Potential of radio telescopes as high-frequency gravitational wave detectors

In the presence of magnetic fields, gravitational waves are converted into photons and vice versa. We demonstrate that this conversion leads to a distortion of the cosmic microwave background (CMB), which can serve as a detector for MHz to GHz gravitational wave sources active before reionization. The measurements of the radio telescope EDGES can be cast as a bound on the gravitational wave amplitude, $h_c < 10^{-21} (10^{-12})$ at 78 MHz, for the strongest (weakest) cosmic magnetic fields allowed by current astrophysical and cosmological constraints. Similarly, the results of ARCADE 2 imply $h_c < 10^{-24} (10^{-14})$ at $3 - 30$GHz. For the strongest magnetic fields, these constraints exceed current laboratory constraints by about seven orders of magnitude. Future advances in 21cm astronomy may conceivably push these bounds below the sensitivity of cosmological constraints on the total energy density of gravitational waves.

Gravitational waves (GWs) produced in the early Universe @Maggiore:2018sht [@Caprini:2018mtu] can traverse cosmic distances without experiencing any interactions, making them a unique probe of very high energy physics. Since the comoving Hubble horizon grows with time, GWs produced at energies around the scale of grand unification have frequencies in the MHz and GHz regime today, far beyond the reach of the laser interferometers LIGO, VIRGO or KAGRA. See @REECE1984341 [@Cruise:2006zt; @Akutsu:2008qv; @Ito:2020wxi; @Cruise_2012; @Ejlli:2019bqj] for some existing laboratory bounds at these frequencies.

Here we focus on searching for high-frequency GWs exploiting the (inverse) Gertsenshtein effect @Gertsenshtein [@Boccaletti1970], which describes the conversion of GWs into photons in the presence of a magnetic field (see e.g. @DeLogi:1977qe [@Raffelt:1987im; @Macedo:1984di; @Fargion:1995mm; @Dolgov:2012be; @Cruise_2012; @Ejlli:2019bqj; @Ejlli:2020fpt]). As an immediate consequence of general relativity and classical electromagnetism, this is a purely SM process. Involving gravity, the conversion probability is extremely small which may, however, be compensated by considering a 'detector' of cosmological size. In fact, magnetic fields with cosmological correlation lengths might well permeate our Universe with certain astrophysical observations strongly suggesting a lower limit of order $\unit[10^{-16}]{G}$ @Neronov:1900zz [@Tavecchio:2010mk; @Takahashi:2013lba], and the CMB setting an upper bound in the pG-nG range @Jedamzik:2018itu [@Ade:2015cva; @Pshirkov:2015tua]. See @Durrer:2013pga for a comprehensive review.

The pioneering study @PhysRevLett.74.634 proposed the inverse Gertsenshtein effect in cosmic magnetic fields to search for GWs but neglected the plasma mass of photons, as pointed out in Ref. @Cillis:1996qy. The idea was revisited in Ref. @Pshirkov:2009sf suggesting an observable effect, however as noted in @Dolgov:2012be decoherence effects were not correctly accounted for. More recently @Fujita:2020rdx studied the production of GWs from CMB photons. In this letter, we focus on CMB distortions arising from the Gertsenshtein effect during the dark ages, i.e. the period between recombination and reionization. Due to the small fraction of free electrons in this period, the effective plasma mass of the photons is suppressed, increasing the conversion probability between GWs and photons. Taking into account inhomogeneities in the thermal plasma and in the cosmic magnetic fields, we demonstrate that existing measurements of the Rayleigh-Jeans tail of the CMB spectrum, performed e.g. by ARCADE 2 @Fixsen_2011 and by EDGES @Bowman:2018yin, can be translated into constraints on GWs in the MHz-GHz regime. These are competitive with, or even exceed, current laboratory constraints, depending on the assumptions on the cosmic magnetic fields.

