# Detecting Planetary-mass Primordial Black Holes with Resonant
Electromagnetic Gravitational Wave Detectors

Nicolas Herman<EMAIL_ADDRESS>Department of Mathematics and Namur
Institute for Complex Systems (naXys), University of Namur, Rue Grafé 2,
B-5000, Namur, Belgium  André Fűzfa<EMAIL_ADDRESS>Department of
Mathematics and Namur Institute for Complex Systems (naXys), University of
Namur, Rue Grafé 2, B-5000, Namur, Belgium  Cosmology, Universe and Relativity
at Louvain, Institute of Mathematics and Physics, Louvain University, 2 Chemin
du Cyclotron, 1348 Louvain-la-Neuve, Belgium  Léonard Lehoucq
<EMAIL_ADDRESS>Department of Mathematics and Namur
Institute for Complex Systems (naXys), University of Namur, Rue Grafé 2,
B-5000, Namur, Belgium  Department of theoretical physics at the ENS Paris-
Saclay, University of Paris-Saclay, avenue des Sciences, 91190, Gif-sur-
Yvette, France  Sebastien Clesse<EMAIL_ADDRESS>Service de
Physique Théorique, Université Libre de Bruxelles (ULB), Boulevard du
Triomphe, CP225, B-1050 Brussels, Belgium  Cosmology, Universe and Relativity
at Louvain, Institute of Mathematics and Physics, Louvain University, 2 Chemin
du Cyclotron, 1348 Louvain-la-Neuve, Belgium

###### Abstract

The possibility to detect gravitational waves (GW) from planetary-mass
primordial black hole (PBH) binaries with electromagnetic (EM) detectors of
high-frequency GWs is investigated. We consider two patented experimental
designs, based on the inverse Gertsenshtein effect, in which incoming GWs
passing through a static magnetic field induce EM excitations inside either a
TM cavity or a TEM waveguide. The frequency response of the detectors is
computed for post-newtonian GW waveforms. We find that such EM detectors based
on current technology may achieve a strain sensitivity down to $h\sim
10^{-30}$, which generates an EM induced power of $10^{-10}$ W. This allows
the detection of PBH binary mergers of mass around $10^{-5}M_{\odot}$ if they
constitute more than $0.01$ percent of the dark matter, as suggested by recent
microlensing observations. We envision that this class of detectors could also
be used to detect cosmological GW backgrounds and probe sources in the early
Universe at energies up to the GUT scale.

###### pacs:

##  I Introduction

Gravitational Waves (GW), first introduced by Einstein in 1916 [1, 2] as a
linear regime of the field equations of General Relativity, have been directly
detected for the first time in 2015 by the LIGO/Virgo collaboration [3]. These
ground-based detectors use laser interferometry techniques, but there also
exists other detection strategies. Einstein’s Equivalence Principle implies
that all types of energies produce and experience gravity in the same way. The
energy of electromagnetic (EM) radiation must therefore source gravity just
like compact objects do and, the other way around, gravity (e.g. gravitational
waves) can manifest itself in the physical characteristics of electromagnetic
radiation. This is the basic principle behind the EM detection (or emission)
of GWs.

The use of EM fields to both generate and detect GWs has been actually
considered for decades, for instance based on the (inverse) Gertsenshtein
effect [4] relying on the coupling between GWs and EM waves in the presence of
a strong static magnetic field. However, the weakness of this coupling makes
any GW detection extremely challenging. Therefore experimental efforts have
mostly concentrated on ground-based detectors based on laser interferometry,
probing a rather low frequency range (typically $1-10\,{\rm kHz}$). But this
technique is not suited for the detection of high-frequency GWs (HFGWs), at
the opposite of EM-based detectors. The Section II presents a short historic
review of HFGW detectors that have been build or proposed. The interested
reader will find more details about detection strategies and potential high-
frequency sources in a recent review [5].

In this paper, we propose two novel patented [6] experimental designs of
resonant high-frequency EM detectors, based on the inverse Gerstenshtein
effect, operating at MHz or GHz frequencies and feasible with current
technology. We compute numerically the EM signal produced by passing HFGWs and
consider planetary-mass primordial black holes (PBHs) as their potential
sources. The detectors are constituted by either a waveguide or a cavity
immersed into a transverse static magnetic field. This outer magnetic field
deserves two purposes in our design. First, a proper (transverse) orientation
of this field is mandatory to convert HFGWs into EM waves through the inverse
Gerstenshtein mechanism. Second, the external magnetic field boosts the output
signal through a resonance mechanism on specific radiation modes that are
excited by the passing HFGWs.

There exists a broad range of hypothetical astrophysical or cosmological
sources of HFGWs with frequencies above kHz (see [5] and references therein),
such as exotic compact objects, PBHs, inflation, reheating, oscillons, cosmic
strings and other topological defects, large curvature fluctuations and phase
transitions in the early Universe. Some sources produce transient signals,
like the merging of compact objects, while cosmological sources typically
induce a permanent stochastic GW background. In particular, HFGW detectors
operating at MHz and GHz frequencies might probe new Physics up to the scale
of the Grand Unified Theories.

In this paper, we focus on the particular case of transient signals from the
merging of light (planetary-mass scale) PBHs. Their existence is still
hypothetical but motivated by several recent observations, such as
microlensing events towards the galactic bulge, recently detected by OGLE [7,
8] and suggesting that such PBHs may constitute between 1% and 10% of the Dark
Matter (DM). LIGO/Virgo observations provide additional motivations for the
existence of PBHs in the stellar range [9, 10, 11] and a unified scenario with
a wide mass distribution imprinted by the known thermal history of the
Universe has been presented in [12]. PBHs may originate from the gravitational
collapse of primordial inhomogeneities in the early Universe. Above a mass
$m_{\rm PBH}\approx 10^{11}$ kg, their evaporation time through the Hawking-
Bekenstein mechanism is much larger than the age of the Universe. Beyond these
motivations, the GW waveform from PBH mergers is well-known, which allows us
to simulate the exact detector response. Our work therefore aims at paving the
road towards an experimental realization of a HFGW detector with a good enough
strain sensitivity, of order $h\sim 10^{-30}$, to detect various GW sources.

The paper is organized as follows: After reviewing previous proposals of EM-
based HFGW detectors in Section II, we introduce in Section III the theory
behind the Einstein-Maxwell system and the Gertsenshtein effects. Section IV
applies the theory to the case of two detector designs based on a resonant
waveguide and a cavity, into an external static magnetic field. Section V is
devoted to the computation of the expected rate and signal from planetary-mass
PBH mergers. In Section VI we gather these calculations to compute the
expected signal in our detectors, in terms of frequency response and induced
EM power, for different realistic designs and different PBH masses. Finally,
we compute forecasted limits on the possible PBH abundance, for two possible
binary formation channels (primordial binaries and tidal capture in clusters).
We discuss our results and present some perspectives in the conclusion
(Section VII).

##  II Brief History of Electromagnetic and HFGW detectors

The use of EM fields to both generate and detect GWs has been actually
considered for decades. While Weber [13] envisioned the importance of both the
generation and detection of GWs as early as 1960, Gertsenshtein [4] discovered
in 1962 a resonance mechanism allowing to produce GWs from EM waves in the
presence of a strong static magnetic field. Later in [14, 15], this mechanism
was studied in greater details using Einstein’s equations, scattering theory
and Feynman perturbation techniques. Gertsenshtein’s mechanism was applied to
astrophysics by Zeldovich [16]. Grishchuk and Sazhin then introduced in [17,
18, 19] purely electromagnetic generators of GWs, using transverse
magnetic/electric (TM/TE) resonant cavities. They envisioned GW emission-
reception laboratory experiments and concluded that they might be
experimentally feasible [18, 19], which would open the road to futuristic
technologies based on GW physics [20]. Resonant cavities and EM waveguides
were then considered as possible detectors of gravitational radiation, either
emitted by natural or artificial sources [21, 22, 18, 19, 20, 23, 24, 25, 26].

EM detectors of GWs would allow exploring higher frequency range than with
laser interferometry, typically from kHz to 100 GHz when using radio-
frequencies or from 100GHz to THz when using microwaves. Investigations of
electromagnetic GW detectors began in the 1970’s with the works [21, 22, 18,
19, 14, 23, 24, 25, 27]. Those detectors either make use of the conversion of
GWs to photons [14], the excitation or modification of resonant modes of EM
cavities and waveguides [21, 22, 18, 19, 23, 24, 25, 28], the change of
polarization plane of an EM wave due to the passing GWs [27], or induced
birefringence of the interior of the cavity [26]. More recently, Ejlli, Cruise
et al. in [29] used available data from experiments designed for the detection
of weakly interacting slim particles to set limits on the stochastic GW
background through the graviton to photon conversion in the ultra-high
frequency band (above 1 THz). There has been also a method proposed by Bentley
et al. [30] to reduce the noise of interferometric GW detectors at high-
frequency (kHz). Other types of HFGW detectors have been recently proposed
[31, 32, 33, 32, 34], using optically trapped dielectric microspheres in a
cavity, resonance between a graviton and a magnon that is based on the Dirac
equation in a curved spacetime or high frequency phonon trapping acoustic
cavities.

##  III Einstein-Maxwell system and the Gertsenshtein effects

The Einstein-Maxwell system models the interplay of gravitation and
electromagnetism by the coupling of their respective field equations (in S.I.
units111The relevant fundamental constants of the Einstein-Maxwell system are
$G$ as Newton’s constant, $c$ as the speed of light and $\mu_{0}$ as the
(vacuum) magnetic permeability.)

$\displaystyle R_{\mu\nu}$ $\displaystyle=$ $\displaystyle\frac{8\pi
G}{c^{4}}T^{\left(\rm em\right)}_{\mu\nu}\;$ (1)
$\displaystyle\nabla_{\mu}F^{\mu\nu}$ $\displaystyle=$
$\displaystyle-\mu_{0}J^{\nu}\;$ (2)

where

$T^{\left(\rm
em\right)}_{\mu\nu}=\frac{1}{\mu_{0}}\left(g^{\alpha\beta}F_{\mu\alpha}F_{\nu\beta}-\frac{1}{4}g_{\mu\nu}F_{\alpha\beta}F^{\alpha\beta}\right)$
(3)

is the Maxwell stress-energy tensor,
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ being the Faraday
tensor of the electromagnetic field, $g_{\mu\nu}$ and $A_{\mu}$ are the metric
and the four-vector potential, $\nabla_{\mu}$ is the covariant derivative with
respect to $g_{\mu\nu}$, $R_{\mu\nu}$ is the Ricci tensor and $J^{\nu}$ the
four-current density. In general relativity, the interplay between gravitation
and electromagnetism is twofold: first, spacetime is curved by the energy of
the electromagnetic field as ruled by Eq. (1) while, at the same time, the
propagation of the electromagnetic field $F_{\mu\nu}$ is governed by the
covariant Maxwell equations in curved spacetime, Eq. (2). For a small
electromagnetic compactness $GE_{\rm EM}/(c^{4}L)\ll 1$, where $E_{\rm EM}$ is
the EM energy stored in some physical system of length $L$, the gravitational
sector of the system can be safely treated in the weak-field limit [35].

