# Growth of Perturbations in Tsallis and Barrow Cosmology

Ahmad Sheykhi<EMAIL_ADDRESS>Department of Physics, College of
Sciences, Shiraz University, Shiraz 71454, Iran  Biruni Observatory, College
of Sciences, Shiraz University, Shiraz 71454, Iran  Bita Farsi
<EMAIL_ADDRESS>Department of Physics, College of Sciences, Shiraz
University, Shiraz 71454, Iran

###### Abstract

We disclose the effects of the entropic corrections to the Friedmann equations
on the growth of perturbations in the early stages of the universe. We
consider two types of corrections to the area law of entropy, known as Tsallis
and Barrow entropies. Using these corrections to entropy, we derive the
modified Friedmann equations and explore the growth of perturbations in a flat
universe filled with dark matter (DM) and cosmological constant. We employ the
spherically symmetric collapse formalism and work in the liner regime for the
perturbations. Interestingly enough, we find out that the profile of density
contrast quite differs from the standard cosmology in Tsallis and barrow
cosmology. We observe that the growth rate of matter perturbations crucially
depend on the values of Tsallis and Barrow parameters. With increasing these
correction parameters to the entropy, the total density contrast increases as
well. This implies that perturbations grow up faster in a universe with
modified entropy corrected Friedmann equations.

##  I Introduction

One of the main challenges of the modern cosmology is understanding the origin
and physics of the growth of perturbations in the early stages of the
universe. Indeed, these perturbations eventually lead to the large scale
structures such as galaxies and clusters of galaxy. It is a general belief
that the large scalae structures of the universe originate from gravitational
instability that amplifies very small initial density fluctuations during the
universe evolution. Such fluctuations then grow slowly over time until they
get robust enough to be detached from the background expansion. Finally, they
collapse into gravitationally-bound systems such as galaxies and clusters of
galaxy. In fact, the primordial collapsed regions serve as the initial cosmic
seeds for which the large scale structures are developed Peebles ; White .
This gives enough motivations to study the perturbations of matter and DE at
the early stages of the universe in the linear and non-linear regimes. It was
argued that the perturbations of DE may form halo structures which can
influence the DM collapsed region nonlinearly Abramo . Studies of the mutual
interactions between dark sectors of the universe, in the perturbation level,
can shed light to understand the nature of both dark components of the
universe. Influence of the DE on structure formation both at the background
level (no fluctuations) and the perturbed level has been widely explored in
the literatures Abramo2 ; Nunes ; Liberato ; Dutta ; Amendola . An appropriate
approach to investigate the growth of perturbations and structure formation is
the so called Top-Hat Spherical Collapse (SC) model Abramo . In this approach
one considers a uniform and spherical symmetric perturbation in an expanding
background and describes the growth of perturbations in a spherical region
using the same Friedmann equations for the underlying theory of gravity
Planelles ; Ziaei1 ; Ziaei2 ; Farsi .

According to thermodynamics-gravity conjecture Jacobson ; Padmanabhan1 ;
Padmanabhan2 ; Paranjape ; Kothawala ; Eling ; Padmanabhan3 ; Frolov ;
Calcagni , one can write the Friedmann equations, in any gravity theory, in
the from of the first law of thermodynamics, $dE=T_{h}dS_{h}+WdV$, on the
apparent horizon and vice versa Cai1 ; Cai2 ; Akbar1 ; Akbar2 ; Akbar3 ;
Sheykhi1 ; Sheykhi2 ; Sheykhi3 ; Sheykhi4 ; Sheykhi5 ; SheyECFE ; Sheyem . For
this purpose, one should pick up the entropy expression of the black hole in
each gravity theory and replace the black hole horizon radius $r{+}$ by the
apparent horizon radius $\tilde{r}_{A}$. While it is more convenient to apply
the Bekenstein-Hawking area law relation defined as $S_{h}=A/(4G)$ for the
black hole entropy (with the black hole horizon area $A=4\pi r_{+}^{2}$), it
should be noted that, there are several types of corrections to the area law
of entropy. Two possible corrections occur from the inclusion of quantum
effects are known as logarithmic and power-law corrections. Logarithmic
corrections, arises from the loop quantum gravity due to thermal equilibrium
fluctuations and quantum fluctuations Das ; Ashtekar ; Zhang ; Banerjee ;
SheyLog  and the power-law corrections appears in dealing with the
entanglement of quantum fields inside and outside the horizon Das2 ; Radicella
; SheyPL .

Another type of correction to the area law of entropy comes from the fact
that, in divergent partition function systems like gravitational systems,
Boltzmann-Gibbs additive entropy should be generalized to non-additive entropy
Wilk ; Gibbs ; RNunes ; Tsallis1 . Based on this, and using the statistical
arguments, Tsallis and Cirto Tsallis2  argued that the microscopic
mathematical expression of the thermodynamical entropy of a black hole does
not obey the area law and get modified as $S_{h}\sim A^{\beta}$, where $\beta$
known as nonextensive parameter, which quantifies the degree of nonextensivity
of the system Tsallis2 . Tsallis definition of entropy plays a crucial role in
studying the gravitational and cosmological systems in the framework of the
generalized statistical mechanics Czinner ; Sayahian ; Moradpour . Some
phenomenological aspects as well as some observational constraints on the
modified Friedmann equations based on Tsallis entropy were explored in Asghari
. It was argued that this model is compatible with observations Asghari .
Modified cosmology through Tsallis holographic DE (THDE) have been also
explored Tavayef ; Abd ; Bram ; Huang ; Bhattacharjee . It was shown that
depending on the value of $\beta$ parameter, THDE model can explain the
current accelerated cosmic expansion and alleviate the age problem Huang .

Recently, Barrow suggested a fractal structure for the black hole horizon and
argued that the area of the horizon may increase due to the quantum-
gravitational deformation Barrow . As a result, the area law of entropy get
modified as $S_{h}\sim A^{1+\Delta/2}$ where $\Delta$ quantifies the quantum-
gravitational deformation. The modified cosmological field equations based on
Barrow entropy have been explored in Sheykhi2 ; Saridakis1 . Later on, new
developments have appeared in the literature aiming to test the performance of
the Barrow entropy in the cosmological framework Saridakis2 . The validity and
the constraints imposed by the generalized second law of thermodynamics,
including the matter-energy content and the horizon entropy, have been
investigated in Saridakis2 . Additionally, the Barrow holographic DE (BHDE)
model was proposed in Saridakis3 ; Sri ; Adh ; Oliv  and has been tested
against the latest cosmological data in Anagnostopoulos ; Dabrowski . It was
found that BHDE describes very efficiently the late accelerated expansion of
the universe, having additionally the correct asymptotic behavior Mamon . It
is worth noting that although the Barrow entropy resembles Tsallis entropy,
however, the physical motivation and origin of them are completely different.
Indeed, Tsallis non-additive entropy correction is motivated by generalizing
standard thermodynamics to a non-extensive one, while Barrow correction to
entropy originates from intricate, fractal structure on horizon induced by
quantum-gravitational effects.

Our aim in this paper is to disclose the influences of the Tsallis and Barrow
entropy corrections on the growth of perturbations in the early stage of the
universe. For this purpose, using thermodynamics-gravity conjecture we extract
the modified Friedmann equations based on Tsallis/Barrow entropy Sheykhi1 ;
Sheykhi2 . We then employ the SC approach, and investigate the evolution of DM
and DE perturbations in the framework of the Tsallis and Barrow cosmology.

The paper is structured as follows. In section II, we provide a review on the
modified Friedmann equations in Tsallis and Barrow cosmology. In section III,
we examine the growth of matter perturbations in Tsallis cosmology with using
the SC approach. In section IV, we explore the growth of perturbations in the
background of Barrow cosmology. We finish our paper with concluding remarks in
section V.