The Gertsenshtein effect
========================

Calculating the conversion rate for this oscillation process requires solving Maxwell's equations for the vector potential, $A^\mu$, describing the electromagnetic radiation, together with the linearized Einstein's equations for the metric $g_{\mu\nu} =\eta_{\mu\nu}+h_{\mu\nu}$, in which $h_{\mu\nu}$ describes the GWs. In this work we will adopt $\eta_{\mu\nu} = \text{diag}(+---)$ and work with natural Heaviside-Lorentz units ($\hbar=c=1$), except in this section where we keep fundamental constants explicitly to emphasize that the Gertsenshtein effect is a classical phenomenon.

Let us ignore the Universe expansion first and consider a GW propagating in the $\mathbf{\hat{e}_3}$ direction inside a fixed box of size $\Delta {\ell}$ that contains a uniform transverse magnetic field, $B$, and a non-negligible uniform density of free electrons, $n_e$. Without loss of generality, we assume that the magnetic field points in the $\mathbf{\hat{e}_1}$ direction. See Fig. \[fig:sketch\]. In this coordinate system we introduce $h_\times = h^{12} =h^{21}$ and $A_\times = A^1$ as well as $h_+ = -h^{22} = h^{11}$ and $A_+ =-A^2$. This is because the aforementioned equations can be elegantly cast as @Dolgov:2012be [@Raffelt:1987im] [^1] $$\begin{aligned}
\left(\Box+\omega^2_\text{pl}/c^2 \right) A_\lambda =- B \partial_{\ell}h_\lambda\,,&&
\Box h_\lambda = \kappa^2 B \partial_{\ell}A_\lambda\,,
\label{eq:waveeq}\end{aligned}$$ where $\lambda \in \{+,\times\}$, ${\ell}$ is the third component, $\Box =\partial_t^2/c^2-\partial_{\ell}^2$, $\kappa =\left(16 \pi G\right)^{1/2}/c^2$. We include the plasma frequency $\omega_\text{pl}= \sqrt{e^2n_e/m_e}$, which acts as an effective mass term and gives electromagnetic waves of frequency $\omega$ a refractive index $\mu = \sqrt{1-{\omega_\text{pl}}^2/\omega^2}$ when $B\to 0$. Eq.  also applies for arbitrary uniform fields with $B$ interpreted as the corresponding transverse component. See the supplementary material for more details. Assuming a plane wave traveling in the positive direction with $\omega \geq {\omega_\text{pl}}$, the exact solution of Eqs.  (see also @Ejlli:2020fpt) can be written as $$\begin{aligned}
\hspace{-8pt}
\psi(t,{\ell})  \equiv
\begin{pmatrix}
\sqrt{\mu}\,\, A_\lambda \\
\frac{1}{\kappa}\, h_\lambda
\end{pmatrix}
=
e^{-i\omega t}
e^{i K {\ell}}
\psi(0,0)\,,\,
\label{eq:sol}\end{aligned}$$ with $K$ being the Hermitian matrix $$\begin{aligned}
K = 
\begin{pmatrix}
 \frac{ \mu}{c}\sqrt{\omega^2+\left(\frac{\kappa B}{1+\mu}\right)^2} \,  &  -i\frac{\sqrt\mu\,\kappa B }{1+\mu}\\
i \frac{\sqrt\mu\, \kappa B }{1+\mu} &  \frac{1}{c}\sqrt{\omega^2+\left(\frac{\kappa B}{1+\mu}\right)^2} 
\end{pmatrix}
 \,.
\label{eq:U}\end{aligned}$$