First, one can consider the gravitational perturbations arising from EM
sources in the Einstein field equations. Considering EM configurations
consisting in a superposition of a static field $F^{(s)}_{\mu\nu}$ and a
varying one $F^{(v)}_{\mu\nu}$, the quadratic terms in Eq. (3) yield to three
general classes of electromagnetically-induced gravitational perturbations
around a Minkowski background

$g_{\mu\nu}=\eta_{\mu\nu}+c_{\mu\nu}+w_{\mu\nu}+h_{\mu\nu}$ (4)

where $\eta_{\mu\nu}=\rm diag(-1,+1,+1,+1)$ is the Minkowski metric and
$c_{\mu\nu},w_{\mu\nu},h_{\mu\nu}\ll 1$ represent the metric perturbations. In
the above equation, (i) $c_{\mu\nu}$ represents the static gravitational field
generated by some external static magnetic or electric field (from a coil or a
capacitor) and arises from the quadratic term in $F^{(s)}_{\mu\nu}$ in (3),
while (ii) $w_{\mu\nu}$ is a gravitational wave generated by the varying EM
[for which the source is the quadratic term in $F^{(v)}_{\mu\nu}$ in Eq. (3)]
and (iii) $h_{\mu\nu}$ is another gravitational wave generated by the coupling
between the external static field $F^{(s)}_{\mu\nu}$ and some EM wave
$F^{(v)}_{\mu\nu}$ [crossed terms in Eq. (3)]. The case (i) has been studied
in [35] but does not give rise to a GW since the outer EM field is static.
Case (ii) has been studied for light propagating pulses in [36, 37] and in
[17, 18, 19, 38] for EM waves in resonant hollow cavities222In [17], a special
case of hollow spherical cavity in an outer radial magnetic field is briefly
considered, giving rise to an admixture of terms $w_{\mu\nu}$ (case (ii)) and
$h_{\mu\nu}$ (resonance - case (iii)) which were not identified as such nor
exploited by the authors.. Case (iii) actually corresponds to what is called
the direct Gertsenshtein effect [4, 16]. This Gertsenshtein effect is a wave
resonance mechanism in which light passing through a region of uniform
magnetic field, perpendicular to the direction of light propagation, produces
GWs. A monochromatic EM wave leads to an outgoing GW of same frequency.
Electromagnetic generation of GWs is a very faint process, due to the extreme
weakness of gravitational coupling. Indeed, the metric perturbations
$h_{\mu\nu}$ produced through the Gerstenshtein mechanism have an amplitude of
order

$h_{\mu\nu}\sim\frac{4GB_{0}E_{0}L^{2}}{c^{5}\mu_{0}}~{},$ (5)

where $L$ is the size of the region in which the magnetic field and the EM
wave interact, and where $B_{0}$ and $E_{0}$ are the amplitudes of the static
magnetic and varying electric fields respectively. To give an idea, in order
to generate a strain $h\approx 10^{-21}$ with $B_{0}\approx 10$ T and
$E_{0}\approx 1\,{\rm MV/m}$, one needs a truly astronomical size of the
interacting region, $L\approx 10^{6}\;{\rm km}$. Therefore, while the direct
Gertsenshtein effect can be used to build electromagnetic GW generators, its
practical application constitutes an extreme experimental challenge.

Second, ripples in spacetime can also interact with a static magnetic field to
produce an outgoing EM wave. This inverse Gertsenshtein effect is described by
the Maxwell equations, Eq. (2), on a perturbed background. An obvious
application of this effect is the detection of GWs passing into a transverse
static magnetic field, yielding to their conversion into EM waves. One
possible way to derive the equations governing the inverse Gertsenshtein
effect is to develop the covariant derivative in Eq. (2) at first order in
metric perturbations, and to treat these ones as an effective current density.
We follow here a different approach, based on a covariant generalization of
the EM wave equations333These are obtained from the two groups of covariant
Maxwell equations $\nabla_{\mu}F^{\mu\nu}=0$ and
$\nabla_{\kappa}F_{\mu\nu}+\nabla_{\mu}F_{\nu\kappa}+\nabla_{\nu}F_{\kappa\mu}=0$
which can be combined to retrieve Eq. (6).

$g^{\alpha\beta}\nabla_{\alpha}\nabla_{\beta}F\indices{{}_{\mu\nu}}+R\indices{{}_{\mu\nu\alpha\beta}}F\indices{{}^{\alpha\beta}}+R\indices{{}^{\alpha}_{\mu}}F\indices{{}_{\nu\alpha}}+R\indices{{}^{\alpha}_{\nu}}F\indices{{}_{\alpha\mu}}=0,$
(6)

where $R\indices{{}_{\mu\nu\alpha\beta}}$ is the Riemann tensor. This set of
equations describes the propagation of EM waves on a curved spacetime. In the
case of a flat Minkowski spacetime, the wave equation Eq. (6) reduces to the
classical wave equation
$g^{\alpha\beta}\nabla_{\alpha}\nabla_{\beta}F\indices{{}_{\mu\nu}}=0$.

Let us now consider a small perturbation $h_{\mu\nu}$ propagating on a
Minkowski background, such that the metric is given by
$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ at first order ($h_{\mu\nu}\ll 1$). If
this perturbation satisfies the Lorenz gauge condition,
$\partial_{\mu}h^{\mu\alpha}=0$, one gets from Eq (6) the linearized wave
equation for the Faraday tensor (see also [19]),

$\displaystyle\begin{split}g^{\alpha\beta}\nabla_{\alpha}\nabla_{\beta}F\indices{{}^{(1)}_{\mu\nu}}&=h\indices{{}^{\alpha\kappa}}\nabla^{(\eta)}_{\alpha}\nabla^{(\eta)}_{\kappa}F\indices{{}^{(0)}_{\mu\nu}}-\partial_{\rho}\left(\partial_{\mu}h\indices{{}_{\alpha\nu}}-\partial_{\nu}h\indices{{}_{\alpha\mu}}\right)F\indices{{}^{(0)\,\rho\alpha}}\\\
&\phantom{=}-\left(\partial^{\gamma}h\indices{{}^{\alpha\beta}}+\partial^{\alpha}h\indices{{}^{\beta\gamma}}-\partial^{\beta}h\indices{{}^{\gamma\alpha}}\right)\left(\eta\indices{{}_{\alpha\mu}}\nabla^{(\eta)}_{\gamma}F\indices{{}^{(0)}_{\nu\beta}}-\eta\indices{{}_{\alpha\nu}}\nabla^{(\eta)}_{\gamma}F\indices{{}^{(0)}_{\mu\beta}}\right)=S_{\mu\nu}.\end{split}$
(7)

In the above equation, we have assumed that the total EM field $F_{\mu\nu}$ is
the superposition of some background EM field $F\indices{{}^{(0)}_{\mu\nu}}$
with which the metric perturbation $h_{\mu\nu}$ interacts to produce an EM
perturbation $F\indices{{}^{(1)}_{\mu\nu}}$. In the following, we focus on the
magnetic conversion of GWs into photons, i.e. the interaction between a
passing GW and an external static magnetic field
($F\indices{{}^{(0)}_{\mu\nu}}$ is purely magnetic) which results in an EM
wave emission. Eq. (7) governs the inverse Gertsenshtein effect.

Under the assumption of a uniform static magnetic field, the first and third
source terms in Eq. (7), both including
$\nabla^{(\eta)}_{\gamma}F\indices{{}^{(0)}_{\alpha\beta}}$, identically
vanish, leaving as the only source term the one with second derivatives of the
metric perturbations,

$S_{\mu\nu}=-\partial_{\alpha}\left(\partial_{\mu}h\indices{{}_{\beta\nu}}-\partial_{\nu}h\indices{{}_{\beta\mu}}\right)F\indices{{}^{(0)\,\alpha\beta}}\cdot$
(8)

We can now apply this theory and conceive a specific experiment for the
detection of HFGWs, produced e.g. by inspiralling PBHs.

##  IV Resonant Electromagnetic detectors of HFGWs

In this section, we describe two detector designs that are based on the
patents [6]. The detection principle is based on the inverse Gertsenshtein
effect, thus a passing GW interacts with an intense static magnetic field. If
the direction of the incoming GW is not colinear with the magnetic field,
faint transverse EM waves are generated and these can be further amplified by
EM resonators. The experimental set-up consists of either a waveguide or a
cavity whose axis of symmetry is orthogonal to the magnetic field. Such a set-
up is similar to haloscopes that are used for the search of axions, like the
ADMX experiment [39, 40], except for the orientation of outer magnetic field.
As shown below, it is mandatory that it is orthogonal to the axis of the
cavity/waveguide in order to detect GWs.

A theorem by Choquet-Bruhat [41] establishes that both direct and inverse
Gertsenshtein effects require the condition of orthogonality between the
external EM field and the direction of GW propagation. This theorem starts
from the hypothesis that incoming or generated GWs have a Wentzel-Kramers-
Brillouin (WKB) form and are of high-frequencies. The equation of propagation
in [41] is then obtained after a development in frequency. We propose below a
variant of Choquet-Bruhat’s demonstration with a development in amplitude
instead of frequency. We first assume that the incoming GW is a plane wave,
$h_{\mu\nu}=a_{\mu\nu}\,\,e^{i\omega\Phi},$ with a general varying phase,
$\Phi=\Phi(x^{\alpha})$. The goal is to show that the constant EM field must
be orthogonal to the direction of propagation of the incoming plane wave, in
order to produce an EM wave. In other words, no EM wave can be generated from
the interaction between a constant EM field and a GW, unless the first one is
orthogonal to the direction of propagation of the second one. In order to show
this, we demonstrate that a vanishing source term $S_{\mu\nu}=0$ in Eqs. (7)
is equivalent to the condition $\Phi_{\alpha}F^{\alpha\mu\,(0)}=q\,\Phi^{\mu}$
with $q$ a real constant and $\Phi_{\alpha}=\partial_{\alpha}\ \Phi$. Since we
assume that $E^{(0)}$ and $B^{(0)}$ are constants in our problem, the source
term is given by Eq.(8). Therefore, the non-zero part of $S_{\mu\nu}$ is the
exterior derivative of an effective 4-current density:
$J^{\mathrm{eff}}_{\mu}=\partial_{\alpha}\,h_{\beta\mu}\,F^{\alpha\beta\,(0)}$.
We can thus rewrite our source term as

$\displaystyle S_{\mu\nu}$
$\displaystyle=\partial_{\nu}\,J^{\mathrm{eff}}_{\mu}-\partial_{\mu}\,J^{\mathrm{eff}}_{\nu}$
(9a)
$\displaystyle=(\Phi_{\nu}\,J^{\mathrm{eff}}_{\mu}-\Phi_{\mu}\,J^{\mathrm{eff}}_{\nu})^{\prime}\quad\mathrm{with}\quad(\cdot)^{\prime}\equiv\partial(\cdot)/\partial\Phi\,.$
(9b)

The last line is obtained as a result of the plane wave approximation (the
amplitude $a_{\mu\nu}$ above is constant), which implies that the partial
derivative $\partial_{\nu}$ is equivalent to
$\Phi_{\nu}\,\frac{\partial}{\partial\Phi}$. Because a GW verifies the eikonal
$\Phi_{\mu}\Phi^{\mu}=0$ (if not, it is inconsistent and vanishes with a
change of coordinate [41]), multiplying the above expression by $\Phi^{\mu}$
gives that $S_{\mu\nu}=0\Leftrightarrow\Phi^{\mu}J^{\mathrm{eff}}_{\mu}=0$,
and thus $\Phi_{\nu}$ is orthogonal to $J^{\mathrm{eff}}_{\mu}$. Since
$\Phi_{\mu}$ does not vanish, this yields to the equivalence
$S_{\mu\nu}=0\Leftrightarrow J^{\mathrm{eff}}_{\mu}=0$. Let us now show the
central result

$S_{\mu\nu}=0\Leftrightarrow
J^{\mathrm{eff}}_{\mu}=0\Leftrightarrow\Phi_{\alpha}F^{\alpha\mu\,(0)}=q\,\Phi^{\mu}~{}.$
(10)

For plane waves $h_{\mu\nu}=a_{\mu\nu}\,\,e^{i\omega\Phi(x^{\alpha})}$, we
have that
$\partial_{\alpha}h_{\lambda\nu}=\Phi_{\alpha}h^{\prime}_{\lambda\nu}$.
Therefore, the effective four-current can be simplified to

$\displaystyle J^{\mathrm{eff}}_{\
\nu}=F^{\alpha\lambda\,(0)}\,\Phi_{\alpha}\,h^{\prime}_{\lambda\nu}$ (11a)

If $\Phi_{\alpha}F^{\alpha\mu\,(0)}=q\,\Phi^{\mu}$ then $J^{\mathrm{eff}}_{\
\nu}=q\,\Phi^{\lambda}\,{h^{\prime}}_{\lambda\nu}.$ In the meantime, the
Lorenz gauge condition, $\partial^{\mu}\,h_{\mu\nu}=0$, is equivalent to
$\Phi^{\mu}\,{h^{\prime}}_{\mu\nu}=0$ in the plane wave approximation. So we
can conclude that the effective 4-current density $J^{\mathrm{eff}}_{\ \nu}$
vanishes . Now, let us now prove the implication in the reverse way.