##  II Modified Friedmann equations in Tsallis and Barrow Cosmology

We start with a spatially homogeneous and isotropic spacetime with line
elements

$ds^{2}={h}_{\mu\nu}dx^{\mu}dx^{\nu}+\tilde{r}^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}),$ (1)

where $\tilde{r}=a(t)r$, $x^{0}=t,x^{1}=r$, and $h_{\mu\nu}$=diag
$(-1,a^{2}/(1-kr^{2}))$ represents the two dimensional metric. The open, flat,
and closed universes corresponds to $k=-1,0,1$, respectively. We assume our
universe is bounded by an apparent horizon, which its radius is Sheykhi3

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

The apparent horizon is a suitable boundary for which the first and second law
of thermodynamics hold on it. The temperature associated with the apparent
horizon has the form Cai2 ; Sheykhi3

$T_{h}=-\frac{1}{2\pi\tilde{r}_{A}}\left(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\right).$
(3)

We further assume the energy content of the universe is in the form of perfect
fluid,

$T_{\mu\nu}=(\rho+p)u_{\mu}u_{\nu}+pg_{\mu\nu},$ (4)

where $\rho$ and $p$ are, respectively, the energy density and pressure of the
matter inside the Universe. The total energy content of the Universe satisfies
the conservation equation, $\nabla_{\mu}T^{\mu\nu}=0$, which yields the
continuity equation,

$\dot{\rho}+3H(\rho+p)=0.$ (5)

In the above relation $H=\dot{a}/a$ stands for the Hubble parameter which
measures the rate of the universe expansion. In an expanding universe, as a
thermodynamic system, we need to define a work term in the first law of
thermodynamics. The work density for an expanding universe may be obtained as
Hay

$W=-\frac{1}{2}T^{\mu\nu}h_{\mu\nu},\ \ \Rightarrow\ \ W=\frac{1}{2}(\rho-p).$
(6)

We further assume the first law of thermodynamics on the apparent horizon is
satisfied and has the form

$dE=T_{h}dS_{h}+WdV,$ (7)

where $V=\frac{4\pi}{3}\tilde{r}_{A}^{3}$ is the volume enveloped by a
3-dimensional sphere, and $T_{h}$ and $W$ are given by (3) and (6). In the
above expression, $S_{h}$ stands for the entropy associated with the apparent
horizon. The total energy of the universe is $E=\rho V$, which after taking
differential and using the continuity equation (5), leads to

$dE=4\pi\tilde{r}_{A}^{2}\rho d\tilde{r}_{A}-4\pi
H\tilde{r}_{A}^{3}(\rho+p)dt.$ (8)

In order to derive the Friedmann equations from the first law of
thermodynamics (7), we need to define the entropy expression associated with
the boundary of the Universe. In this paper, we consider two modification for
the area law, known as Tsallis and Barrow entropy.

###  II.1 Modified Friedmann equations through Tsallis entropy

The Tsallis entropy associated with the boundary of the Universe is given by
Tsallis2

$\displaystyle S_{h}=\gamma A^{\beta},$ (9)

where $A$ is the apparent horizon area, $\gamma$ is an unknown constant and
$\beta$ known as Tsallis parameter or nonextensive parameter, which is a real
parameter which quantifies the degree of non-extensivity Tsallis2 . When
$\beta=1$ and $\gamma=1/(4L_{p}^{2})$, the well-known area law of entropy is
recovered. Using Tsallis entropy in the first law of thermodynamics on the
apparent horizon, we derive the differential form of the modified Friedmann
equation Sheykhi1

$-\frac{2}{\tilde{r}_{A}^{3}}\left(4\pi\tilde{r}_{A}^{2}\right)^{\beta-1}d\tilde{r}_{A}=\frac{2\pi}{3\gamma\beta}d\rho.$
(10)

Integrating, we arrive at the modified Friedmann equation based on Tsallis
entropy Sheykhi1

$\left(H^{2}+\frac{k}{a^{2}}\right)^{2-\beta}=\frac{8\pi
L_{p}^{2}}{3}\rho+\frac{\Lambda}{3},$ (11)

where $\Lambda$ is the integration constant which can be interpreted as the
cosmological constant, and we have defined

$\gamma\equiv\frac{2-\beta}{4\beta L_{p}^{2}}\left(4\pi\right)^{1-\beta}.$
(12)

The above Friedmann equation can be rewritten as

$\left(H^{2}+\frac{k}{a^{2}}\right)^{2-\beta}=\frac{8\pi
L_{p}^{2}}{3}(\rho+\rho_{\Lambda}),$ (13)

where $\rho_{\Lambda}=\Lambda/(8\pi L_{p}^{2})$ is the energy density of the
cosmological constant. Since $\gamma>0$, thus from Eq. (12) we have $\beta<2$.
The second modified Friedmann equation can be easily derived by combining the
continuity equation (5) with the first Friedmann equation (13). We find
Sheykhi1

$\displaystyle(4-2\beta)\frac{\ddot{a}}{a}\left(H^{2}+\frac{k}{a^{2}}\right)^{1-\beta}+(2\beta-1)\left(H^{2}+\frac{k}{a^{2}}\right)^{2-\beta}$
$\displaystyle=-8\pi L_{p}^{2}(p+p_{\Lambda}).$ (14)

where $p_{\Lambda}=-\Lambda/(8\pi L_{p}^{2})$ is the vacuum (cosmological
constant) pressure. As usual, we define the density parameters as

$\displaystyle\Omega_{m}=\frac{\rho_{m}}{\rho_{c}},\ \ \
\Omega_{\Lambda}=\frac{\rho_{\Lambda}}{\rho_{c}},\ \ \
\rho_{c}=\frac{3H^{4-2\beta}}{8\pi L_{p}^{2}}.\ \ $ (15)

Therefore, in terms of the density parameters, the first Friedmann equation
(13) can be written as

$\displaystyle\Omega_{m}+\Omega_{\Lambda}=(1+\Omega_{k})^{2-\beta},$ (16)

where the curvature density parameter is defined as usual,
$\Omega_{k}=k/(a^{2}H^{2})$, and we have assumed $\rho=\rho_{m}$ by neglecting
the radiation.

###  II.2 Modified Friedmann equations through Barrow entropy

The Barrow entropy associated to the apparent horizon is given by Barrow

$\displaystyle S_{h}=\left(\frac{A}{A_{0}}\right)^{1+\Delta/2},$ (17)

where $A$ is apparent horizon area and $A_{0}$ is the Planck area. The
exponent $\Delta$ ranges as $0\leq\Delta\leq 1$ and stands for the amount of
the quantum-gravitational deformation effects Barrow . When $\Delta=0$, the
area law is restored and $A_{0}\rightarrow 4G$, while $\Delta=1$ represents
the most intricate and fractal structure of the horizon. Using the
thermodynamics-gravity conjecture, the differential form of the Friedmann
equation derived from the first law of thermodynamics, based on Barrow
entropy, is given by Sheykhi2

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

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

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

where $\Lambda$ is a constant of integration which can be interpreted as the
cosmological constant, and $G_{\rm eff}$ stands for the effective Newtonian
gravitational constant,

$G_{\rm
eff}\equiv\frac{A_{0}}{4}\left(\frac{2-\Delta}{2+\Delta}\right)\left(\frac{A_{0}}{4\pi}\right)^{\Delta/2}.$
(20)

If we define $\rho_{\Lambda}={\Lambda}/(8\pi G_{\rm eff})$, Eq. (19), can be
rewritten as

$\left(H^{2}+\frac{k}{a^{2}}\right)^{1-\Delta/2}=\frac{8\pi G_{\rm
eff}}{3}(\rho+\rho_{\Lambda}).$ (21)

Combining the modified Friedmann equation (21) with the continuity equation
(5), we arrive at

$\displaystyle(2-\Delta)\frac{\ddot{a}}{a}\left(H^{2}+\frac{k}{a^{2}}\right)^{-\Delta/2}+(1+\Delta)\left(H^{2}+\frac{k}{a^{2}}\right)^{1-\Delta/2}$
$\displaystyle=-8\pi G_{\rm eff}(p+p_{\Lambda}),$ (22)

where $p_{\Lambda}=-{\Lambda}/(8\pi G_{\rm eff})$. This is the second modified
Friedmann equation governing the evolution of the universe based on Barrow
entropy. In the limiting case where $\Delta=0$ ($G_{\rm eff}\rightarrow G$),
Eqs. (21) and (22) reduce to the Friedmann equation in standard cosmology.