It is convenient to introduce $\psi$ because its magnitude, $|\psi(t,{\ell})|^2$ , is conserved. This easily follows from the unitarity of the matrix ${\cal U({\ell})}=e^{i K {\ell}} $. In particular, $\psi(0,0)=(0,h_{\lambda,0}/\kappa)$ for a pure GW state entering the box, and consequently $ \psi(t,\Delta {\ell}) = e^{-i\omega t}\left({\cal U}_{12} (\Delta{\ell})  ,{\cal U}_{22} (\Delta{\ell})\right) h_{\lambda,0}/\kappa$ after leaving the box. Since $|{\cal U}_{12} (\Delta{\ell})|^2 + |{\cal U}_{22} (\Delta{\ell})|^2 =1$, the quantity $P(\Delta {\ell})\equiv |{\cal U}_{12} (\Delta {\ell})  |^2$ can be interpreted as the probability of GW conversion after traversing a distance $\Delta {\ell}$. Simple algebra shows $$P (\Delta {\ell}) =|K_{12}|^2 \,{{\ell}_\text{osc}}^2\, \sin^2(\Delta{\ell}/{{\ell}_\text{osc}})\,,
\label{eq:P0}$$ with ${{\ell}_\text{osc}}^{-1}  = 
\sqrt{\omega^2(1-\mu)^2/c^2+ \kappa^2 B^2}/2$. These expressions reduce to the approximated formulae previously found (see e.g. @Raffelt:1987im [@Ejlli:2018hke]).

Although cosmic magnetic fields are not expected to be perfectly homogeneous, coherent oscillations take place in highly homogeneous patches, for which ${{\ell}_\text{osc}}\ll \Delta {\ell}$ and therefore $P(\Delta {\ell})  = |K_{12}|^2 {{\ell}_\text{osc}}^2/2$ on average. Taking into account inhomogeneities in $n_e$ [^2] and $B$, the coherence of the $g \leftrightarrow \gamma$ oscillations is lost on distances larger than $\Delta {\ell}$, that is, the smallest distance on which $B$ and $n_e$ are uniform. Denoting the total distance traveled by the GW as $D$, this corresponds to traversing $N = D/\Delta {\ell}$ independent regions with a conversion probability $P(\Delta {\ell})$ each. As long as $N \cdot P(\Delta {\ell}) \ll 1$, this gives a total conversion probability of $P(D) = D |K_{12}|^2 {{\ell}_\text{osc}}^2/(2\Delta {\ell})$  @Cillis:1996qy [@Pshirkov:2009sf], corresponding to an average conversion rate (i.e. probability per time) [^3] given by $$\langle\Gamma_{g\leftrightarrow\gamma}\rangle =\frac{c\,|K_{12}|^2 {{\ell}_\text{osc}}^2}{2   \Delta {\ell}}\,.
\label{eq:rate}$$ In the supplementary material we demonstrate that this simple estimate correctly captures the essential features of a more involved computation based on the expected power spectrum of the magnetic field. Note that any additional inhomogeneities would further enhance the conversion rate by limiting the coherence of the $g \leftrightarrow \gamma$ oscillations.

\[t\]

We now include the effect of the Universe expansion during the dark ages. This is the period between photon decoupling and reionization, $z_{\text{dec}}\simeq 1100 \gtrsim z \gtrsim  z_{\text{rei}}\simeq 10$, beginning with the formation of the CMB and ending when the first stars were formed. During this time, the refractive index of MHz-GHz CMB photons is determined by the tiny electron density, with the contributions of neutral hydrogen, helium and birefringence being subdominant @Chen:2013gva [@Kunze:2015noa; @Mirizzi:2009iz]. This allows us to adopt Eq , after a few modifications. The conversion probability in an adiabatic expanding Universe is simply the line-of-sight integral of the rate $${{\cal P}}\equiv \int_{l.o.s.} \langle \Gamma_{g\leftrightarrow \gamma}  \rangle dt = \int^{z_{\text{ini}}}_{0} \frac{\langle\Gamma_{g\leftrightarrow \gamma}\rangle}{ (1+z)\,H} dz \,,
\label{eq:integratedP}$$ where we use null-geodesics $Hdt= dT/T =dz/(1+z) $. Also, $z_{\text{ini}}\leq z_{\text{dec}}$ is an initial condition to be specified below and $H =  H_{\text{dec}}\left(T/T_{\text{dec}}\right)^{3/2}$ is the Hubble parameter during the dark ages, which are matter dominated. Furthermore, the average magnetic energy density of the Universe $\rho_B= B^2/2$ redshifts as $\rho_B=\rho_{B0}\left(1+z\right)^4$ [^4]. Additionally, such a field is associated with a coherence length, $\lambda_B=\lambda^0_B/(1+z)$, because it is not expected to be homogeneous everywhere. Concerning these two quantities we emphasize three important facts here and refer the reader to @Durrer:2013pga for a more comprehensive discussion: i) a recent CMB analysis gives $B_0\lesssim \unit[47]{pG}$ @Jedamzik:2018itu ii) Blazar observations strongly suggest a lower limit on $B_0$ @Caprini:2015gga because otherwise their gamma-ray spectra can not be explained under standard cosmological assumptions @Neronov:1900zz [@Tavecchio:2010mk; @Takahashi:2013lba; @Chen:2014rsa], and iii) magnetohydrodynamic turbulence damps out large magnetic fields at small distances, imposing an additional (theoretical) upper limit @Durrer:2013pga. Fig. \[fig:hc\] show these constraints.