Let us assume $J^{\mathrm{eff}}_{\ \nu}=0$ and move to radiative coordinates,
that is to say that $\Phi=x^{0}$ so we have directly $\Phi_{0}=1$ and
$\Phi_{i}=0$. The GW obey to the eikonal, thus $\eta^{00}=0$ and
$\eta^{0i}=\Phi^{i}$. In such a comobile coordinate system, the significant
component of the wave are the $h_{ij}$. The source term becomes

$\displaystyle J^{\mathrm{eff}}_{\ \nu}$
$\displaystyle=\Phi_{0}\,F^{0\lambda\,(0)}\,h^{\prime}_{\lambda\nu}+\Phi_{i}\,F^{i\lambda\,(0)}\,h^{\prime}_{\lambda\nu}$
(12a) $\displaystyle=F^{0\lambda\,(0)}\,h^{\prime}_{\lambda\nu}$ (12b)
$\displaystyle=F^{0j\,(0)}\,h^{\prime}_{ij}$ (12c)

So the fact that $J^{\mathrm{eff}}_{\ \nu}=0$ implies that
$F^{0j\,(0)}\,h^{\prime}_{ij}=0$. At any point of space time, we can choose a
spatial coordinate system such that $F^{0j\,(0)}=A\,\delta^{j}_{1}$. Then we
have $F^{0j\,(0)}\,h^{\prime}_{ij}=0,$ that leads to $A\,{h^{\prime}}_{i1}=0$
so $A=0$ since the GW is non-zero. So $F^{0j\,(0)}=0=q\,\Phi^{j}$, because
$\Phi^{j}=\eta^{0j}=0$. Knowing that $\Phi_{0}=1$ and $\Phi_{i}=0$, we can
show that $F^{0j\,(0)}=\Phi_{\alpha}\,F^{\alpha j\,(0)}$. We can also consider
that $\Phi^{0}=0$ and $F^{\alpha 0}=-F^{0\alpha}$ to conclude that
$\Phi_{\alpha}F^{\alpha\mu\,(0)}=q\,\Phi^{\mu}$. This completes our
demonstration. Let us now particularise the final result
$\Phi_{\alpha}F^{\alpha\mu\,(0)}=q\,\Phi^{\mu}$ in a illustrating case. In
cartesian coordinates, we can write down the null vector
$\Phi_{\alpha}=(k,0,0,k)$ with $k$ the wave vector of the incident GW which is
therefore propagating along the $z$-direction. The above-mentioned condition
Eq.(10) now leads now to two constraints on the EM field : $E^{x}+B^{y}=0$ and
$E^{y}-B^{x}=0$. If one considers the case when there is no electric field,
then this condition implies that the components of the outer magnetic field
that are transverse to the direction of GW propagation vanish:
$B^{x}=B^{y}=0\cdot$ In other words, any longitudinal magnetic field $B^{z}$
does not produce any EM wave by inverse Gertsenshtein effect (since
$S_{\mu\nu}=0$). To produce GW by this mechanism, one needs $S_{\mu\nu}\neq 0$
or, equivalently, a non vanishing magnetic field in the direction transverse
to the GW propagation ($B_{\perp}\neq 0$).

This is the reason why experiments like ADMX [39] do not have the right
configuration to detect GW. Indeed, they use a longitudinal outer magnetic
field which can therefore only interact with GW propagating transversely to
it. This interaction can only produce EM waves that are in the same direction
as the constant magnetic field but this is forbidden in the TM cavity they are
using (since only transverse excitation modes are allowed, not longitudinal
ones). To turn a haloscope into a HFGW detector, then one simply needs to
rotate the outer magnetic field by a quarter of turn.

Let us now present our proposed experimental set-ups. One can either consider
the resonance of the induced EM waves inside a cylindrical cavity of radius
$R$ or inside a waveguide made of two (or more) concentric open cylinders with
inner radius $R_{1}$ and outer radius $R_{2}$. We denote by $L$ the length of
the resonators and by $B^{(0)}_{\rm ext}$ the external magnetic field, assumed
of constant magnitude for simplicity. A schematic representation of our
cavities can be found in the figure 1.

Figure 1: Schematic representation of the experimental designs: a cylindrical
TM cavity (top) and TEM waveguide (bottom), into an external static and
transverse magnetic field.

We briefly present here the responses of these cavities to an incoming GW
signal. The interested reader will find the details of the computations at the
end of this paper, in the Appendix. In the following, we will assume $c=1$.
The starting point is the induction of EM waves when the GW passes
perpendicularly to the static magnetic field, as described by Eq. (7).
Considering the outer magnetic field along the x-direction :
$\vec{B^{(0)}_{\rm ext}}=B^{(0)}_{\rm ext}\vec{e_{x}}\cdot$, we obtain the
following wave equation for the induced magnetic field $\vec{B}^{(1)}$

$\left(-\frac{\partial^{2}}{\partial
t^{2}}+\vec{\Delta}\right)\vec{B}^{(1)}=B^{(0)}_{\rm
ext}\begin{pmatrix}\frac{\partial^{2}h_{+}}{\partial
z^{2}}\cos(\phi)+\frac{\partial^{2}h_{\times}}{\partial z^{2}}\sin(\phi)\\\
-\frac{\partial^{2}h_{+}}{\partial
z^{2}}\sin(\phi)+\frac{\partial^{2}h_{\times}}{\partial z^{2}}\cos(\phi)\\\
0\end{pmatrix}$ (13)

where $h_{+}$ and $h_{\times}$ are the usual polarizations of the incoming GW
in the traceless-transverse gauge. Although there is also an induced electric
field $\vec{E}^{(1)}$, the response of the detector is dominated by the
induced magnetic field, as we shall see below.

We can then project this Eq.(13) on the proper functions of the laplacian
operator in cylindrical coordinates. This spectral decomposition is given by

$B^{(1)}_{r,\phi}(t,r,\phi,z)\approx\sum_{k,m,n}\hat{b}^{r,\phi}_{k,m,n}(t)\cdot\psi^{r,\phi}_{kmn}(r,\phi,z)$
(14)

where $\psi^{r,\phi}_{kmn}(r,\phi,z)$ are the cylindrical harmonics that
satisfy the boundary conditions of our EM cavities. The result of this
spectral decomposition allows reducing the above wave equation Eq.(13) to an
ordinary differential equation describing a forced harmonic oscillator for
each spectral mode $\hat{b}^{r,\phi}_{k,m,n}$ in our cavity:

$\frac{d^{2}\hat{b}^{r,\phi}_{k,m,n}}{dt^{2}}+\Omega^{2}_{kn}\hat{b}^{r,\phi}_{k,m,n}=\hat{s}^{r,\phi}_{k,m,n}(t)$
(15)

where $\Omega^{2}_{kn}$ are the proper frequencies of the resonant cavities
and $\hat{s}^{r,\phi}_{k,m,n}(t)$ are the spectral coefficients of the source
of the wave equation Eq.(13).

The energy variation $\Delta\mathcal{E}$ inside the cavity is given by, at the
leading order (see our Appendix for details),

$\Delta\mathcal{E}\approx\frac{2\pi
B_{0}}{\mu_{0}}\cdot\sum_{k}\mathcal{I}_{k}\hat{b}_{k,1,0}(t)$ (16)

where the coefficients $\mathcal{I}_{k}$ arise from the spatial averaging in
the transverse directions (these adimensional quantities depends on the cavity
geometry only). Let us recall that the numbers $k$, $m$ label the transverse
decomposition in the radial and azimuthal direction, while number $n$ labels
the different longitudinal modes. We can see that only the $(k,1,0)$ modes,
which are constant in the longitudinal direction $z$ (since $n=0$), are
contributing to the energy variation at first order in $B^{(1)}$. Since these
$(k,1,0)$ modes do not propagate along $z$, one can see that there is no
spatial phase shift with this energy variation at first order. These $(k,1,0)$
modes of the induced magnetic field are sourced by

$\hat{s}_{k,1,0}^{r,\phi}(z,t)=\pi
B_{0}L^{2}\mathcal{I}_{k}\int_{-L/2}^{L/2}\frac{\partial^{2}h_{+}(z,t)}{\partial
z^{2}}dz$ (17)

The detailed computations from Eq.(7) to Eqs.(15-17) is available at the end
of this paper, in the Appendix.

A dimensional analysis of Eq.(16) leads to the following estimation for the
order of magnitude of the induced energy variation inside the resonator

$\Delta\mathcal{E}\approx\frac{2\pi B_{0}^{2}L^{3}}{\mu_{0}}\mathcal{H}_{\rm
GW}\mathcal{F}$ (18)

where $\mathcal{H}_{\rm GW}$ is the (dimensionless) amplitude of the strain of
the GW and where $\mathcal{F}$ is a adimensional geometrical factor444In other
words, we extract all the dimensional factors in the equation (16), letting
just one adimensional expression that depends on the geometry of the detector
and its frequency sensitivity. accounting for the shape of the resonator
(length and diameter). This factor $\mathcal{F}$ is of the order of unity when
the spectrum of the incoming GW matches the resonance bandwidth. In the other
cases, we must consider a frequency dependant geometrical factor
$\mathcal{F}(\omega)$.

Please note that with our model there is no temporal phase shift in the
conversion process. Indeed there is no imaginary part in the Fourier transform
of $\Delta E$ and so the complex argument is null. This is mainly due to the
fact that Eq.(15) is purely harmonic, without any dissipation, and therefore
no phase shift can happen. Instead of a derivation in the time domain, one
could also use a frequency approach of the cavities responses to demonstrate
this, but this goes beyond the scope of this paper. However, the ohmic losses
of energy in the sidewalls of the cavities could be represented by a
dissipative term in equation (15), and so a phase shift could appear in the
conversion process for resistive cavities. Although ohmic losses will lower
the efficiency of the resonance mechanism investigated here, this can be
avoided by working with superconducting cavities.

##  V HFGWs from planetary-mass primordial black hole mergers

PBH binaries may have formed through two different channels. First, in the
early Universe, when two PBHs form sufficiently close to each other for their
dynamics to decouple from the expansion of the Universe, before the matter-
radiation equality [42, 11]. Second, by tidal capture in dense environments
[9, 10], such as ultra-faint dwarf galaxies. In this section, we review the
motivations to consider planetary-mass PBHs binaries and we estimate their
expected merging rate and gravitational-wave signal, for those two channels.
We then calculate the astrophysical range of resonant electromagnetic
detectors as a function of their strain sensitivity. Finally, we compute for
each formation channel the limits that could be set on the abundance of
planetary-mass PBHs.

###  V.1 Motivations

The progenitor masses and low effective spins of the black hole mergers
detected by LIGO/Virgo have revived the interest for PBHs in the
$[1-100]M_{\odot}$ range [9, 10, 11, 43]. However it is debated if PBHs could
constitute only a small fraction, or up to the totality of the DM in the
Universe. In this context, detecting a sub-solar black hole would almost
clearly point to a primordial origin555See however [44, 45] for another
subsolar black hole formation channel, with different spin predictions, in a
specific dark matter scenario.. Going beyond the simplest but unrealistic
assumption of a monochromatic mass function, the distribution of PBHs could
span several decades of masses, as it is the case if curvature fluctuations at
the origin of PBH formation are nearly scale invariant – a generic prediction
of inflation – or come from a broad peak in their power spectrum. Then the
known thermal history of the Universe, in particular the QCD transition at
$\sim 100$ MeV and the electroweak epoch at $\sim 100$ GeV, should have left
imprints in the PBH mass function [46, 12], whatever is the mechanism at the
origin of those curvature fluctuations. These features take the form of a high
peak at the solar mass scale and two bumps at $\sim 30M_{\odot}$ and $\sim
10^{-5}M_{\odot}$. Such an extended mass function could explain a series of
puzzling observations (see [47, 12] and references therein) such as unexpected
microlensing events, LIGO/Virgo black hole mergers, some properties of ultra-
faint dwarf galaxies, unexpected correlations in X-ray and infrared cosmic
backgrounds, and super-massive black holes at high redshifts. The bump in the
planetary-mass range is consistent with recent detections of star and quasar
microlensing events [7, 48, 49], which suggest a DM fraction of $f_{\rm
PBH}\sim 0.01$ made of compact objects, quite much than one can expect for
floating planets, but that could be expected for PBHs in the unified scenario
presented in [12]. Recently, the possible detection of a stochastic GW
background at nanoHerz frequencies by NANOGrav [50] may also hint at the
existence of PBHs with planetary [51] or stellar [52, 53, 54] masses, and Wang
et al. [55, 56] puts some constraints on the PBH abundance for the current
detectors to probe a stochastic GW background made of PBHs. However all these
observations could have another origin and the derived limits are still
subject to large astrophysical uncertainties. In the future, it is therefore
important to find complementary ways to probe the existence of such objects,
and to distinguish their nature and origin.