In this case, we define the critical density parameter as
$\rho_{c}=3H^{2-\Delta}/(8\pi G_{\rm eff})$, and thus the first Friedmann
equation (21) can be rewritten as

$\displaystyle\Omega_{m}+\Omega_{\Lambda}=(1+\Omega_{k})^{1-\Delta/2}.$ (23)

Now we are going to show that the modified Friedmann equations based on
Tsallis/Barrow entropy derived in Eqs. (13) and (21) can describe the late
time accelerated expansion. For a flat universe filled with pressureless
matter ($p=p_{m}=0$) and cosmological constant, we have
$\Omega_{m}+\Omega_{\Lambda}=1.$ The total equation of state parameter can be
written

$\displaystyle\omega_{\rm
t}=\frac{p_{\Lambda}}{\rho_{m}+\rho_{\Lambda}}=-\frac{1}{\rho_{m}/\rho_{\Lambda}+1},$
(24) $\displaystyle\Rightarrow\omega_{\rm
t}(z)=-\frac{\Omega_{\Lambda,0}}{(1-\Omega_{\Lambda,0})(1+z)^{3}+\Omega_{\Lambda,0}},$

where $\rho_{\Lambda}=\rho_{\Lambda,0}$, and $\rho_{m}=\rho_{m,0}(1+z)^{3}$.
If we take $\Omega_{\Lambda,0}\simeq 0.7$ and $\Omega_{m,0}\simeq 0.3$, we
have

$\displaystyle\omega_{\rm t}(z)=-\frac{0.7}{0.7+0.3(1+z)^{3}}.$ (25)

At the present time where $z\rightarrow 0$, we have $\omega_{\rm t}=-0.7$,
while at the early universe where $z\rightarrow\infty$, we get $\omega_{\rm
t}=0$. This implies that at the early stages, the universe undergoes a
decelerated phase while at the late time it experiences an accelerated phase.

Figure 1: The behavior of $\omega_{\rm t}$ in terms of redshift $z$ for
different values of $\Omega_{\Lambda,0}$.

In Fig. 1 we have plotted the $\omega_{t}$ versus $z$ for different values of
$\Omega_{\Lambda,0}$. This figure confirms an accelerated universe at the late
time.

##  III Growth of perturbation in Tsallis cosmology

In a spatially flat universe, the modified Friedmann equation (13) can be
written as

$\displaystyle H^{4-2\beta}=\dfrac{1}{3}\left(\rho_{m}+\Lambda\right),$ (26)

where we have taken $8\pi L_{p}^{2}=1$ and hence $\rho_{\Lambda}=\Lambda$. The
second Friedmann equation (14) based on Tsallis entropy, is given by

$\displaystyle(4-2\beta)\dfrac{\ddot{a}}{a}H^{2-2\beta}+(2\beta-1)H^{4-2\beta}=\Lambda,$
(27)

where $p_{m}=0$ and $p_{\Lambda}=-\Lambda$. The Hubble expansion rate can be
obtained as

$H^{2}=\left(\dfrac{\rho_{m}+\Lambda}{3}\right)^{1/(2-\beta)}.$ (28)

We can also define the normalized Hubble parameter as

$E(z)=\frac{H(z)}{H_{0}}=\left[(1-\Omega_{\Lambda,0})(1+z)^{3}+\Omega_{\Lambda,0}\right]^{1/(4-2\beta)}.$
(29)

The evolution of the normalized Hubble parameter versus $z$ for different
values of $\beta$ is plotted in Fig. 2. As we can see, in Tsallis cosmology
the Hubble parameter is smaller than $\Lambda$CDM model, implying that in this
model, our Universe expands slower. Also the Hubble parameter decreases with
decreasing the non-extensive parameter $\beta$.

Figure 2: The behavior of the normalized Hubble rate $E(z)$ for different
values of $\beta$ in Tsallis cosmology, where we have taken
$\Omega_{\Lambda,0}=0.70$.

The deceleration parameter in terms of the redshift can be written as

$\displaystyle
q=-1-\dfrac{\dot{H}}{H^{2}}=-1+\dfrac{(1+z)}{H(z)}\dfrac{dH(z)}{dz}$
$\displaystyle=-1+\dfrac{3}{2}\dfrac{(1+z)^{3}(1-\Omega_{\Lambda,0})}{(2-\beta)[(1-\Omega_{\Lambda,0})(1+z)^{3}+\Omega_{\Lambda,0}]}.$
(30)

We have plotted the behavior of the deceleration parameter $q(z)$ for
different $\beta$ parameter in Fig. 3. We observe that the universe
experiences a transition from a decelerating phase ($q>0$) to an accelerating
phase ($q<0$), at redshift around $z_{tr}\approx 0.6$ compatible with
observations. It is seen that $z_{tr}$ depends on the non-extensive parameter
$\beta$ and decreases with increasing $\beta$.

Figure 3: The behavior of the deceleration parameter $q(z)$ as a function of
redshift for different $\beta$ ($\beta=1$ corresponds to $\Lambda$CDM model).
Here we have taken $\Omega_{\Lambda,0}=0.70$.

Another quantity which is helpful in understanding the phase transitions of
the universe is called the jerk parameter. This is a dimensionless quantity
obtained by taking the third derivative of the scale factor with respect to
the cosmic time, provides a comparison between different DE models and the
$\Lambda$CDM $(j=1)$ model. The jerk parameter is defined as

$j=\dfrac{1}{aH^{3}}\dfrac{d^{3}a}{dt^{3}}=q(2q+1)+(1+z)\dfrac{dq}{dz}.$ (31)

For the $\Lambda$CDM model, the value of $j$ is always unity. A
non-$\Lambda$CDM model occurs if there is any deviation from the value of
$j=1$. From Fig. 4 we observe that the jerk parameter is smaller (larger) than
$\Lambda$CDM for $\beta<1$ ($\beta>1$).

Figure 4: The evolution of jerk parameter with respect to redshift for
different values of $\beta$ parameter.

Combining Eqs. (26) and (27), we arrive at

$\displaystyle\dfrac{\ddot{a}}{a}=\dfrac{1}{4-2\beta}\Bigg{\\{}(1-2\beta)\left(\dfrac{\rho_{m}+\Lambda}{3}\right)^{1/(2-\beta)}$
$\displaystyle+\Lambda\left(\dfrac{\rho_{m}+\Lambda}{3}\right)^{(\beta-1)/(2-\beta)}\Bigg{\\}}.$
(32)

The conservation equations take the form

$\dot{\rho}_{m}+3H\rho_{m}=0.$ (33)

This equation describes the density evolution of pressureless matter labeled
by $m$ with background density $\rho_{m}$. The SC model describes a spherical
region with uniform density so that at time $t$,
$\rho_{m}^{c}=\rho_{m}(t)+\delta\rho_{m}$. We can write the conservation
equation for spherical perturbed region of radius $a_{p}$ as

$\dot{\rho}_{m}^{c}+3h\rho_{m}^{c}=0,$ (34)

where $h=\dot{a_{p}}/a_{p}$ is the local expansion rate of the spherical
perturbed region of radius $a_{p}$. Since Eq. (32) valid for all spacetime
region, therefore we further assume it holds for the spherical perturbed
region with radius $a_{p}$, namely,

$\displaystyle\dfrac{\ddot{a}_{p}}{a_{p}}=\dfrac{1}{4-2\beta}\Bigg{\\{}(1-2\beta)\left(\dfrac{\rho_{m}^{c}+\Lambda}{3}\right)^{{1}/(2-\beta)}$
$\displaystyle+\Lambda\left(\dfrac{\rho_{m}^{c}+\Lambda}{3}\right)^{(\beta-1)/(2-\beta)}\Bigg{\\}},$
(35)

where for simplicity, we assume $\beta^{c}=\beta$. The density contrast of a
single fluid labeled by $\delta_{m}$, is defined as Ziaei1

$\delta_{m}=\dfrac{\rho_{m}^{c}}{\rho_{m}}-1=\dfrac{\delta\rho_{m}}{\rho_{m}}.$
(36)

Taking the time derivative of Eq. (36) and using Eqs. (33) and (34), yields

$\dot{\delta_{m}}=3(1+\delta_{m})(H-h).$ (37)

If we take the second derivative with respect to time, we find

$\ddot{\delta}_{m}-\dfrac{\dot{\delta}_{m}^{2}}{1+\delta_{m}}=3(\dot{H}-\dot{h})(1+\delta_{m}),$
(38)

where we have used Eq. (37). Combining Eqs. (32), (III) and (36), we can find
the following equation

$\displaystyle\dot{H}-\dot{h}$ $\displaystyle=$
$\displaystyle-H^{2}+h^{2}+\dfrac{2\beta-1}{2(2-\beta)^{2}}\left(\dfrac{\rho_{m}}{3}\right)^{1/(2-\beta)}\delta_{m}$
(39)
$\displaystyle+\dfrac{\Lambda(1-\beta)}{2(2-\beta)^{2}}\left(\dfrac{\rho_{m}}{3}\right)^{(\beta-1)/(2-\beta)}\delta_{m},$

where we have expanded the first and second terms in matter dominated era and
in the linear term of $\delta_{m}$. This is due to the fact that we work in
the early universe which $\rho_{\Lambda}/\rho_{m}<1$ and in the linear regime
where $\delta_{m}<1$. Therefore, Eq. (38) with using Eqs. (37) and (39) in
linear regime can be rewritten as