In addition, the electron number density during this epoch is $n_e(z) = n_{b0} (1+z)^3 X_e(z)
$, where $n_{b0}=\unit[0.251]{m^{-3}}$ is the baryon number density today @Aghanim:2018eyx and $X_e(z)$ is the ionization fraction, taking values $1,\, 0.68,\,0.0002$ and $0.15$ at $z=0,\,10,\,20$ and $1100$, respectively [^5]. This gives plasma frequencies today, $\omega_{\text{pl},0}$, lying in the Hz range, which allows us to take $1-\mu(z) =(1+z)X_e(z)\omega_{\text{pl},0}^2/(2\omega_0^2)\ll 1$, for waves of frequency $\omega= \omega_0 (1+z)$ with $\omega_0 \sim$ GHz. Moreover, $B_0\lesssim \unit[47]{pG}$ results in the oscillation length being numerically dominated by the plasma frequency so that $
{{\ell}_\text{osc}}^{-1} = (1+z)^2 X_e(z)\omega_{\text{pl},0}^2/(4\omega_0c)
$. This gives ${{\ell}_\text{osc}}\ll\unit[1]{pc}\ll \Delta {\ell}$, as anticipated above. Here, in order to account for electron inhomogeneities we conservatively take $\Delta {\ell}= \Delta {\ell}_0/(1 + z)$ to be given by $\Delta {\ell}_0 = \text{min}[\lambda_\text{EQ}, \lambda_B^0]$ where $\lambda_{EQ}/(2\pi) = \unit[95]{Mpc}$ is the characteristic comoving scale for the onset of structure formation (corresponding to the perturbation mode entering the horizon at matter-radiation equality). Putting all this together, we obtain $$\begin{aligned}
{{\cal P}}\simeq \,6.3\times 10^{-19} \left( \frac{B_0}{\text{nG}} \right)^2
 \left( \frac{\omega_0}{T_0} \right)^2
\hspace{-3pt}
 \left( \frac{\text{Mpc}}{\Delta {\ell}_0}\right)  
\left( \frac{{\cal I}(z_{\text{ini}})}{ 10^6}\right),
\label{eq:intf}\end{aligned}$$ with $T_0/(2 \pi) = 2.725~\text{K}/(2 \pi) = 56.78$ GHz and ${\cal I}(z_{\text{ini}})=\int_0^{z_{\text{ini}}}dz\,(1+z)^{-3/2}X_e^{-2}(z)$. The left panel of Fig. \[fig:hc\] displays contours of $(T_0/\omega_0)^2{{\cal P}}$ in the parameter space of cosmic magnetic fields. The inset shows ${\cal I}'(z_{\text{ini}})$, explaining the weak redshift-dependence of ${\cal I}(z_{\text{ini}})$, with the largest contribution arising from $z\sim 10$.