As we show in this paper, HFGW detectors will have the ability to detect or
set new limits on the abundance of light, subsolar PBHs, of mass $m_{\rm
PBH}\sim 10^{-5}M_{\odot}$. HFGWs are indeed produced during the merging phase
of such light PBHs. The frequency associated to the innermost stable circular
orbit (ISCO), when the GW emission is close to maximal, is given by

$f_{\rm ISCO}=\frac{4400\,{\rm Hz}}{(m_{1}+m_{2})/M_{\odot}}~{}.$ (19)

with $m_{1}$ and $m_{2}$ the masses of the two binary components. A frequency
of $200$ MHz thus corresponds to a PBH mass of $10^{-5}M_{\odot}$, the same
order than the mass of the lenses at the origin of the microlensing events
reported in [7]. Nevertheless, for being an interested HFGW signal, one needs
to investigate if the merging rate of such PBHs can lead to at least
$\mathcal{O}(1)$ mergers per year within the HFGW detector range.

PBHs therefore constitute a target of much interest for our experimental
concept of EM detection of HFGWs. From the amplitude and spectral response of
the resonant detectors, we will characterize the expected signals from PBHs
mergers for a large range of progenitor masses in the interval
$[10^{-8};10^{-3}]\,M_{\odot}$, located at $1$ Gpc distance. Then, for a given
detector sensitivity, we will compute the expected limits on the PBH
abundance.

###  V.2 Gravitational-waves from inspiraling binaries

A good estimation of the GW strain produced at a given frequency $f_{\rm GW}$
during the inspiralling phase of a black hole binary is provided by the Post-
Newtonian approximation [57],

$h\approx\frac{2}{D}\left(\frac{G\mathcal{M}}{c^{2}}\right)^{5/3}\left(\frac{\pi
f_{\rm GW}}{c}\right)^{2/3}~{},$ (20)

where $\mathcal{M}\equiv(m_{1}m_{2})^{3/5}/(m_{1}+m_{2})^{1/5}$ is the binary
chirp mass and $D$ is the distance to the observer. The GW emission is close
to maximal at the ISCO frequency and, for a given chirp mass, for equal mass
binaries, $m_{1}=m_{2}=m_{\rm PBH}$. For an experiment with a detector strain
sensitivity $h_{\rm det}$ at this maximal frequency, the corresponding
astrophysical reach $D_{\rm max}$ is given by

$D_{\rm max}\approx 1.6\times\frac{(m_{\rm PBH}/M_{\odot})}{h_{\rm det}\times
10^{20}}{\rm Mpc}.$ (21)

For instance, for a strain sensitivity of $h_{\rm det}\sim 10^{-25}$ and
$m_{\rm PBH}\sim 10^{-5}M_{\odot}$, eventual mergers towards the galactic
center, in the Milky Way DM halo, or in satellite ultra-faint dwarf galaxies,
could be detected. For a better sensitivity down to $h_{\rm det}\sim
10^{-30}$, corresponding to the optimal sensitivity of the proposed designs of
resonant EM detectors, then one would probe planetary-mass PBH mergers in more
distant galaxies.

###  V.3 Merging rate of primordial binaries

If PBHs are spatially randomly distributed at formation, it happens that two
PBHs form so close to each other that their gravitational attraction
overpasses the effect of the Hubble-Lemaître expansion at some point before
matter radiation equality. In such a case, they directly form a binary whose
orbital parameters and lifetime do not only depend on the two black hole
masses but also on the mass and distance of the nearest PBHs. Eventually, it
takes of the order of the age of the Universe for the PBH binary to merge. The
merging rates $\tau$ today associated with this binary formation channel and
an arbitrary mass function have been evaluated in [58, 59, 60, 61] as

$\displaystyle R^{\rm prim}(m_{1},m_{2})$ $\displaystyle\equiv$
$\displaystyle\frac{{\rm d}\tau}{{\rm d}\ln m_{1}{\rm d}\ln m_{2}}$ (22)
$\displaystyle\approx$ $\displaystyle\frac{1.6\times 10^{6}}{\rm
Gpc^{3}yr}f_{\rm PBH}^{2}f(m_{1})f(m_{2})f_{\rm sup}$ $\displaystyle\times$
$\displaystyle\left(\frac{m_{1}+m_{2}}{M_{\odot}}\right)^{-\frac{32}{37}}\left[\frac{m_{1}m_{2}}{(m_{1}+m_{2})^{2}}\right]^{-{\frac{34}{37}}}$

where $f(m)$ is the today density distribution of PBHs normalized to one
($\int f(m){\rm d}\ln m=1$) and $f_{\rm PBH}$ is the integrated DM fraction
made of PBHs. We also define an effective parameter

$\tilde{f}_{\rm PBH}(m_{\rm PBH})\equiv f_{\rm PBH}f(m_{\rm PBH})f_{\rm
sup}^{1/2}$ (23)

that includes a rate suppression factor ($f_{\rm sup}$) to take into account
the possible rate suppression due to binary disruption by early-forming
clusters, an effect put in evidence by N-body simulations when $f_{\rm
PBH}\gtrsim 0.1$ [62]. In such a case, one can recover the LIGO/Virgo merging
rates inferred from the recent detections of GW190425, GW190521 and GW190814
involving at least one BH in the mass gaps, with $f_{\rm PBH}=1$ and $f_{\rm
sup}\simeq 0.0025$ [63]. In the opposite case, $f_{\rm sup}=1$ and
$\tilde{f}_{\rm PBH}$ simply represents the DM density fraction made of PBHs
at a given mass and within a unit logarithmic mass interval.

If one considers the merging rates of equal-mass binaries that produce the
largest strain signal, one gets

$\displaystyle R^{\rm prim}(m_{\rm PBH})$ $\displaystyle\approx$
$\displaystyle\frac{3.1\times 10^{6}}{\rm Gpc^{3}yr}\tilde{f}_{\rm
PBH}^{2}\left(\frac{m_{\rm PBH}}{M_{\odot}}\right)^{-0.86}.$ (24)

In turn, one can determine the radius of the sphere in which one expects one
event per year,

$D^{\rm prim}_{1}=\left(\frac{4\pi}{3}R^{\rm prim}\right)^{-1/3}\approx
4.2\,{\rm Mpc}\times\tilde{f}_{\rm PBH}^{-2/3}\left(\frac{m_{\rm
PBH}}{M_{\odot}}\right)^{0.29}~{}.$ (25)

For simplicity we neglected the effects of redshift that are anyway
insignificant for most of the considered cases. For instance, in the scenario
of [12, 63] with $f_{\rm PBH}=1$, $f_{\rm sup}=0.0025$ and
$f(10^{-5}M_{\odot})\simeq 10^{-2}$, one gets $D_{1}^{\rm
prim}(10^{-5}M_{\odot})\approx 23\,{\rm Mpc}$. Using Eqs. (20) and (21), one
then obtains the required GW strain sensitivity to detect one of these merger
events per year,

$\displaystyle h_{1}^{\rm prim}$ $\displaystyle\approx$ $\displaystyle
3.8\times 10^{-21}\tilde{f}_{\rm PBH}^{2/3}\left(\frac{m_{\rm
PBH}}{M_{\odot}}\right)^{0.7}$ (26) $\displaystyle\approx$ $\displaystyle
8.3\times 10^{-19}\tilde{f}_{\rm PBH}^{2/3}\left(\frac{\rm
Hz}{f}\right)^{0.7}~{},$ (27)

which can be typically targeted by GW experiments operating at frequencies
from kHz up to GHz. This relation can be inverted to obtain a limit on the DM
fraction at a given mass (if $f_{\rm PBH}<0.1$) in case of null detection, as
a function of the strain sensitivity,

$\displaystyle\tilde{f}_{\rm PBH}\lesssim 9.1\left[\frac{h_{\rm
det}}{10^{-20}}\right]^{3/2}\left(\frac{m_{\rm
PBH}}{M_{\odot}}\right)^{-1.07}.$ (28)

However, the strain sensitivity of the detector depends on the waveform and
signal duration, which depend on the PBH mass. It is thus more adequate to
compute a limit on the PBH abundance taking these effects into account and
instead assuming an EM power sensitivity, which we do in the next section. For
instance, we obtain with the two proposed experimental designs and a power
sensitivty of $10^{-10}$W, limits that are competitive with the current
microlensing limits at the same mass scale. These are represented in our final
Figure 7.

Finally, we point out that when our analysis was being finalized, the authors
of [64] have claimed that the rates from primordial binaires are highly
suppressed compared to previous calculation. The reason is a subtle general
relativistic effect arising when one considers geodesics in black hole
exterior spacetime metrics that are FLRW asymptotic. If this claim is correct,
primordial binaires are by far outside the reach of EM detectors, but one can
nevertheless consider the merging rates inside PBH clusters, which we detail
thereafter.

###  V.4 Merging rate from PBH clusters

The second binary formation channel is through dynamical capture in dense PBH
halos. As any other DM candidate, PBHs are expected to form halos during the
cosmic history, and their clustering properties determine the overall merging
rate. For instance, for a monochromatic mass spectrum and a standard Press-
Schechter halo mass function, one gets a rate [9]

$R^{\rm capt}\sim f_{\rm PBH}^{2}\times\mathcal{O}(1-100)\,{\rm
yr^{-1}\,Mpc}^{-3}$ (29)

that is independent of the PBH mass. For more realistic extended mass
functions, the abundance, size and evolution of DM halos, partially or
entirely made of PBHs, is impacted by several effects, see e.g. [65, 66, 67,
68, 69, 70, 71, 72, 73, 74, 75]. Let us mention a Poissonian noise from the
discrete nature of PBHs, a seeding effect from heavy PBHs, the enhancement of
the primordial power spectrum at the origin of PBH formation, the dynamical
heating and evaporation of clusters, etc. These effects can either boost or
suppress the merging rates from clusters and make the whole clustering
dynamics a rather complex and model-dependent process, subject to large
uncertainties. Invoking clustering is also crucial to evade microlensing
limits on stellar-masses [76, 77] in scenarios with $f_{\rm PBH}=1$. As an
alternative of using uncertain theoretical predictions, on can instead infer
an upper limit on the PBH merging rate from LIGO/Virgo observations, see e.g
[63]. The merging rate from tidal capture in PBH clusters for an arbitary mass
function is given by [10, 63]

$\displaystyle R^{\rm capt}(m_{1},m_{2})$ $\displaystyle\equiv$
$\displaystyle\frac{{\rm d}\tau}{{\rm d}\ln m_{1}{\rm d}\ln m_{2}}$ (30)
$\displaystyle\approx$ $\displaystyle R_{\rm clust}f_{\rm PBH}^{2}\times
f(m_{1})f(m_{2})$ $\displaystyle\times$
$\displaystyle\frac{(m_{1}+m_{2})^{10/7}}{(m_{1}m_{2})^{5/7}}\rm{yr^{-1}Gpc^{-3}},$

where $R_{\rm clust}$ is an effective parameter encompassing the clustering
properties. For $f_{\rm PBH}=1$ and $R_{\rm clust}\approx 450$, these rates
are consistent with the latest LIGO/Virgo observations, for a broad PBH mass
function impacted by the transient reduction of the critical threshold of PBH
formation at the QCD epoch. As already mentioned, this effect is unavoidable
and may have induced a peak around $2.5M_{\odot}$ and a bump around
$30\,M_{\odot}$ in the PBH mass function. In this scenario, around one percent
of the DM could be made of planetary-mass PBHs around $10^{-5}M_{\odot}$. Like
for primordial binaries, we define an effective parameter

$\tilde{f}_{\rm PBH}\equiv\left(\frac{R_{\rm clust}}{450}\right)\times f_{\rm
PBH}f(m_{\rm PBH})$ (31)

representing the DM density fraction at a given mass and per logarithmic mass
interval, in the above mentioned scenario. One then obtains the merging rate
for equal-mass binaries,

$R^{\rm capt}(m_{\rm PBH})=1.2\times 10^{3}\tilde{f}_{\rm PBH}^{2}~{},$ (32)

the corresponding source distance $D_{1}^{\rm capt}$,

$D_{1}^{\rm capt}\approx 58\,{\rm Mpc}\times\tilde{f}_{\rm PBH}^{-2/3},$ (33)

and the required experimental strain sensitivity to detect one event per year,

$h_{\rm 1}^{\rm capt}\approx 2.7\times 10^{-22}\tilde{f}_{\rm
PBH}^{2/3}\left(\frac{m_{\rm PBH}}{M_{\odot}}\right).$ (34)

This GW signal is therefore typically lower than for PBH binaires on
planetary-mass scales.