$\displaystyle\ddot{\delta}_{m}+2H\dot{\delta}_{m}-\dfrac{3}{2}\dfrac{2\beta-1}{(2-\beta)^{2}}\left(\dfrac{\rho_{m}}{3}\right)^{1/(2-\beta)}\delta_{m}$
$\displaystyle-\dfrac{3}{2}\dfrac{(1-\beta)\Lambda}{(2-\beta)^{2}}\left(\dfrac{\rho_{m}}{3}\right)^{(\beta-1)/(2-\beta)}\delta_{m}=0,$
(40)

where the matter energy density is given by $\rho_{m}=\rho_{m,0}a^{−3}$. In
order to study the evolution of the density contrast $\delta_{m}$ in terms of
the redshift parameter, $1+z=a^{-1}$, we first replace the time derivatives
with the derivatives with respect to the scale factor $a$. It is a matter of
calculations to show that

$\displaystyle\dot{\delta}_{m}=aH\delta^{\prime}_{m},$
$\displaystyle\ddot{\delta}_{m}=aH^{2}\left(\dfrac{1-2\beta+\Lambda
H^{2\beta-4}}{4-2\beta}\right)\delta^{\prime}_{m}+a^{2}H^{2}\delta^{\prime\prime}_{m}$
(41)

where the prime stands for the derivative with respect to the scale factor
$a$. Combining Eqs. (III) and (41), we arrive at

$\displaystyle\delta^{\prime\prime}_{m}+\dfrac{1}{a(4-2\beta)}\left[(9-6\beta)+\Lambda
H^{2\beta-4}\right]\delta^{\prime}_{m}$
$\displaystyle+\Bigg{\\{}(1-2\beta)\rho^{{1}/(2-\beta)}_{m}-3(1-\beta)\Lambda\rho^{(\beta-1)/(2-\beta)}_{m}\Bigg{\\}}$
$\displaystyle\times\dfrac{3^{(1-\beta)/(2-\beta)}}{a^{2}H^{2}}\dfrac{\delta_{m}}{2(2-\beta)^{2}}=0.$
(42)

It should be noted that in standard cosmology ($\beta=1$) and in the absence
of the cosmological constant ($\Lambda=0$), Eq. (42) reduces to

$\delta^{\prime\prime}_{m}+\dfrac{3}{2a}\delta^{\prime}_{m}-\dfrac{3}{2a^{2}}\delta_{m}=0,$
(43)

which is the result obtained in standard cosmology Abramo . In Fig. 5, we plot
the matter density contrast $\delta_{m}$ as a function of redshift for
different values of $\beta$ parameter and for redshifts $10<z<100$.

Figure 5: Growth of matter perturbations (density contrast $\delta_{m}$) for
different values of $\beta$ in Tsallis cosmology, where the dashed-line for
$\beta=1.06$, solid-line for $\Lambda$CDM ($\beta=1$), and dash-dotted line
for $\beta=0.94$. Here we take $\delta_{m}(z_{i})=0.0001$.

We can see that non-extensive parameter $\beta$ affects the growth of matter
perturbations, in particular in the lower redshifts. Indeed the growth of
perturbations increases with increasing the non-extensive Tsallis parameter
$\beta$. Further, we find out that, in lower redshifts, the matter
perturbations for $\beta<1$ grows slower compared to standard cosmology
($\beta=1$), while for $\beta>1$ it grows up faster.

##  IV Growth of perturbation in Barrow cosmology

In a flat FRW universe, the modified Friedmann equations, based on Barrow
entropy, can be rewritten

$\displaystyle H^{2-\Delta}=\Gamma(\rho_{m}+\rho_{\Lambda}),$ (44)
$\displaystyle\left(2-\Delta\right)\dfrac{\ddot{a}}{a}H^{-\Delta}+\left(1+\Delta\right)H^{2-\Delta}$
$\displaystyle=-3\Gamma(p_{m}+p_{\Lambda}),$ (45)

where

$\Gamma\equiv\dfrac{1}{3}\left(\dfrac{2-\Delta}{2+\Delta}\right)\left(\dfrac{1}{8\pi^{2}}\right)^{{\Delta}/{2}}.$
(46)

Note that here we work in the unit where $k_{B}=c=\hslash=1$, and thus
$A_{0}=4G$. We also assume $8\pi G=1$, for simplicity. In terms of the density
parameters, the first Friedmann equation (44) can be written
$\Omega_{m}+\Omega_{de}=1$, where the dimensionless density parameters are now
defined as

$\Omega_{m}\equiv\dfrac{\Gamma\rho_{m}}{H^{2-\Delta}},\ \ \ \ \
\Omega_{de}\equiv\dfrac{\Gamma\rho_{de}}{H^{2-\Delta}}.$ (47)

We can write the normalized Hubble parameter as

$E(z)=\dfrac{H(z)}{H_{0}}=\left[(1-\Omega_{\Lambda,0})(1+z)^{3}+\Omega_{\Lambda,0}\right]^{1/(2-\Delta)}.$
(48)

The evolution of the normalized Hubble parameter versus $z$ for different
values of $\Delta$ is plotted in Fig. 6. It is seen that the Hubble parameter
is larger that $\Lambda$CDM $(\Delta=0)$ model, indicating that in Barrow
cosmology the universe expands faster. Also, the Hubble parameter decreases
with decreasing the Barrow parameter $\Delta$.

Figure 6: The behavior of the normalized Hubble rate for different values of
$\Delta$ where we have taken $\Omega_{\Lambda,0}=0.7$.

The deceleration parameter in terms of the redshift can be written as

$\displaystyle
q=-1-\dfrac{\dot{H}}{H^{2}}=-1+\dfrac{(1+z)}{H(z)}\dfrac{dH(z)}{dz}$
$\displaystyle=-1+\dfrac{3(1+z)^{3}(1-\Omega_{\Lambda,0})}{(2-\Delta)\left((1-\Omega_{\Lambda,0})(1+z)^{3}+\Omega_{\Lambda,0}\right)}.$
(49)

We have plotted the behavior of the deceleration parameter $q(z)$ for
different $\Delta$ parameter in Fig. 7. We observe that the universe
experiences a transition from decelerating phase ($z>z_{tr}$) to accelerating
phase ($z<z_{tr}$). Also we can see that with increasing $\Delta$, the phase
transition between deceleration and acceleration take place at lower
redshifts.

Figure 7: The behavior of the deceleration parameter as a function of redshift
for different $\Delta$. Here we take $\Omega_{\Lambda,0}=0.7$.

Similar to previous section, we can find the jerk parameter in this model.
From Fig. 8 we observe the jerk parameter for all values of $\Delta$ parameter
is greater than $\Lambda$CDM model.

Figure 8: The evolution of jerk parameter with respect to redshift for
different values of $\Delta$ parameter.

Combining Eqs. (44) and (45), we find

$\displaystyle\dfrac{\ddot{a}}{a}=\frac{\Gamma^{2/(2-\Delta)}}{\Delta-2}\Bigg{\\{}(1+\Delta)\left(\rho_{m}+\Lambda\right)^{2/(2-\Delta)}$
$\displaystyle-3\Lambda\left(\rho_{m}+\Lambda\right)^{\Delta/(2-\Delta)}\Bigg{\\}}.$
(50)

In order to study the growth of perturbations in Barrow cosmology, following
the approach of the previous section, we employ the SC method. We can write
down for the spherical perturbed region of radius $a_{p}$,

$\displaystyle\frac{\ddot{a}_{p}}{a_{p}}=\frac{\Gamma^{2/(2-\Delta)}}{\Delta-2}\Bigg{\\{}(1+\Delta)\left(\rho_{m}^{c}+\Lambda\right)^{2/(2-\Delta)}$
$\displaystyle-3\Lambda\left(\rho_{m}^{c}+\Lambda\right)^{\Delta/(2-\Delta)}\Bigg{\\}},$
(51)

