a function of
frequency for different Barrow exponent $\delta$.

From the subfigures in Fig. 5, one can see that the effect of Barrow entropy
suppresses the frequency of the SGWB signal $\nu$ but has no significant
influence on its energy density ${\Omega_{{\text{gw}}}}{h^{2}}$. With these
results, one can explore the possibility of the detection of SGWB generated
from the first-order cosmological QCD phase in the framework of Barrow
entropy. In Fig. 5, the black solid curve represents the original SGWB
spectrum without Barrow entropy, whose peak frequency of the SGWB is around
$100$ nHz. In addition, the curve lies within the sensitive region of SKA,
IPTA, and NANOGrav 12.5-yr observation, which implies the SGWB signal can be
detected by SKA, IPTA, and NANOGrav. However, when considering the effect of
Barrow entropy, it is clear that the SGWB spectrum moves to the lower
frequency region. From Fig. 5, one can see that the peak frequency of the SGWB
with $\delta=0.5$ drops to around $10^{-1}$ nHz, which results in its signal
being detected only by SKA and EPTA. In the limiting case $\delta=1$, the Fig.
5 shows the peak frequency of the SGWB reduces to around $10^{-4}$ nHz, and
the probability of detection of the SGWB signal becomes very low since only
the tail of the signal reaches the projection sensitivity of the SKA. As a
result, the signal of SGWB with $\delta=1$ is the most challenging to detect,
which require more sensitive detectors for discovery.

In light of the importance role of Barrow entropy for the study of the
properties of black holes and the early universe, an increasing number of
investigations in recent years have been carried out to constrain the the
range of Barrow exponent $\delta$. According to the observational date of
Supernovae (SNIa) and Hubble parameter, Anagnostopoulos, Basilakos and
Saridakis cha42  constrained on the scenario of Barrow holographic dark energy
found outcomes of $\delta=0.094_{-0.101}^{+0.094}$. By using the Big Bang
Nucleosynthesis data, the upper bound of the exponent of Barrow entropy is
constrained to be $\delta\leqslant 1.4\times{10^{-4}}$ cha43 . Furthermore,
the data from M87* central black hole shadow and the S2 star observations have
been used to extract constrained on Barrow exponent and the result showed that
$\delta\leqslant 0.0828$ at 1$\sigma$ cha44 . More recently, based on the
modified Friedman equations due to Barrow entropy and the gravitational
baryogenesis, the authors in Ref. cha45  pointed out that the range of Barrow
exponent should be $0.005\leqslant\delta\leqslant 0.008$. Now, according to
the above results, e. g., $\delta=0.094$, we can analyze the observability of
SGWB in the framework of Barrow entropy. From Fig. 5, one can see that the
peak frequency of SGWB has shifted left to $\nu=3\times{10^{-8}}$ Hz and the
SGWB signal (purple dashed-dotted curve) lies within the observation-sensitive
region of SKA, IPTA, EPTA and NANOGrav 12.5-yr observation, which indicates
that the SGWB signal in the formwork of Barrow entropy can be detected by
detectors such as the SKA, IPTA, EPTA, and NANOGrav. In turn, one can analyze
the properties of QG and the fractal structure of the universe via the
detection of SGWB.

##  V Conclusion

In this paper, in a framework where Barrow entropy are present, we
investigated the SGWB generated by the first-order cosmological QCD phase
transition of the early univers. This work started with the assumption the
universe has expanded adiabatically, in this case the effect of Barrow entropy
can be found to be significantly influence the time variation of universe
temperature
${{{\text{d}}T}\mathord{\left/{\vphantom{{{\text{d}}T}{{\text{d}}t}}}\right.\kern-1.2pt}{{\text{d}}t}}$,
the relation between the scale factor and redshift
${{{{a_{*}}}\mathord{\left/{\vphantom{{{a_{*}}}{{a_{0}}}}}\right.\kern-1.2pt}{{a_{0}}}}}$,
the redshift in the SGWB frequency peak relative to the corresponding value at
current epoch
${{{{\nu_{0{\text{peak}}}}}\mathord{\left/{\vphantom{{{\nu_{0{\text{peak}}}}}{{\nu_{*}}}}}\right.\kern-1.2pt}{{\nu_{*}}}}}$,
as well as the density parameter of the gravitational waves at current epoch
${\Omega_{{\text{gw}}}}$. Then, based on recent lattice calculations, we
analyzed the impact of the trace anomaly on the equation of state during the
QCD phase transition and argued for itsnon-negligible contribution to the
generation of the SGWB signal. Finally, by taking account of Bubble wall
collisions, sound waves and magnetohydrodynamic turbulence as the sources of
stochastic gravitational wave, an analysis of the influence of Barrow entropy
on the total energy density and the peak signal of stochastic gravitational
wave signal is carried out. Our results showed that the effect of Barrow
entropy can effectively suppress
${{{{\nu_{0{\text{peak}}}}}\mathord{\left/{\vphantom{{{\nu_{0{\text{peak}}}}}{{\nu_{*}}}}}\right.\kern-1.2pt}{{\nu_{*}}}}}$,
the ratio of SGWB spectrum measured today to the the time of phase transition
${{{\Omega_{{\text{gw}}}}}\mathord{\left/{\vphantom{{{\Omega_{{\text{gw}}}}}{{\Omega_{{\text{gw}}*}}}}}\right.\kern-1.2pt}{{\Omega_{{\text{gw}}*}}}}$,
and the total peak signal of SGWB today ${\nu_{{\text{total}}}}$, but increase
the the evaluation of Hubble parameter from phase transition epoch to the
current time
${{{H_{*}}}\mathord{\left/{\vphantom{{{H_{*}}}{{H_{0}}}}}\right.\kern-1.2pt}{{H_{0}}}}$.
Importantly, we find that effect of Barrow entropy reduce the frequency of the
SGWB signal, so that different detectors (SKA, IPTA, NANOGrav, and EPTA) can
detect gravitational wave signals with different $\delta$. Furthermore,
according to the previous constrained works on the Barrow exponent $\delta$,
we suppose that SGWB with $\delta=0.094$ can be detected at least by SKA,
IPTA, NANOGrav and EPTA. This in turn helps people study the properties of QG
and the fractal structure of the universe.

Acknowledgements

We thank Mohamed Moussa for his enthusiastic help.

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