Finally, like for the case of primordial binaries, we have derived the
expected limits on $\tilde{f}_{\rm PBH}$ for a given experimental strain
sensitivity,

$\displaystyle\tilde{f}_{\rm PBH}\lesssim 2.5\times 10^{4}\left[\frac{h_{\rm
det}}{10^{-20}}\right]^{3/2}\left(\frac{m_{\rm
PBH}}{M_{\odot}}\right)^{-3/2}.$ (35)

As an example, if $h_{\rm det}=10^{-30}$ and $m_{\rm PBH}=10^{-5}$, one gets
$\tilde{f}_{\rm PBH}\lesssim 7\times 10^{-4}$, which is better than the
current microlensing limits. Again, it is more accurate to assume a power
sensitivity rather than a strain sensitivity. Doing so, the corresponding
limits on $\tilde{f}_{\rm PBH}$ have been represented in our final Figure 7.

##  VI Probing PBH mergers with resonant EM detectors of HFGWs

Let us consider GW trains produced during the final inspiraling phase of PBHs
binaries of different masses and passing through the resonator and let us
analyze the induced EM radiation. For simplicity, we consider a GW propagation
colinear with the longitudinal axis of the EM resonator and perpendicular to
the outer magnetic field. In this case, the GW can be approximated by a plane
wave, i.e. $h_{+,\times}=h_{+,\times}(z,t)$ since the radius of the detector
is much smaller than the incoming wavefront for a distant source. Some
inclination of the direction of the incoming GW would result in a signal of
lower amplitude, since only the component of the GW in the direction of the
outer magnetic field contributes to the inverse Gertsenshtein effect. Yet,
this ideal case allows to illustrate the physical process and to estimate the
expected response of the detector and the output signal. We have computed the
typical EM signals for resonators of various shapes. Post-Newtonian time-
domain GW waveforms are generated by using the LALSuite library [78], assuming
a 4PN approximation. For all the simulations, we choose a signal sampling
frequency that is four time the ISCO frequency [Eq. (19)]. The initial
frequency of the signal is set to $f_{\rm ISCO}/25$. On Fig. 2, we provide an
example of the GW waveform produced by the merging of a PBH binary with
component masses of $10^{-5}M_{\odot}$ and located at a distance of $1\,\rm
Gpc$. The amplitude of the GW strain at reception, denoted $\mathcal{H}_{\rm
GW}$, is of order $10^{-28}$ and the signal duration is of order
$10^{-5}s\cdot$.

Figure 2: Left: GW waveform $h_{+}$ for the final inspiraling phase of a PBH
binary with component masses of $10^{-5}M_{\odot}$, at a distance of $1\,\rm
Gpc$, in the post-newtonian (4PN) approximation. Right: corresponding GW
signal power spectrum.
---
Figure 3: Top: time evolution of the induced EM power in the two resonators.
Bottom: GW signal power spectrum (blue) and induced power spectrum (green).
The detectors are 1-meter long, in a 5 T constant magnetic field. The outer
radius is 5 m and for TEM coaxial case, with an inner radius of 10 cm. The
peaks in the power spectrum correspond to the resonance frequencies,
represented by the red dots.

Then, by using Eq. (16) and solving the forced harmonic oscillator equations,
Eqs. (15), we compute the induced EM power for the two designs of resonators.
Our results are displayed on Fig. 3, for 1-meter long resonators in a 5 T
constant magnetic field. The radius of the resonators is set to 5 m and for
the TEM coaxial design, we consider a 10 cm inner radius. As expected, the
induced power increases near the merger. In each case, we have also computed
the frequency spectrum of the induced power, which exhibits peaks
corresponding to the resonance frequencies, coming from Eqs. (X) and (XI) in
the Appendix. The power rms values are $1.00\times 10^{-10}$ W for the TEM
resonator and $1.03\times 10^{-10}$ W for the TM resonator.

Figure 4: The first five proper frequencies of the cavity (TM resonator,
straight lines) and the waveguide (TEM resonator, dashed line) as a function
of their outer radius (assuming a length of $L=1$ m and an inner radius of
$R_{1}=0.1$ m for the waveguide).

Fig. 4 shows the first five proper frequencies of the TEM and TM cavities as a
function of their outer radius, within the range $10$ cm to $10$ m, for a
detector of one meter length. For the TM resonator, these frequencies range
from 10 MHz to GHz, while the TEM waveguide can reach higher frequencies for
the same detector size. In the case of an outer radius almost equal to the
inner one (thin case), the resonance frequencies can be one order of magnitude
larger than for a TM resonator. We point out that it would even be possible to
use several concentric TEM waveguides to further extend the range of resonance
frequencies. Fig 4 therefore illustrates how one could play with the geometry
of the experimental set-up in order to optimize the detector response to some
motivated GW signal. The PBH mass is another parameter to consider. For the
same detectors as in Fig. 3, we have simulated the detection of the signal for
two additional PBH masses, of $10^{-3}$ and $10^{-7}M_{\odot}$, the
corresponding power spectra are shown in Fig. 5. For $10^{-3}\,M_{\odot}$, the
resonance frequencies are higher than the ISCO frequency. The resulting EM
power spectrum in the detector exhibits a continuous set of frequencies that
matches the incoming GW power spectrum followed by the excitation of the
detector resonance frequencies that constitutes the highest values in the
induced radiation power spectrum. The continuous set of induced frequencies
comes from the particular solution of the forced oscillator Eq. (15), while
the excitation of the detector resonance frequencies comes from the general
solution.

In the $10^{-7}\,M_{\odot}$ case, the detector resonance frequencies are lower
than the main GW signal frequencies. Therefore we have a clear separation
between the EM power spectrum that is induced by the resonance frequencies of
the detector and the one induced by a range of frequencies inherited from the
incoming GW signal. In Fig. 5, the part of the power spectrum induced by the
resonance frequencies is overestimated. Indeed the numerical FFT of the
incoming GW leads to a plateau at low frequencies as shown in Fig. 2. As a
result this plateau will be excited by the proper frequencies of the cavity
leading to an overestimation of this part of the induced power spectrum. We
show below how to avoid this numerical limitation.

---
Figure 5: Same as the bottom panel of Fig. 3, but for a PBH mass of
$10^{-3}\,M_{\odot}$ (top pannel) and $10^{-7}\,M_{\odot}$ (bottom panel). In
the first/second case, the GW frequency is below/above the resonance
frequencies (represented by the red points) of the detector.

$m_{\text{PBH}}$ ($M_{\odot}$) time (s) $\mathcal{H}_{\rm GW}$
$f_{\text{ISCO}}$ (Hz) $P_{\text{RMS}}$ TEM (W) $P_{\text{RMS}}$ TM (W)
$10^{-3}$ 5.43e-03 1.15e-26 2.20e+06 2.37e-14 3.19e-14 $10^{-4}$ 5.43e-04
1.37e-27 2.20e+07 2.98e-12 4.96e-12 $10^{-5}$ 5.43e-05 1.40e-28 2.20e+08
1.00e-10 1.03e-10 $10^{-6}$ 5.43e-06 1.17e-29 2.20e+09 1.51e-11 6.31e-12

Table 1: GW signal duration, maximal strain, ISCO frequency and the
corresponding induced power in the resonant cavity or waveguide, for different
values of the PBH mass. The detectors are 1 m long for an outer radius of 5 m.
The inner radius in the TEM case is 10 cm. The transverse static magnetic
field is set to $5$ T. The minimal frequency of the GW waveform signal is set
to $f_{\rm ISCO}/25$. The bandwidth of the resonant frequencies considered is
$\left[2\cdot 10^{7},2\cdot 10^{8}\right]$ Hz  Figure 6: RMS values of the
induced power (left) and energy variation (right) in the TEM (blue) and TM
(red) detectors as in Table 1, as a function of the PBH mass, for mergers
located at a distance of one Gpc. The excitation of resonant frequencies
boosts the signal in the range between $10^{-6}$ and $10^{-4}M_{\odot}$. Below
$10^{-6}M_{\odot}$ we plot in dashed lines the expected behaviour due to
numerical limitation in our simulations. Asymptotic behaviour coming from our
analysis has been plotted in black dotted lines.

Finally, we have computed the power in a 1-meter long resonator of 5-meter
radius in a 5T magnetic field, induced by the merging of PBHs with a broad
range of masses. Our results are summarized in Table 1. We find that the
induced power is maximal for a PBH mass around $10^{-5}\,M_{\odot}$, i.e. when
the corresponding GW frequency lies within the range of the resonance
frequencies. We have shown in Fig. 6 the rms power as a function of the PBH
mass, for the same experimental configuration. In order to do so, we have
considered 41 simulated signals of PBH mergers for 61 different PBH masses
(i.e., 41 signals per discretization point in mass, and each of these signals
corresponding to different initial frequency ranging from $f_{\rm ISCO}/30$ to
$f_{\rm ISCO}/10$). Because of the numerical problem at masses beyond
$10^{-6}M_{\odot}$(poor resolution of the FFT of the GW signal at low
frequencies), we extrapolate our data with the expected behaviour that we will
develop further in this section. When the GW spectrum covers the fundamental
resonance modes of the detector, its response is maximal and this occurs for
PBH masses covering two orders of magnitude around $10^{-5}M_{\odot}$.

In addition, we have also shown on Fig. 6 the rms energy variation inside the
cavity, as a function of the PBH mass. The induced energy typically
corresponds to millions of photons. It is well above the sensitivity of the
ADMX experiment, which has achieved an effective instrumental noise
temperature of order $\mathcal{O}(1)$K at similar frequencies [39, 40] and
with a similar value of the magnetic field. Further work is however needed in
order to quantify more accurately the energy or power sensitivity that could
be achieved with a similar technology.

For masses larger than $10^{-4}\,M_{\odot}$, the frequencies associated to the
GWs are smaller than the proper frequencies of the cavity and so the amplitude
of the total solution of (15) is proportional to the amplitude of the source
term, divided by the square of the proper frequencies of the cavity. So the
behavior of the particular solution is the same as the source term. From
Eq.(13), we know that the source term is proportional to the second time
derivative of the strain. On one hand, the GW strain is proportional to PBH
mass (see Eq.(20)). On the other hand, the maximal GW frequency is inversely
proportional to the PBH mass. As a result, the energy released in the detector
goes like $1/m_{\rm PBH}$, which explains the linear decreases observed at
large mass.