Combining Eqs. (36), (50) and (IV), we arrive at

$\displaystyle\dot{H}-\dot{h}+H^{2}-h^{2}=\Bigg{\\{}\dfrac{2(1+\Delta)}{(2-\Delta)^{2}}(\Gamma\rho_{m})^{2/(2-\Delta)}$
$\displaystyle-\dfrac{3\Delta}{(2-\Delta)^{2}}\Lambda(\Gamma\rho_{m})^{\Delta/(2-\Delta)}\Bigg{\\}}\delta_{m}.$
(52)

Using Eqs. (37), (38) and (52), after a simple calculations, we find

$\displaystyle\ddot{\delta}_{m}+2H\dot{\delta}_{m}-\Bigg{\\{}\dfrac{6(1+\Delta)}{(2-\Delta)^{2}}(\Gamma\rho_{m})^{2/(2-\Delta)}$
$\displaystyle-\dfrac{9\Delta}{(2-\Delta)^{2}}\Lambda(\Gamma\rho_{m})^{\Delta/(2-\Delta)}\Bigg{\\}}\delta_{m}=0.$
(53)

Changing our variable from time to the scale factor, we can rewrite the above
equations as

$\displaystyle\delta^{\prime\prime}_{m}+\dfrac{3}{a(2-\Delta)}\left[(1-\Delta)+\Gamma\Lambda
H^{\Delta-2}\right]\delta^{\prime}_{m}$
$\displaystyle-\Bigg{\\{}2(1+\Delta)(\Gamma\rho_{m})^{2/(2-\Delta)}-3\Delta\Lambda(\Gamma\rho_{m})^{\Delta/(2-\Delta)}\Bigg{\\}}$
$\displaystyle\times\dfrac{3}{a^{2}H^{2}}\dfrac{\delta_{m}}{(2-\Delta)^{2}}=0.$
(54)

It should be noted that in standard cosmology ($\Delta=0$) and with the
absence of the cosmological constant ($\Lambda=0$), this equation reduces to
Eq. (43). In Fig. 9, we have plotted the matter density contrast as a function
of redshift for different values of $\Delta$ parameter and for redshifts
$10<z<100$.

Figure 9: Growth of matter perturbations (density contrast $\delta_{m}$) for
different values of $\Delta$, where dashed-line for $\Delta=0.1$, dash-dotted
line for $\Delta=0.05$ and solid-line for $\Lambda CDM$. Here we take
$\delta_{m}(z_{i})=0.0001$

We can see that Barrow exponent $\Delta$ affects the growth of total
perturbations, in particular in the lower redshifts. Indeed, in low redshifts,
the total density contrast increases with increasing $\Delta$ parameter. This
implies that in Barrow cosmology, the perturbations grow up faster compared to
standard cosmology. This is due to the fact that in Barrow cosmology, the
spacetime has quantum-gravitational deformation, and intricate fractal
structure, which can support the growth of perturbations in energy density.

##  V Concluding remarks

In this paper, we have investigated the modified cosmological field equations
of flat FRW universe when the entropy associated with the apparent horizon is
in the form of Tsallis/Barrow entropy. We have explored the growth of
perturbations in the context of Tsallis/Barrow cosmology. We have considered
the energy/matter content of the universe in the form of matter (baronic
matter and DM) and cosmological constant. Employing the SC approach for the
growth of perturbations, we extracted the differential equation for the
evolution of the matter perturbations. Further, for the matter density
contrast, we observed that the profile of the growth of perturbations differ
from standard cosmology. Indeed, in Tsallis cosmology and in lower redshifts,
the perturbations grow slower for $\beta<1$ compared to the standard cosmology
($\beta=1$), while they grow faster for $\beta>1$. In Barrow cosmology,
however, the growth rate of the total perturbations always became faster
compared to the standard cosmology.

Also we find that both Tsallis and Barrow cosmology can explain the phase
transition from a decelerating to an accelerating universe. In contrast to
Tsallis cosmology, we observed that Barrow cosmology predict that the universe
expands with greater rate compared to $\Lambda$CDM model.

It is important to note that although the modified Friedmann equations based
on Tsallis and Barrow entropy are similar to each other, but the growth of
perturbation differs in some cases in these models. This is due to the fact
that the non-extensive Tsallis parameter $\beta$ takes different value from
Barrow exponent $\Delta$. Indeed non-extensive Tsallis parameter has an upper
bound, $\beta<2$, while Barrow exponent ranges as $0\leq\Delta\leq 1$.
Besides, the origin of these two corrections to the entropy are completely
different.

Finally, it is worth mentioning that Barrow entropy correction to the area law
is just a first approximation on the subject of quantum gravitational
implications of the black hole horizons. In reality the underlying spacetime
foam deformation will be complex, wild and dynamical. So one could think of an
exponent $\Delta$ that depends on time and scale, as it has already been done
with Tsallis entropy exponent S. Nojiri . In order to constrain the Barrow
exponent $\Delta$ and Tsallis nonextensive parameter $\beta$, one may perform
a full observational confrontation using data from Supernovae type Ia data
(SNIa), Cosmic Microwave Background (CMB) shift parameters, Baryonic Acoustic
Oscillations (BAO), growth rate and Hubble data observations. These issues are
beyond the scope of the present work and we leave them for the future
investigations.

###### Acknowledgements.

We are grateful to the referee for very helpful and constructive comments
which helped us improve our paper significantly.

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11institutetext:  PRISMA+ Cluster of Excellence, Institute of Physics,
Johannes Gutenberg University, Mainz, Germany,
Department of Physics, Stony Brook University, USA

# The Global Network of Cavities to Search for Gravitational Waves (GravNet):
A novel scheme to hunt gravitational waves signatures from the early universe

Kristof Schmieden 11 Matthias Schott 11221122

###### Abstract

The idea of searching for gravitational waves using cavities in strong
magnetic fields has recently received significant attention. Specifically,
discussions focus on cavities with relatively small volumes, which are
currently employed in the search for axions. In this context, we propose a
novel experimental scheme that enables the detection of gravitational waves in
the GHz regime, which could originate, for example, from primordial black hole
mergers. The scheme is based on synchronous measurements of cavity signals
from multiple devices operating in magnetic fields at distant locations. While
gravitational wave signatures might be detectable in individual cavities,
distinguishing them from noise is highly challenging. By analyzing the
correlation among signals from several, possibly geographically separated
cavities, it is not only possible to significantly enhance the signal-to-noise
ratio but also to investigate the source of those gravitational wave
signatures. In the context of this proposal, a first demonstration experiment
with one superconducting cavity is currently conducted, which is the basis of
the proposed data-analysis approaches. On this basis the prospects of GravNet
(Global Network of Cavities to Search for Gravitational Waves) are outlined in
the paper.

###### Contents

  1. 1 Introduction
  2. 2 Sources of high frequency gravitational waves
  3. 3 Basic Considerations on Cavity Designs
    1. 3.1 The SUPAX Cavity Design
    2. 3.2 Combination of several cavities with a coherent signal
    3. 3.3 Single Photon Counting in Cavities
  4. 4 GravNet as Resonant Cavity Experiment (GravNet-1)
    1. 4.1 Setup
    2. 4.2 Sensitivities
    3. 4.3 Possible Extensions
  5. 5 GravNet as Photon Counting Experiment (GravNet-2)
    1. 5.1 Setup
    2. 5.2 Sensitivities
  6. 6 Conclusion

##  1 Introduction

The detection of gravitational waves (GWs) by the LIGO and Virgo
interferometers [1] marked the beginning of a new era in astronomy.
Gravitational waves, with frequencies spanning from super massive binary black
hole systems in the nHz regime to kHz for compact binary objects and up to GHz
for GWs from the cosmic gravitational wave background [2], are an essential
part of our understanding of the universe.

Interferometers, like LIGO and Virgo, have proven to be highly successful in
detecting GWs, and future generations, such as the Einstein Telescope [3], are
in the design phase. An alternative concept for GW detection exploits their
coupling to the electromagnetic field, using radio frequency cavities, either
pumped or placed in a static magnetic field. Recently, the latter approach has
been discussed in more detail [4, 5, 6], especially in the context of searches
for axion-like particles [7, 8, 9].

The basic principle behind the cavity-based experiments is simple: a
gravitational wave distorts the cavity’s shape, altering the magnetic flux
through the cavity and generating an electric signal that can be detected.
Additional, the GW couples directly to the EM field via the inverse
Gertsenshtein effect. Hence, a gravitational wave that is passing through a
cavity within a static magnetic field, creates an effective current in
Maxwell’s equations, leading to an electromagnetic field that oscillates at
the same frequency as the gravitational wave. The induced electromagnetic
field can be resonantly enhanced using microwave cavities and the generated
radio frequency power detected.

The sensitivity of such experiments depends on the GW frequency, incoming
direction, the cavity’s resonance frequencies, and the external magnetic field
strength. The sensitivity to gravitational waves using a cavity-based
experiment has been derived in [4] and can be summarised by the signal power

$P_{sig}=\frac{1}{2}Q\omega_{g}^{3}V^{5/3}(\eta_{n}h_{0}B_{0})^{2}\frac{1}{\mu_{0}c^{2}},$
(1)

with $\omega_{g}$ denoting the GW frequency and $h_{0}$ the magnitude of the
GW strain. The cavity is described by its volume $V$, its quality factor $Q$
as well as the external magnetic field $B_{0}$. The dimensionless coupling
constant $\eta_{n}$ is given by the overlap of the induced current with the
excited cavity mode:

$\eta_{n}=\frac{|\int_{V}d^{3}xE^{*}_{n}\cdot\hat{j}_{+,\times}|}{V^{1/2}(\int_{V}d^{3}x|E_{n}|^{2})^{1/2}},$
(2)

where $E_{n}$ denotes a resonant mode of the cavity and $\hat{j}_{+,\times}$
describe spatially-dependent dimensionless functions for the spatial profile
and polarization of the GW signal. We refer to [4] for further details. It is
important to note that the sensitivity increases quadratically with the
magnetic field strength $B_{0}$ and to the power of 5/3 with the cavity volume
V.

Unlike axion searches, where each axion mass corresponds to a single resonance
frequency, GW signals may be broadband and not localized at a specific
frequency. Even more importantly, GW are expected to be coherent over large
distances, in contrast to axion halo signatures. This opens up a completely
new approach to search for GW: instead of constructing one dedicated cavity
based experiment, a global network of several cavity experiments over the
whole world could coherently combine their measurements and boost
significantly the sensitivity. In particular, it is sufficient to use off-
shelf high field magnet systems, which can be also operated at smaller
laboratories.

We present in the following the basic idea of a Global Network of Cavities to
Search for Gravitational Waves (GravNet). After a brief discussion on the
potential sources of the high frequency gravitational waves (Section 2), we
briefly discuss the baseline cavity experiment SUPAX (Section 3), which will
be used to estimate the overall baseline sensitivity of GravNet. Extensions of
GravNet are summarized in Section 4.

##  2 Sources of high frequency gravitational waves

Various sources have the potential to produce high-frequency gravitational
waves (GWs), and they have been studied extensively [4]. Notably, mergers
involving sub-solar mass objects, including primordial black holes (PBHs) and
other exotic compact objects, are typical examples. Additionally, GWs can be
generated by boson clouds via black hole superradiance. Also exotic compact
objects, such as boson and fermion stars, gravitino stars, gravistars, and
dark matter blobs, can have significantly lower masses than a solar mass,
enabling them to emit GWs at high frequencies.

During the inspiral phase of a binary merger, the frequency of the emitted GWs
increases as the compact objects approach each other. There exists an upper
bound on the GW frequency emitted by a binary in a quasi-circular orbit, which
corresponds to the frequency at the innermost stable circular orbit (ISCO).
For a binary with equal-mass compact objects ($M_{b}$) during the early stages
of the merger, the GW frequency evolves as:

$\omega_{g}\approx
14\,\text{GHz}\times\left(\frac{10^{-6}M_{\odot}}{M_{b}}\right)\left(\frac{r_{\text{ISCO}}}{r_{b}}\right)^{3/2},$

where $M_{\odot}$ denotes the solar mass. Therefore, only very light binaries,
such as sub-Earth mass PBHs, can generate GW signals in the GHz regime well
before reaching the ISCO.

However, GW signals from such light binaries become highly transient near the
ISCO, as the emitted GW frequency evolves over time according to:

$\frac{d\omega_{g}}{dt}\approx\left(\frac{M_{b}}{r_{b}}\right)^{11/6}.$

This time dependence limits the sensitivity of resonant experiments. The
number of orbital cycles ($N_{\text{cyc}}$) a binary spends emitting GWs
within the resonator bandwidth $\omega_{g}/Q$ for frequencies in the GHz
regime can be estimated as:

$N_{\text{cyc}}\approx
10^{-3}\times\left(\frac{10^{6}M_{\odot}}{M_{b}}\right)^{5/3}\left(\frac{10^{5}}{Q}\right).$

Requiring at least $N_{\text{cyc}}>10^{5}$ for the detection of a GW signal
implies $M_{b}<10^{-11}M_{\odot}$. Remarkably, this is the regime where PBHs
could constitute a significant fraction of the cosmological dark matter
abundance [4]. The dependence of the time the PBH merger signal spends within
a cavity bandwidth is shown in Figure 5.

Figure 1: Left: Maximum feasible integration time given by the duration the
frequency change is less than $f_{0}/Q$, assuming $Q=10^{6}$, for a PBH
merger. Right: typical strains of PBH mergers in dependence of their mass for
distances of 1pc and 10pc from the earth.

However, other sources such as GWs from superradiance of boson clouds in the
vicinity of PBHs and other more exotic objects are non-transient and remain at
constant frequencies over long time intervals, order of months, allowing for
long measurement times [10].

The typical distance of high frequency gravitational wave sources is in the
order of 1 pc or more. Hence the GW wavefront at earth can be assumed to be
coherent across distances at earth, in particular when the origin of the wave
can be determined, as the phase-differences at different experimental
locations at earth can be calculated.

##  3 Basic Considerations on Cavity Designs

In order to estimate the expected signal over noise ratio for a typical cavity
of the GravNet proposal, we use the existing numbers of the SUPAX experiment,
which is currently in operation at the University of Mainz [11]. The SUPAX
cavity and its readout will be discussed in section 3.1. We then discuss the
advantages of combining several cavities in a coherent manner in section 3.2,
before we discuss single photon detection approaches in cavities in section
3.3.

###  3.1 The SUPAX Cavity Design

SUPAX is an experiment which aims for the search for dark photons and axion-
like particles with masses around $34\,\mu\text{eV}$ using a cavity in a high
magnetic field [11].

Figure 2: Cavity of the SUPAX experiment with a length of 16cm.

Figure 3: Schematic overview of the SUPAX experimental setup.

SUPAX uses a rectangular cavity which is suspended from the top of a liquid
helium (LHe) dewar and immersed in the LHe bath, as illustrated in Figure 3.
The dimensions of the cavity are carefully chosen to maximize its effective
volume while ensuring that the final assembly fits within the designated
cryostat, which has a diameter of $52\text{\,}\mathrm{m}\mathrm{m}$.
Additionally, the mode next to the investigated $\text{TM}_{010}$ mode is kept
at least $>$100\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$$ away. This leads to
inner dimensions of the cavity of $150\times 22.8\times
30$\mathrm{m}\mathrm{m}^{3}$$. These dimensions yield a resonance frequency
of:

$f=$8.47\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$\hskip
5.69054pt\Rightarrow\hskip 5.69054ptm_{A^{\prime}}=2\pi\hbar\cdot
f\approx$35\text{\,}\mu\mathrm{e}\mathrm{V}$.$

at room temperature. The cavity depicted in Figure 3, was milled from a solid
copper block.

The unloaded quality factor is measured to be $Q_{0}=40726\pm 4250$ in good
agreement with the simulation result of $Q_{0}^{\text{sim}}=40059\pm 128$.

The readout is based on a real-time spectrum analyser which is capable of
streaming IQ time-series data with a bandwidth of 40MHz around the 8 GHz
centre frequency to a readout computer. For the purpose of the axion (dark
photon) search the data is immediately converted into the frequency domain and
the resulting spectra stored for offline analysis. To study the offline
combination of multiple cavity signals the IQ data is stored.

The next generation cavity of the SUPAX experiment is coated with niobium-
nitrite superconductor. This will lead to an expected improvement of the
quality factor by a factor of 10 to 100. The RF behaviour of the coating in
the magnetic field is currently under investigation.

The magnet to be used is a fast-ramping superconducting solenoid from
CRYOGENIC LTD with a central field of up to 14T over a length of 20cm111The
magnet is currently installed at the Helmholtz institute at Mainz by the group
of Prof. Budker. The room temperature bore has a diameter of $89\,\text{mm}$
which limits the diameter of the used cryostat.