As mentionned earlier, because of a numerical limitation we do not plot the
induced energy and RMS power for masses smaller than $10^{-6}\,M_{\odot}$.
Nonetheless we can extrapolate the expected behaviour of the mass dependence.
The analysis is the same as in the large mass regime except that this time it
is the proper frequencies of the cavity that are smaller than the frequencies
associated to the GWs. Thus the amplitude of the total solution of (15) is now
proportional to the amplitude of the source term, divided by the square of the
GW frequencies. As these frequencies are inversely proportional to the PBH
mass, the behaviour of the energy released in the detector is $m_{\rm
PBH}^{2}$ times what it was in the previous case. So the energy variation goes
like $m_{\rm PBH}$. We will show this in a forthcoming paper by using a
frequency domain approach.

Finally, we observe that there is no significant difference between the TEM
and the TM detectors, highlighting some robustness with respect to the choice
of the experimental set-up.

As a final step, we have combined our calculations of the PBH merging rates
and of the induced power as a function of the PBH mass for mergers at a fixed
distance of 1 Gpc, in order to forecast limits on the PBH abundance for a
fixed detector sensitivity of $10^{-10}$W and a survey of one year. For doing
so, we have used the calculated values of $D_{1}$ for the two binary formation
channels. Given that the GW strain is inversely proportional to the distance
of the source, the EM power released in the detector is also inversely
proportional to the distance, see Eq. (18). Our final results are displayed in
Fig. 7. We can clearly see that the resonance plays a role by boosting the
detection limits to be as low as $\tilde{f}_{\rm PBH}\lesssim 10^{-8}$ for
primordial binaries, and $\tilde{f}_{\rm PBH}\lesssim 10^{-4}$ for tidal
capture in clusters. . Resonant EM-based HFGW detectors could therefore set
unprecedented and independent limits on the abundance of planetary-mass PBHs.

Figure 7: Expected limits on the effective parameter $\tilde{f}_{\rm PBH}$
corresponding to the dark matter density fraction in PBHs at a given mass and
per logarithmic mass interval in the two models described in the text:
primordial binaries (blue limit) and tidal capture in halos (red limit). These
limits are computed for the proposed experimental designs of EM resonant HFGW
detectors, assuming a power sensitivity of $10^{-10}$W, achievable with
current technology. The orange curve shows the possible abundance of
planetary-mass PBHs inferred form recent microlensing observations towards the
galactic bulge [7].

##  VII Conclusion

The detection of dozens of black hole or neutron star mergers by LIGO/Virgo
and, more recently, the plausible observation of a stochastic GW background
with pulsar timing arrays (PTAs) by NANOGrav, have revealed the bright future
of GW astronomy. LIGO/Virgo probe GW frequencies between $10$ and $10^{4}$ Hz
and PTAs of $10^{-9}$ Hz. In the future, ground-based and space-detectors like
Einstein Telescope, Cosmic Explorer and LISA will fill the gap between these
two ranges and probe other GW sources like intermediate mass black hole
binaries. On the other side of the GW spectrum, HFGW detectors might be
equally interesting and probe cosmological stochastic backgrounds or exotic
planetary-mass compact objects like PBHs. Motivated by recent claims that PBHs
with a wide mass distribution could explain, among others, the masses and
spins of LIGO/Virgo black holes, the NANOGrav observation and the dark matter
in the Universe, we envision their detectability with two designs of resonant
EM detectors based on the inverse Gertsenshtein effect and operating at
frequencies of order $100$ MHz.

The released energy in a EM resonator has been calculated for fourth-order
post-Newtonian GW waveforms corresponding to the final inspiraling phase of
light PBH binaries. The experimental apparatus either consists in a one-meter
long cylindrical resonant cavity (TM) or in a conducting waveguide (TEM),
together with a static 5 Tesla magnetic field orthogonal to their axis of
symmetry. For a $5$ meters radius resonator or a waveguide of $0.1$ meter
inner radius, we find a typical power variation of order of $10^{-10}$ W,
detectable with present technology. Similar resonant cavities have been
already built for axion searches, but their collinear magnetic field prevents
the detection of HFGWs. We have also studied the expected signal and its power
spectrum for different binary masses. Their merging rates have been estimated
for two PBH binary formation channels, at formation in the early Universe or
due to tidal capture in dense clusters. The optimal sensitivity is obtained
for PBH masses of order $10^{-5}M_{\odot}$. For the two considered models, we
forecast the stringent limits on the PBH abundance that could be set with such
experiments, of order $\tilde{f}_{\rm PBH}\lesssim 10^{-8}$ and
$\tilde{f}_{\rm PBH}\lesssim 10^{-4}$, respectively. Finally, one should keep
in mind that we can tune the parameters of the detector to match the frequency
range of highest interest.

Therefore HFGW therefore have the ability to set new constraints on the
fraction of dark matter made of light PBHs that would be complementary to
microlensing limits. In particular, they could be used to distinguish between
PBH and planetary origins of recently detected microlensing events towards the
galactic center, suggesting that about one percent of the DM could be made of
such objects.

HFGW detectors with similar designs could also be used to detect cosmological
stochastic GW backgrounds. Frequencies around $100$ MHz are of particular
interest in this context, because these are characteristic to GW sources at
energies close to the GUT scale, like (p)reheating, oscillons, phase
transitions or evaporating PBHs. In particular, we point out that a strain
sensitivity down to $h\sim 10^{-30}$ for achievable TM and TEM detectors is of
order of the typical amplitude of such cosmological backgrounds. Ongoing
research focuses on using a frequency analysis of the resonance mechanim to
get direct information about the frequency sensitivity of our detector.
Another ongoing work is computing the power variation in the detectors
generated by a stochastic, almost isotropic GW background. Other future
research topic is to characterize the expected experimental noise with such
detectors. The results for axion haloscope detection [39, 40] could be an
interesting starting point.

In summary, resonant electromagnetic HFGW detectors are ideal to probe various
aspects of fundamental physics, from early Universe cosmology to exotic
compact objects like planetary-mass primordial black holes. Our work
contributes to pave the way in this direction and provides strong motivations
to start the experimental development of such detectors, with currently
available technology.

Acknowledgments
This research used resources of the ”Plateforme Technologique de Calcul
Intensif (PTCI)” (http://www.ptci.unamur.be) located at the University of
Namur, Belgium, which is supported by the F.R.S.-FNRS under the convention No.
2.5020.11. The PTCI is member of the ”Consortium des Équipements de Calcul
Intensif (CÉCI)” (http://www.ceci-hpc.be).

## Appendix : Details of computations from Eq.(7) to Eqs.(15-17)

In the Lorenz gauge, an incoming gravitational plane wave can be written down,
in cylindrical coordinates $(t,r,\phi,z)$

$\displaystyle h_{rr}(\phi,z,t)$ $\displaystyle=$ $\displaystyle-
h_{\phi\phi}=h_{+}(z,t)\cos(2\phi)+h_{\times}(z,t)\sin(2\phi)$ $\displaystyle
h_{r\phi}(\phi,z,t)$ $\displaystyle=$ $\displaystyle-
h_{+}(z,t)\sin(2\phi)+h_{\times}(z,t)\cos(2\phi)$

with $h_{rr},h_{\phi\phi},h_{r\phi}$ are the metric perturbations in the non-
coordinate basis $(cdt,dr,r\,d\phi,dz)$ and where $h_{+}$ and $h_{\times}$ are
the usual polarizations of the incoming GW in the traceless-transverse gauge.

The incoming GW will interact with the component of the magnetic field that is
perpendicular to its direction of propagation. We therefore consider the outer
magnetic field along the x-direction : $\vec{B^{(0)}_{\rm ext}}=B^{(0)}_{\rm
ext}\vec{e_{x}}\cdot$ Any component of the magnetic field that would be
collinear to the direction of the propagation of the gravitational plane wave,
$\vec{B^{(0)}_{\rm ext}}\approx\vec{e_{z}}$, does not produce any EM wave
through the inverse Gertsenshtein effect. Indeed, since
$F\indices{{}^{(0)}_{xy}}=B_{z}$, one has that, from Eq.(8)

$S_{\mu\nu}=-\partial_{x}\left(\partial_{\mu}h\indices{{}_{y\nu}}-\partial_{\nu}h\indices{{}_{y\mu}}\right)F\indices{{}^{(0)\,xy}}+\left(x\leftrightarrow
y\right)\cdot$

Each term in the above equation will identically vanish in the case of a plane
gravitational wave propagating along the direction of the magnetic field
($h_{+,\times}=h_{+,\times}(z,t)$).

Let now assume our EM resonators are axially symmetric along the $z-$direction
and examine Eq. (7) in cylindrical coordinates. The external constant magnetic
field along the $x-$direction gives rise to only two components of the zeroth
order electromagnetic tensor

$\displaystyle F\indices{{}^{(0)}_{rz}}$ $\displaystyle=B^{(0)}_{\rm
ext}\sin(\phi)$ $\displaystyle F\indices{{}^{(0)}_{\phi z}}$
$\displaystyle=B^{(0)}_{\rm ext}\cos(\phi)$

in the non-coordinate basis $(cdt,dr,r\,d\phi,dz)\cdot$ For an incoming
gravitational plane wave interacting with a constant magnetic field along the
x-direction, Eq.(7) can be written down in terms of the induced electric and
magnetic fields $\vec{E}^{(1)},\vec{B}^{(1)}$

$\displaystyle\left(-\frac{\partial^{2}}{\partial
t^{2}}+\vec{\Delta}\right)\vec{E}^{(1)}=\vec{S}_{E}$ (I)
$\displaystyle\left(-\frac{\partial^{2}}{\partial
t^{2}}+\vec{\Delta}\right)\vec{B}^{(1)}=\vec{S}_{B},$ (II)

with the following source terms

$\vec{S}_{E}^{\perp}=B^{(0)}_{\rm ext}\frac{\partial^{2}h_{+}}{\partial
t\partial z}\begin{pmatrix}-\sin(\phi)\\\
-\cos(\phi)\end{pmatrix}+B^{(0)}_{\rm
ext}\frac{\partial^{2}h_{\times}}{\partial t\partial
z}\begin{pmatrix}\cos(\phi)\\\ -\sin(\phi)\end{pmatrix},$ (III)
$\vec{S}_{B}^{\perp}=B^{(0)}_{\rm ext}\frac{\partial^{2}h_{+}}{\partial
z^{2}}\begin{pmatrix}\cos(\phi)\\\ -\sin(\phi)\end{pmatrix}+B^{(0)}_{\rm
ext}\frac{\partial^{2}h_{\times}}{\partial z^{2}}\begin{pmatrix}\sin(\phi)\\\
\cos(\phi)\end{pmatrix}\cdot$ (IV)

These Eqs.(II) and (IV) leads to the Eq.(13) of the paper.