###  3.2 Combination of several cavities with a coherent signal

Typical distances of sources of high frequency gravitational waves are 1 pc or
more, hence their sources can be thought of point-like at earth, yielding
coherent GW wave-fronts.

Before discussing the combination of locally separated cavities, we discuss
first the combination of N cavity signals, recorded at the same location. This
gives a factor of N improvement in the SNR over the individual cavity, and not
as trivially assumed an improvement of $1/\sqrt{N}$. To understand this
feature, we follow the discussion in [12] and consider an N-port power
combiner/divider as in Figure 4. The middle picture shows how N voltages with
frequency $\omega$ and phase differences $\phi$ combine. Assuming the same
phase and amplitude at each input the output Voltage behaves like
$V_{out}=\sqrt{N}V_{in}$, hence the power scales linearly with $N$.

Having $N$ cavities distributed across earth, one needs at least three
cavities which define the incoming direction of the gravitational wave. This
defines the phase-difference at any further cavity, thus yielding an increase
on the SNR by $N-3$ for $N$ similar cavities that are distributed across
earth. As one single setup will not be sensitive enough to reliably detect a
GW, the direction of the incoming GW, and hence the phase alignment of the
various setups, will be stepped through all possible directions in the offline
data analysis.

Figure 4: Left: Equivalent circuit for one mode of a single-port cavity,
Middle: Schematic of an N-Port Power Combiner / Divider, Right: Equivalent
circuit for one mode of a two-port cavity

###  3.3 Single Photon Counting in Cavities

Single photon counting in cavities is a cutting-edge technique that has
revolutionized the field of quantum optics and precision measurements. It
involves detecting and quantifying individual photons that are confined within
cavities. For RF photons the problem is much more difficult than for optical
wavelengths as the energy per photon is lower. However, recently sensitivity
to single RF photons was experimentally shown and several further techniques
are promising to reach single RF photon sensitivity soon.

Among the currently discussed detection methods are current biased Josephson
Junctions (CBJJ) [13], Kerr Josephson parametric amplifiers [14] and transmon
Q-bits [15].While CBJJs do not yet reach single photon sensitivity, the latter
two approaches do with the transmon Q-bit showing a detection efficiency of
$43\%$ for $7\,\text{GHz}$ photons at a dark count rate below $90\,Hz$.

Similar to the resonant cavity readout, one would also expect a huge advantage
when combining several cavities, which are distributed across the earth.
Assuming a single photon detection efficiency of 50% and an improved
background rate of 10Hz for one cavity one would get $3\times 10^{8}$
background events per year, while a signal is expected only once per year.
Since the background is uncorrelated between independent cavities, one can
require a coincidence signal across N cavities. Assuming a timing resolution
of 0.2 ms the background is suppressed by a factor of $(1/0.2,\text{ms}\times
10\,Hz)^{N-1}=500^{N-1}$. Requiring at least N=5 cavities to record a
coincident signal suppresses the background therefore by $\approx 6\times
10^{10}$, i.e. yielding a background free search. The problem is that 5
coincident cavities also have a resulting signal efficiency of $0.5^{5}\approx
0.03$ corresponding to an average time for one recorded signal event of 32
years. However, if one combines e.g. $N>5$ cavities, the likelihood to record
a signal in at least 5 cavities increases according to a binomial likelihood
distribution. Clearly, at least three cavities are required to define the
direction of the incoming GW, i.e. to define the coincidence window for any
other cavity. Assuming $N=20$, of which $N=17$ are actually available for the
analysis, and a single signal efficiency of $p=0.5$, a total signal efficiency
of $>90\%$ is be expected. The optimisation of the coincidence setup is
detailed in section 5.

##  4 GravNet as Resonant Cavity Experiment (GravNet-1)

The first implementation of GravNet will most likely be based on the usage of
the resonant cavity approach, as the underlying technologies are all well
tested and understood.

###  4.1 Setup

As discussed in the previous section, typical commercially available (and
therefore cost-efficient) high-field magnets have a cylindrical volume of a
high constant magnetic field. Typical dimensions range between radii of 2 to
$5\,\text{cm}$ with a height of 10 to $40\,\text{cm}$. The coupling constant
of GW signals to the cavity is in the order of $\approx 0.1$ for cylindrical
cavities, but $1$ for spherical cavities. Assuming a constant magnetic field
in a cylindrical volume of $r=4\,\text{cm}$ and $h=24\,\text{cm}$, one can
either fit one cylindrical cavity with those dimensions or three spherical
cavities of $r=4\,\text{cm}$. Comparing the effect on $P_{sig}$ between one
cylindrical and one spherical cavity, this implies a factor on
$P^{\text{cylinder}}_{sig}/P^{\text{spherical}}_{sig}$ of $\approx 12$ due to
larger volume of the cylinder, but a factor of $\eta^{2}=0.01$ due to the
coupling. The overall effect on $P_{sig}$ is roughly up to a factor $10$ in
favor of a spherical cavity. Using three coupled cavities instead will
increase the effective volume three fold and the SNR by a factor of 6, as can
be seen from eq. 1.

According to those considerations, we foresee three cavities for one
experimental setup of GravNet, each with a radius of $4\,\text{cm}$, placed in
a constant magnetic field of 14 T with a height of $24\,\text{cm}$. Finally,
we assume that the readout system for GravNet will be based on Josephson
Parametric Amplifiers (JPAs) or similar low noise amplifiers with system
temperatures of $0.1\,\text{K}$.

An overview of all critical parameters of one experimental GravNet setup is
given in Table 1.

Setup | Supax | GravNet
---|---|---
Shape | cyl. | spher.
$f_{0}$ [GHz] | 8.3 | 5.0
Volume [l] | $0.128$ | $0.21$
$Q_{0}$ | 39600 | $10^{6}$
$\eta$ | 0.08 | 0.6
$T_{\text{sys}}$ [K] | 5 | 0.1
$B$ [T] | 14
int. time | 1 s
n cavities | 1 | 3
noise power [W] | $1.5\cdot 10^{-21}\,W$ | $6.2\cdot 10^{-23}\,W$
$h_{0}(P_{\text{sig}}=P_{\text{noise}})$ | $7.1\cdot 10^{-21}$ | $5.2\cdot 10^{-23}$
Table 1: Parameters of the experimental setup defining the signal and noise
power. The measured values were obtained using the Supax Cu cavity in LHe. The
expected values assume a superconducting, spherical cavity with 4 cm radius.

###  4.2 Sensitivities

We think it is realistic to assume that the final GravNet experiment will
combine $N=10$ individual experimental locations across the globe, each
hosting three cavities as detailed in the previous section. Assuming 1s
integration time and the sensitivity for each setup as listed in Table 1, the
sensitivity on $h_{0}$ will improve by a factor $\sqrt{N}$ to $h_{0}<1.7\cdot
10^{-23}$. This requires a phase aligned combination of the time-series data
of each of the setups, yielding a linear increase in the SNR with increasing
number of setups. To this end the direction of the incoming GW has to be known
to be able to calculate the relative phase differences between the setups
depending on their geographic location. This can be achieved by assuming any
direction of the GW, combining the data and searching for signal and then
scanning through both angles defining the direction of the GW.

The sensitivity can be trivially increased by increasing the integration time
of the cavity signals. While integration times over hours are feasible, this
requires a GW source with stable frequency over at least the intended
integration time. Bosonic clouds exhibiting gravitational superradiance a
superb candidate.

Integrating for two hours instead of 1 second leads to an improved by one
order of magnitude on the GW strain, reaching $h_{0}<5.6\cdot 10^{-24}$ for a
single setup and consequently about $1.7\cdot 10^{-24}$ for ten setups. The
dependence of the sensitivity on $h_{0}$ on the integration time is shown in
Figure 5.

Figure 5: Shown in the sensitivity on the GW strain $h_{0}$ in dependence on
the integration time for the resonant cavity setup with parameters assumed as
shown in Table 1.

Figure 6: Schematic drawing one experimental GravNet-1 setup with resonant
cavities.

Figure 7: Schematic drawing one experimental GravNet-2 setup as single photon
experiment.

###  4.3 Possible Extensions

It should be noted that one could in principle also combine different cavity
layouts and magnet systems into GravNet, which are then testing different
resonance frequencies at different sensitivities. Since such a setup would
certainly introduce a certain model dependence in particular when searching
for transient signals, its final efficiency still has to be studied in detail.

##  5 GravNet as Photon Counting Experiment (GravNet-2)

###  5.1 Setup

As discussed in Section 3.3, the shape of the cavity does not increase the
likelihood of a conversion, but only the active volume of the cavity within
the magnetic field is relevant. Given that the cost driving factor is always
the magnet system, but not the design of the cavities, we assume the same
magnet setup as in GravNet-1 but assume two independent cylindrical cavities
with dimensions of $r=4\,cm$ and $h=12\,cm$ instead of three spherical
cavities. While the volume increases the sensitivity with $V^{5/3}$, one gains
significantly more due to the Binomial probabilities, discussed in Section
3.3.

###  5.2 Sensitivities

Similar to GravNet-1, we assume again N=10 different experimental setups, with
a total of $N=20$ operational single and independent cavities, as depicted in
Figure 7. The cavities operate at a resonance frequency around $5\text{GHz}$
and exhibit a volume of $0.6\,l$ each. The single RF photon detection
efficiency is taken to be $50\%$, a dark count rate of $10\,Hz$ and a time
resolution of 0.2 ms are assumed, as discussed in section 3, the following
sensitivities are expected.

A naive sensitivity estimate yields, assuming a coincidence time window of 0.2
ms and each setup consisting of 2 independent cavities, a coincidence dark
count rate of 1.2 counts per minute. Requiring a coincidence of 5 cavities in
total a dark count rate of 1 in 190 years is expected. The efficiency to
detect the coincident production of RF photons in at least 5 out of 20
cavities is calculated using the binomial distribution with $n=20$ and $p=0.5$
where $P(x\geq 5)=99.4\%$.

The question is however how this translates to a sensitivity on the GW strain
$h_{0}$. The photon flux from thermal noise at $0.1$ K and a sensitive
bandwidth of 1 kHz is about 10 photons per second at a photon energy of 5 GHz.
At 1 MHz sensitive bandwidth would increase the photon flux to 400 Hz.
Decreasing the temperature to $0.01$ K would reduces the thermal photon flux
by one order of magnitude. Hence, we assume for the following calculation a
photon flux of 10 Hz from thermal radiation and a negligible contribution to
the dark count rate from the detector itself. Clearly, to be able to
discriminate the thermal noise photons to a signal from a BPH merging event a
coincidence measurement is needed, as indicated above.

The photon flux $\Phi$ generated by a GW can be estimated by dividing the
signal power by the photon energy $\Phi=P_{sig}/h\nu$. Using eq. 1 and
assuming $Q_{0}=10^{6}$ and $\eta=0.1$ the photon flux generated in one cavity
in dependence on the GW strain is shown in Fig. 8. Two cavity dimensions are
shown: GravNet-a and GravNet-b, whose parameters are summarized in Table 2.
The smaller cavity (GravNet-a) shows a signal photon flux comparable to the
thermal noise of $10\,Hz$ at $h_{0}=1.7\times 10^{-21}$ while the larger
cavity (GravNet-b) reaches that flux at $h_{0}=1.6\times 10^{-24}$.

Figure 8: Photon flux in dependence of the GW strain $h_{0}$ at 5 GHz for GravNet a and b setups detailed in Table 2. Setup | GravNet-a | GravNet-b |
---|---|---|---
radius | 40 mm | 40 cm |
length | 12cm | 50 cm |
Volume [$m^{3}$] | $6\times 10^{-4}$ | 0.25 |
$Q_{0}$ | $10^{6}$ | $10^{5}$ |
$T_{\text{sys}}$ [K] | 0.1 | 0.1 |
$B$ [T] | 14 | 9 |
noise power [W] | $4.4\cdot 10^{-23}\,W$ | $4.4\cdot 10^{-23}\,W$ |
$h_{0}(P_{\text{sig}}=P_{\text{noise}})$ | $1.6\cdot 10^{-22}$ | $3.4\cdot 10^{-24}$ |
$\gamma$-flux [1/s] | 10 | 10 |
Table 2: Parameters of the experimental setup defining the signal and noise
power. The measured values were obtained using the Supax Cu cavity in LHe. The
expected values assume a superconducting, spherical cavity with 4 cm radius.

The target rate of accidental coincidences (ac) from the thermal noise are set
to one per year. This defines the length of the allowed coincidence window
$\Delta t$ in dependence on the number of required coincidences $k$ and the
background rate $bkg$:

$1/\Delta t=(\text{bkg}\cdot\text{secPerYear})^{1/(k-1)}\cdot\text{bkg}$

This dependence is shown in Figure 10. Knowing the needed coincidence interval
we can continue and calculate the efficiency to detect one photon from a GW in
k detectors within the coincidence window. The result is shown for various
assumptions on the signal photon flux in Figure 10, assuming 20 independent
detectors in total. A photon flux of $\Phi=30\,Hz$ is not reliable detected
any more, while for a photon flux of $\Phi=40\,Hz$ a detection efficiency of 1
is still reached using a coincidence of 18 out of 20 cavities with a
coincidence window of $31\,ms$. This photon flux corresponds to a GW strain of
$h_{0}=3\times 10^{-22}$ ($h_{0}=3\times 10^{-24}$) for the GravNet-a
(GravNet-b) setups, respectively, as is shown in Figure 8.

Figure 9: Maximum allowed coincidence interval to achieve a background
coincidence rate of 1 per year given the number of coincidences and assuming a
dark count rate (thermal noise + detector noise) of 10 Hz.

Figure 10: Signal efficiency in dependence of the number of coincidences and
the coincidence window set according to Figure 10 for various assumptions on
the signal photon rate. A total of 20 independent detectors is assumed.

It is worth stressing the fundamental difference of a counting experiment to
the resonance based experiment. In the resonant cavity excitation experiment
one relies on the signal to be monochromatic w.r.t. to the cavity bandwidth,
at least long enough to ring up the cavity so that the maximal RF energy can
be read out, which is still below a ms for very high Q cavities. Note that
typical integration times for such setups is O(100) seconds, and even in the
sensitivity estimate in Section 4 1s integrating time is assumed. Then a phase
aligned combination of the signals from multiple cavities is required, which
may be challenging.

Contrary, in the counting experiment we rely on simultaneous measurements of
single photons within a time interval of $ms$ making the experiment sensitive
to very short, transient signals. The cavity signals don’t need to be phase
aligned and even measurements from setups with variations in frequencies can
easily be combined. Using order of 20 individual detectors the experiment
becomes essentially background free given the parameters of the detectors as
stated above, and can be sensitive to strains down to $10^{-22}$ to
$10^{-24}$, depending on the size of the conversion volume. Those results are
summarized in Figure 11 in comparison to other existing experiments.

Figure 11: Estimated Sensitivity of the GravNet 1 and 2 experiments on the GW
strain $h$, assuming 10 setups each. For GravNet-1 two integration times of 1
second and 2 hours are shown. For GravNet-2 sensitivities for two cavity
dimensions, a and b as detailed in Table 2, are shown. The range of PBH merger
signals as well as the sensitivity of similar experiments are adapted from
[16]. Note that the integration times for the shown sensitivities are much
shorter for GravNet than that of typical axion search experiments.

##  6 Conclusion

In this paper we propose to setup a world-wide network of cavity based
experiments to search for high frequency gravitational waves (GravNet) with
frequencies in the 5 GHz regime. Assuming ten participating laboratories,
sensitivities on the gravitational wave strain of $h_{0}<10^{-24}$ for
monochromatic sources and long integration times and $h_{0}<10^{-22}$ for
$~{}10\,ms$ transient signals can be reached, depending on the chosen
experimental setup. Using larger magnet setups with lower field would even
extend the sensitive to transient signals to $h_{0}<10^{-24}$. This would
allow for a first search of several exotic sources of high frequency
gravitational waves as well as for primordial black hole models with masses
larger than $3\times 10^{-7}\,M_{\odot}$. In particular it should be noted,
that the experimental concept is based on commercial magnet systems, which are
rather cost efficient compared to specialized dedicated high field magnets.
Moreover, a worldwide collaboration such as GravNet would share costs
automatically with local lab-based experiments and boost the R&D effort within
many groups, since new developments at one site can be easily integrated in
all other experimental locations.

This paper should be seen as a starting point for discussions towards a
collaborative world-wide effort in the context of gravitational wave physics
beyond the regime of classic astrophysical objects.

## Acknowledgement

We thank Pedro Schwaller, Sebastian Schenk and Tim Schneemann for the helpful
discussions and comments during the preparation of this paper. This work would
have not been possible without the continuous support from the PRISMA+ Cluster
of Excellence at the University of Mainz.

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