Our demonstration earlier is illustrated here, indeed the longitudinal
component $(E,B)^{(1)}_{z}$ are not sourced since $S_{E,z}=S_{B,z}=0$ so that
the induced radiation is purely transverse electromagnetic (TEM):
$(E,B)^{(1)}_{z}=0$. In the above equations, $\vec{\Delta}$ is the vector
laplacian operator in cylindrical coordinates, which is given by

$\vec{\Delta}\vec{E}=\begin{pmatrix}\displaystyle{\nabla^{2}E_{r}-\frac{E_{r}}{r^{2}}-\frac{2}{r^{2}}\partial_{\phi}E_{\phi}}\\\
\\\
\displaystyle{\nabla^{2}E_{\phi}-\frac{E_{\phi}}{r^{2}}+\frac{2}{r^{2}}\partial_{\phi}E_{r}}\\\
\\\ \nabla^{2}E_{z}\end{pmatrix}$ (V)

where $\nabla^{2}$ is the scalar laplacian :
$\nabla^{2}f=\partial_{r}^{2}f+\frac{1}{r}\partial_{r}f+\frac{1}{r^{2}}\partial^{2}_{\phi}f+\partial_{z}^{2}f\cdot$.
Any EM field inside an ideal resonator must verify the boundary conditions
along any perfect conducting surfaces:
$\vec{E}^{(1)}_{\parallel}=\vec{B}^{(1)}_{\perp}=\vec{0}\cdot$ Therefore, any
waveguide with two concentric open cylinders will host EM field configurations
with $E^{(1)}_{z}|_{r=R1,R_{2}}=B^{(1)}_{r}|_{r=R_{1},R_{2}}=0$ while a
cylindrical conducting cavity will host EM fields satisfying
$E^{(1)}_{z}|_{r=R}=B^{(1)}_{r}|_{r=R}=0$ and $E^{(1)}_{r,\phi}|_{z=\pm
L/2}=B^{(1)}_{z}|_{z=\pm L/2}=0\cdot$. The inhomogeneous wave equations
Eqs.(I,II) can be efficiently solved using spectral methods, i.e. by
approximating the unknown field, $(\vec{E},\vec{B})^{(1)}(t,r,\phi,z)$ by a
decomposition over a basis of orthogonal functions that satisfy the above-
mentioned boundary conditions. Here, a natural choice of such basis is given
by the following set of cylindrical harmonics (omitting the useless
longitudinal component $\psi^{z}_{kmn}$)

$\psi^{r}_{kmn}=C_{kmn}\cdot\mathcal{R}_{km}(r)\cdot\begin{Bmatrix}\cos\\\
\sin\end{Bmatrix}\left(m\phi\right)\cdot\begin{Bmatrix}\cos\\\
\sin\end{Bmatrix}\left(\frac{2\pi nz}{L}\right)$ (VI) $\psi^{\phi}_{kmn}=\mp
C_{kmn}\cdot\mathcal{R}_{km}(r)\cdot\begin{Bmatrix}\sin\\\
\cos\end{Bmatrix}\left(m\phi\right)\cdot\begin{Bmatrix}\cos\\\
\sin\end{Bmatrix}\left(\frac{2\pi nz}{L}\right)$ (VII)

where $C_{kmn}$ are normalization coefficients and where the radial
eigenfunctions $\mathcal{R}_{km}(r)$ are such that the boundary conditions are
satisfied. This is the case when

$\displaystyle\mathcal{R}_{km}(r)$ $\displaystyle=$ $\displaystyle
A_{k}.J_{m-1}(\alpha_{k}.r)+Y_{m-1}(\alpha_{k}.r)$ (VIII) $\displaystyle=$
$\displaystyle J_{m-1}(\alpha_{k}.r),$ (IX)

for the waveguide and the cavity, respectively. In the above equation,
$J_{n}(r),Y_{n}(r)$ are Bessel functions of the first and second kind,
respectively, and the constants $A_{km}$ and $\alpha_{k}$ are solutions of the
following system

$\displaystyle A_{k}.J_{m-1}(\alpha_{k}.R_{1,2})+Y_{m-1}(\alpha_{k}.R_{1,2})$
$\displaystyle=$ $\displaystyle 0,$ (X)

for the case of the waveguide with two open conducting cylinders and

$\displaystyle J_{m-1}(\alpha_{k}.R)$ $\displaystyle=$ $\displaystyle 0,$ (XI)

in the case of the cylindrical cavity.

Those cylindrical harmonics
$\left(\vec{\psi}_{kmn}\right)^{T}=\left(\psi^{r}_{kmn},\psi^{\phi}_{kmn},\psi^{z}_{kmn}\right)$
are eigenfunctions of the vector laplacian operator $\vec{\Delta}$ in
cylindrical coordinates with the following eigenvalues:

$\vec{\Delta}\vec{\psi}_{kmn}=-\left(\alpha^{2}_{k}+\frac{4\pi^{2}n^{2}}{L^{2}}\right)\vec{\psi}_{kmn}=-\Omega^{2}_{kn}\vec{\psi}_{kmn}$
(XII)

These eigenfunctions $\psi^{r,\phi}_{kmn}$ constitute a complete orthonormal
set with scalar product

$\left(f,g\right)=\int_{R_{1}(0)}^{R_{2}(R)}\int_{0}^{2\pi}\int_{-L/2}^{L/2}f(r,\phi,z)g(r,\phi,z)r.dr.d\phi.dz$

where the different bounds of the integral over the radius correspond to the
case of a waveguide or a cavity.

Moving back to Eqs.(I,II), one can use a truncated expansion on this basis as
an approximation of the unknown field. For instance, let us set

$B^{(1)}_{r,\phi}(t,r,\phi,z)\approx\sum_{k,m,n}\hat{b}^{r,\phi}_{k,m,n}(t)\cdot\psi^{r,\phi}_{kmn}(r,\phi,z)$
(XIII)

with

$\hat{b}^{r,\phi}_{k,m,n}(t)=\left(B^{(1)}_{r,\phi},\psi^{r,\phi}_{k,m,n}\right)$

so that the inhomogeneous wave equation Eq.(II) now reduces to a harmonic
oscillator ODE for each mode $\hat{b}^{r,\phi}_{k,m,n}(t)$, Eq.(15) in the
paper.

$\frac{d^{2}\hat{b}^{r,\phi}_{k,m,n}}{dt^{2}}+\Omega^{2}_{kn}\hat{b}^{r,\phi}_{k,m,n}=\hat{s}^{r,\phi}_{k,m,n}(t)$
(XIV)

where $\hat{s}^{r,\phi}_{k,m,n}(t)$ are the spectral coefficients of the
source $S^{r,\phi}_{B}$:

$S_{B}^{r,\phi}(t,r,\phi,z)\approx\sum_{k,m,n}\hat{s}^{r,\phi}_{k,m,n}(t)\cdot\psi^{r,\phi}_{kmn}(r,\phi,z)\cdot$

In this approach, each mode $\hat{b}^{r,\phi}_{k,m,n}(t)$ that composes the
induced magnetic field $\vec{B}^{(1)}$ behaves as an harmonic oscillator
driven by the corresponding mode of the source $\hat{s}^{r,\phi}_{k,m,n}$. The
general solution of such system describes a resonance process through a
superposition of oscillations at proper frequency $\Omega_{kn}$ and a
particular solution of the inhomogeneous equation. This last can easily be
obtained with a Fourier series method. Since the source of the induced
magnetic field is directly related to the second derivative of the metric
perturbation, resonance will be achieved when the incoming GW share some
common frequencies with the coaxial detector, as we have seen in the main
text.

Fortunately, there are some simplifications allowing to restrict to specific
modes of the cylindrical harmonic decomposition of the induced EM fields
$\left(\vec{E}^{(1)},\vec{B}^{(1)}\right)\cdot$ First, in the case of a plane
gravitational wave with a direction of propagation perpendicular to the one of
the outer magnetic field $\vec{B}^{(0)}$, the source terms are given by
Eqs.(III-IV) and therefore the induced EM fields will only have an azimuthal
mode number equal to one: $m=1$. Second, if ones focuses on the detection of
the induced EM energy inside the resonator then it suffices to consider the
components of the induced magnetic fields that are in the same direction than
the outer strong magnetic field. Indeed, the EM radiation that is induced into
the resonators will modify their EM energy content. The total electromagnetic
energy $\mathcal{E}$ inside the waveguide is given by the classical formula

$\mathcal{E}=\frac{1}{2}\int_{V}\left(\epsilon_{0}||\vec{E}||^{2}+\frac{||\vec{B}||^{2}}{\mu_{0}}\right)dV$

with $V$ the volume of the waveguide. The dominant contribution to the energy
variation $\Delta\mathcal{E}$ is given by the coupling between the external
magnetic field $\vec{B}^{(0)}$ and the induced one $\vec{B}^{(1)}$

$\Delta\mathcal{E}=E_{\rm
tot}-E^{(0)}\approx\frac{1}{\mu_{0}}\int_{V}\left(\vec{B}^{(0)}\bullet\vec{B}^{(1)}\right)dV$
(XIV)

neglecting the terms in $||\vec{E}^{(1)}||^{2}$ and in
$||\vec{B}^{(1)}||^{2}\cdot$ Since we have assumed here the external magnetic
field lies in the $x-$direction, one only needs to compute

$B^{(1)}_{x}=B^{(1)}_{r}\cos\phi-B^{(1)}_{\phi}\sin\phi$

from Eqs.(II,IV). In the case of incoming plane GW orthogonal to the outer
magnetic field, only the modes with $m=1$ (such that
$B^{(1)}_{r}\approx\cos\phi$ and $B^{(1)}_{\phi}\approx-\sin\phi$, see above)
and $n=0$, i.e. constant in the $z-$direction, will give a non-vanishing
contribution to the integral over the volume in the computation of the energy.
The important terms are therefore sourced by the Eq.(17),

$\hat{s}_{k,1,0}^{r,\phi}(z,t)=\pi
B_{0}L^{2}\mathcal{I}_{k}\int_{-L/2}^{L/2}\frac{\partial^{2}h_{+}(z,t)}{\partial
z^{2}}dz$ (XV)

with the dimensionless quantity $\mathcal{I}_{k}$ given by

$\mathcal{I}_{k}=\int_{r_{1}(0)}^{r_{2}(R/L)}\mathcal{R}_{k,1}(L\rho)\rho
d\rho$ (XVI)

with $\rho=r/L$, $r_{1,2}=R_{1,2}/L$ and for a magnetic field along the
$x-$direction.

In summary, the energy variation is given by, at the leading order, the Eq.
(16)

$\Delta\mathcal{E}\approx\frac{2\pi
B_{0}}{\mu_{0}}\cdot\sum_{k}\mathcal{I}_{k}\hat{b}_{k,1,0}(t)$ (XVII)

with
$\hat{b}_{k,1,0}(t)=\hat{b}_{k,1,0}^{r}(t)+\hat{b}_{k,1,0}^{\phi}(t)=2\hat{b}^{r,\phi}_{k,1,0}(t)\cdot$
Those modes $\hat{b}_{k,1,0}(t)$ are solutions of the forced harmonic
oscillator equation (XIV) with source term given by (twice) Eq.(XV). This
completes the detailed explanation to obtain Eqs.(15-17).

## References

  * Einstein [1916] A. Einstein, Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften (Berlin , 688 (1916).
  * Einstein [1918] A. Einstein, Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften (Berlin , 154 (1918).
  * Abbott  _et al._ [2016] B. P. Abbott, R. Abbott, T. D. Abbott, _et al._ , Physical Review Letters 116, 10.1103/PhysRevLett.116.061102 (2016), arXiv:1602.03837 .
  * Gertsenshtein [1960] M. Gertsenshtein, Journal of Experimental and Theoretical Physics 14, 113 (1960).
  * Aggarwal  _et al._ [2020a] N. Aggarwal, O. D. Aguiar, A. Bauswein, G. Cella, S. Clesse, _et al._ ,  (2020a), arXiv:2011.12414 .
  * [6] A. Füzfa, Devices for the directional emission and reception of gravitational waves, Patents PCT/EP2018/086758, PCT/EP2018/086760.
  * Niikura  _et al._ [2019] H. Niikura, M. Takada, S. Yokoyama, T. Sumi, and S. Masaki, Physical Review D 99, 83503 (2019), arXiv:1901.07120 .
  * Mróz  _et al._ [2017] P. Mróz, A. Udalski, J. Skowron, R. Poleski, S. Kozłowski, M. K. Szymański, I. Soszyński, Ł. Wyrzykowski, P. Pietrukowicz, K. Ulaczyk, D. Skowron, and M. Pawlak, Nature 10.1038/nature23276 (2017), arXiv:1707.07634 .
  * Bird  _et al._ [2016] S. Bird, I. Cholis, J. B. Muñoz, Y. Ali-Haïmoud, _et al._ , Physical Review Letters 116, 201301 (2016), arXiv:1603.00464 .
  * Clesse and García-Bellido [2017] S. Clesse and J. García-Bellido, Physics of the Dark Universe 15, 142 (2017), arXiv:1603.05234 .
  * Sasaki  _et al._ [2016] M. Sasaki, T. Suyama, T. Tanaka, and S. Yokoyama, Physical Review Letters 117, 61101 (2016), arXiv:1603.08338 .
  * Carr  _et al._ [2019] B. Carr, S. Clesse, J. García-Bellido, and F. Kühnel, arXiv (2019), arXiv:1906.08217 .
  * Weber [1960] J. Weber, Physical Review 117, 306 (1960).
  * Boccaletti  _et al._ [1970] D. Boccaletti, V. De Sabbata, P. Fortini, and C. Gualdi, Il Nuovo Cimento B Series 10 70, 129 (1970).
  * De Logi and Mickelson [1977] W. K. De Logi and A. R. Mickelson, Physical Review D 16, 2915 (1977).
  * Zeldovich [1974] Y. B. Zeldovich, Soviet Physics JETP 38, 652 (1974).
  * Grishchuk and Sazhin [1973] L. P. Grishchuk and M. V. Sazhin, Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki 65, 441 (1973).
  * Grishchuk and Sazhin [1975] L. P. Grishchuk and M. V. Sazhin, Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki 68, 1569 (1975).
  * Grishchuk [1977] L. P. Grishchuk, Soviet Physics - Uspekhi 20, 319 (1977).
  * Grishchuk [2003] L. P. Grishchuk, arXiv preprint gr-qc/0306013 (2003), arXiv:0306013 [gr-qc] .
  * Braginskii and Menskii [1971] V. B. Braginskii and M. B. Menskii,  _JETP Letters_ , Tech. Rep. 11 (Moscow State Univ., 1971).
  * Braginskii  _et al._ [1973] V. B. Braginskii, L. P. Grishchuk, A. G. Doroshkevich, Y. B. Zel’dovich, I. D. Novikov, and M. V. Sazhin,  _Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki_ , Tech. Rep. 5 (Moscow State Univ., 1973).
  * Pegoraro  _et al._ [1978a] F. Pegoraro, L. A. Radicati, P. Bernard, and E. Picasso, Physics Letters A 68, 165 (1978a).
  * Pegoraro  _et al._ [1978b] F. Pegoraro, E. Picasso, and L. A. Radicati, Journal of Physics A: Mathematical and General 11, 1949 (1978b).
  * Caves [1979] C. M. Caves, Physics Letters B 80, 323 (1979).
  * Gerlach [1992] U. H. Gerlach, Physical Review D 46, 1239 (1992).
  * Cruise [1983] A. M. Cruise, Monthly Notices of the Royal Astronomical Society 204, 485 (1983).
  * Ballantini  _et al._ [2003] R. Ballantini, P. Bernard, E. Chiaveri, A. Chincarini, G. Gemme, R. Losito, R. Parodi, and E. Picasso, Classical and Quantum Gravity 20, 3505 (2003).
  * Ejlli  _et al._ [2019] A. Ejlli, D. Ejlli, A. M. Cruise, G. Pisano, and H. Grote, European Physical Journal C 79, 1 (2019).
  * Bentley  _et al._ [2019] J. Bentley, P. Jones, D. Martynov, A. Freise, and H. Miao, Physical Review D 99, 102001 (2019).
  * Arvanitaki and Geraci [2013] A. Arvanitaki and A. A. Geraci, Physical Review Letters 110, 71105 (2013), arXiv:1207.5320 .
  * Aggarwal  _et al._ [2020b] N. Aggarwal, G. P. Winstone, M. Teo, M. Baryakhtar, S. L. Larson, V. Kalogera, and A. A. Geraci, (2020b), arXiv:2010.13157 [gr-qc] .
  * Ito  _et al._ [2020] A. Ito, T. Ikeda, K. Miuchi, and J. Soda, Probing GHz gravitational waves with graviton–magnon resonance (2020), arXiv:1903.04843 .
  * Goryachev and Tobar [2014] M. Goryachev and M. E. Tobar, Physical Review D - Particles, Fields, Gravitation and Cosmology 10.1103/PhysRevD.90.102005 (2014), arXiv:1410.2334 .
  * Füzfa [2016] A. Füzfa, Physical Review D 93, 24014 (2016), arXiv:1504.00333 .
  * Tolman  _et al._ [1931] R. C. Tolman, P. Ehrenfest, and B. Podolsky, Physical Review 37, 602 (1931).
  * Rätzel  _et al._ [2016] D. Rätzel, M. Wilkens, and R. Menzel, New Journal of Physics 18, 23009 (2016), arXiv:1511.01023 .
  * Meng-Xi  _et al._ [1997] T. Meng-Xi, L. Fang-Yu, and L. Jun, Acta Physica Sinica (overseas Edition) 6, 161 (1997).
  * Asztalos  _et al._ [2010] S. J. Asztalos, G. Carosi, C. Hagmann, D. Kinion, K. Van Bibber, M. Hotz, L. J. Rosenberg, G. Rybka, J. Hoskins, J. Hwang, P. Sikivie, D. B. Tanner, R. Bradley, and J. Clarke, Physical Review Letters 104, 41301 (2010), arXiv:0910.5914 .
  * Du  _et al._ [2018] N. Du, N. Force, R. Khatiwada, E. Lentz, _et al._ (ADMX Collaboration), Phys. Rev. Lett. 120, 151301 (2018).
  * Choquet-Bruhat [2009] Y. Choquet-Bruhat,  _General Relativity and the Einstein Equations_, Oxford Mathematical Monographs (OUP Oxford, 2009).
  * Nakamura  _et al._ [1997] T. Nakamura, M. Sasaki, T. Tanaka, and K. S. Thorne, Astrophys. J. Lett. 487, L139 (1997), arXiv:astro-ph/9708060 .
  * Kashlinsky [2016] A. Kashlinsky, Astrophys. J. Lett. 823, L25 (2016), arXiv:1605.04023 [astro-ph.CO] .
  * Kouvaris  _et al._ [2018] C. Kouvaris, P. Tinyakov, and M. H. Tytgat, Physical Review Letters 121, 221102 (2018), arXiv:1804.06740 .
  * Dasgupta  _et al._ [2021] B. Dasgupta, R. Laha, and A. Ray, Phys. Rev. Lett. 126, 141105 (2021).
  * Byrnes  _et al._ [2018] C. T. Byrnes, M. Hindmarsh, S. Young, and M. R. S. Hawkins, JCAP 08, 041, arXiv:1801.06138 [astro-ph.CO] .
  * Clesse and García-Bellido [2018] S. Clesse and J. García-Bellido, Phys. Dark Univ. 22, 137 (2018), arXiv:1711.10458 [astro-ph.CO] .
  * Hawkins [2020] M. R. Hawkins, Astronomy and Astrophysics 633, A107 (2020), arXiv:2001.07633 .
  * Bhatiani  _et al._ [2019] S. Bhatiani, X. Dai, and E. Guerras, The Astrophysical Journal 885, 77 (2019).
  * Arzoumanian  _et al._ [2020] Z. Arzoumanian _et al._ (NANOGrav), (2020), arXiv:2009.04496 [astro-ph.HE] .
  * Domènech and Pi [2020] G. Domènech and S. Pi, (2020), arXiv:2010.03976 [astro-ph.CO] .
  * De Luca  _et al._ [2020a] V. De Luca, G. Franciolini, and A. Riotto, (2020a), arXiv:2009.08268 [astro-ph.CO] .
  * Vaskonen and Veermäe [2020a] V. Vaskonen and H. Veermäe, (2020a), arXiv:2009.07832 [astro-ph.CO] .
  * Kohri and Terada [2020] K. Kohri and T. Terada, (2020), arXiv:2009.11853 [astro-ph.CO] .
  * Wang  _et al._ [2019] S. Wang, T. Terada, and K. Kohri, Physical Review D 10.1103/PhysRevD.99.103531 (2019).
  * Wang  _et al._ [2020] S. Wang, T. Terada, and K. Kohri, Erratum: Prospective constraints on the primordial black hole abundance from the stochastic gravitational-wave backgrounds produced by coalescing events and curvature perturbations (Physical Review D (2019) 99 (103531) DOI: 10.1103/PhysRevD.99.103531) (2020).
  * Antelis  _et al._ [2018] J. M. Antelis, J. M. Hernández, and C. Moreno, Journal of Physics: Conference Series 1030, 12005 (2018).
  * Kocsis  _et al._ [2018] B. Kocsis, T. Suyama, T. Tanaka, and S. Yokoyama, The Astrophysical Journal 854, 41 (2018).
  * Raidal  _et al._ [2019] M. Raidal, C. Spethmann, V. Vaskonen, and H. Veermäe, Journal of Cosmology and Astroparticle Physics 2019 (2), 18, arXiv:1812.01930 .
  * Gow  _et al._ [2020] A. D. Gow, C. T. Byrnes, A. Hall, and J. A. Peacock, Journal of Cosmology and Astroparticle Physics 2020 (1), 31, arXiv:1911.12685 .
  * Liu  _et al._ [2019] L. Liu, Z. K. Guo, and R. G. Cai, Physical Review D 10.1103/PhysRevD.99.063523 (2019), arXiv:1812.05376 .
  * Vaskonen and Veermäe [2020b] V. Vaskonen and H. Veermäe, Phys. Rev. D 101, 043015 (2020b), arXiv:1908.09752 [astro-ph.CO] .
  * Clesse and Garcia-Bellido [2020] S. Clesse and J. Garcia-Bellido, (2020), arXiv:2007.06481 [astro-ph.CO] .
  * Boehm  _et al._ [2020] C. Boehm, A. Kobakhidze, C. A. O’Hare, Z. S. Picker, and M. Sakellariadou, (2020), arXiv:2008.10743 [astro-ph.CO] .
  * Chisholm [2006] J. R. Chisholm, Phys. Rev. D 73, 083504 (2006), arXiv:astro-ph/0509141 .
  * Chisholm [2011] J. R. Chisholm, Phys. Rev. D 84, 124031 (2011), arXiv:1110.4402 [astro-ph.CO] .
  * Belotsky  _et al._ [2019] K. M. Belotsky, V. I. Dokuchaev, Y. N. Eroshenko, E. A. Esipova, M. Y. Khlopov, L. A. Khromykh, A. A. Kirillov, V. V. Nikulin, S. G. Rubin, and I. V. Svadkovsky, Eur. Phys. J. C 79, 246 (2019), arXiv:1807.06590 [astro-ph.CO] .
  * Carr and Silk [2018] B. Carr and J. Silk, Mon. Not. Roy. Astron. Soc. 478, 3756 (2018), arXiv:1801.00672 [astro-ph.CO] .
  * Suyama and Yokoyama [2019] T. Suyama and S. Yokoyama, PTEP 2019, 103E02 (2019), arXiv:1906.04958 [astro-ph.CO] .
  * Moradinezhad Dizgah  _et al._ [2019] A. Moradinezhad Dizgah, G. Franciolini, and A. Riotto, JCAP 11, 001, arXiv:1906.08978 [astro-ph.CO] .
  * Matsubara  _et al._ [2019] T. Matsubara, T. Terada, K. Kohri, and S. Yokoyama, Phys. Rev. D 100, 123544 (2019), arXiv:1909.04053 [astro-ph.CO] .
  * Young and Byrnes [2020] S. Young and C. T. Byrnes, JCAP 03, 004, arXiv:1910.06077 [astro-ph.CO] .
  * Padilla  _et al._ [2020] N. Padilla, J. Magana, J. Sureda, and I. Araya, (2020), arXiv:2010.06470 [astro-ph.CO] .
  * De Luca  _et al._ [2020b] V. De Luca, V. Desjacques, G. Franciolini, and A. Riotto, JCAP 11, 028, arXiv:2009.04731 [astro-ph.CO] .
  * Jedamzik [2020] K. Jedamzik, JCAP 09, 022, arXiv:2006.11172 [astro-ph.CO] .
  * García-Bellido and Clesse [2018] J. García-Bellido and S. Clesse, Phys. Dark Univ. 19, 144 (2018), arXiv:1710.04694 [astro-ph.CO] .
  * Calcino  _et al._ [2018] J. Calcino, J. Garcia-Bellido, and T. M. Davis, Mon. Not. Roy. Astron. Soc. 479, 2889 (2018), arXiv:1803.09205 [astro-ph.CO] .
  * LIGO Scientific Collaboration [2018] LIGO Scientific Collaboration, {LIGO} {A}lgorithm {L}ibrary - {LALS}uite, free software (GPL) (2018).