CMB Distortions
===============

The CMB photon distribution, $f_\gamma (\omega,T)$, retains its equilibrium form during the dark ages, i.e. is given by a black-body spectrum, $f_{\text{eq}}= 1/(e^{\omega/T}- 1)$ with $\omega/T=\omega_0/T_0$. Our aim here is to calculate deviations from such a spectrum, $\delta f_\gamma = f_{\gamma} -f_{\text{eq}}$.

The spectrum of GWs is commonly characterized by $\Omega_\text{GW}$, which parametrizes the corresponding energy density per logarithmic frequency bin. This quantity can be used to introduce – in an analogous manner to $f_\gamma$– the distribution function for GWs, $f_g$. More precisely, in terms of it, the energy density is given by $$\begin{aligned}
 \rho_{g}(T) \hspace{-2pt} =\hspace{-2pt} \int \frac{d\ln\omega}{ \pi^2} \, \omega^4 f_g  \equiv \rho_c(T) \hspace{-2pt}\int d \ln\omega \,\, \Omega_\text{GW}\left(\frac{\omega}{2 \pi},T\right) \label{eq:rhog2},\end{aligned}$$ with $\rho_c(T)$ denoting the Universe total energy density. Both distributions satisfy the Boltzmann equation $ \hat{L} f_{\gamma/g} =\pm \langle\Gamma_{g\leftrightarrow\gamma}\rangle (f_g -f_\gamma)$, where $\hat{L} \equiv\partial_t - H \omega \partial_\omega = - H \left( T \partial_T + \omega \partial_\omega \right)$ is the corresponding Lioville operator. Its solution leads to $$\begin{aligned}
\delta f_\gamma (\omega_0,T_0)
=\left( f_g (\omega_\text{ini},T_\text{ini})- f_{\text{eq}}\right)
{{\cal P}}+ {\cal O}({{\cal P}}^2)\,,
\label{eq:master}\end{aligned}$$ with ${{\cal P}}$ defined as in Eq. . We solve the Boltzmann equations from an initial temperature $T=T_{\text{ini}}$ – when the photon distribution is a black-body spectrum, i.e $f_\gamma(\omega,T_{\text{ini}})= f_{\text{eq}}(\omega/T_{\text{ini}})$ – until today. If decoupling is prior to the GW emission, the latter fixes $T_{\text{ini}}$. Otherwise, we set $T_\text{ini} =T_\text{dec}$ because the ionization fraction sharply drops after $z \sim z_{\text{dec}}$ rendering any prior contribution negligible. This is illustrated in the inset of Fig. \[fig:hc\] (left panel), which also shows that the conversion rate is anyways largely insensitive to the precise value of $T_{\text{ini}}$.

Eq.  can alternatively be derived by considering the density-matrix formalism. In that case case, $f_\gamma$ and $f_g$ are proportional to the diagonal entries of such a matrix, which evolves by means of the Hamiltonian associated with Eq. . See the supplementary material for details. The fact that using both methods we obtain the same result –i.e. Eq. – is reassuring and indicates that decoherence effects are properly taken into account @Dolgov:2012be. Due to this as well as the way we treat inhomogeneities, our results differ from those of @Pshirkov:2009sf.

Constraints on the stochastic GW background
===========================================

In this letter we focus on the Rayleigh-Jeans part of the CMB, i.e. $\omega \ll T$ implying $f_\text{eq} \simeq T/\omega$. In this regime, a subdominant GW contribution to the total radiation energy density is compatible with $f_g \gg f_\gamma~$[^6], and can thus produce an enhancement of the low-frequency CMB tail through the first term of Eq. . More precisely, the assumption $f_g > f_\gamma$ translates to $\Omega_\text{GW}/\Omega_\gamma > 15/\pi^4 (\omega/T)^3$ as can be seen by rewriting Eq.  as $
 \Omega_\text{GW} =\omega^4  f_g\left(\omega,T\right)/(\pi^2 \rho_c) = (15/\pi^4) \, \left(\omega/T\right)^4 f_g(\omega,T) \,  \Omega_\gamma  \,
$ with $\Omega_\gamma = \rho_\gamma/\rho_c =   \pi^2 T^4/(15\,\rho_c)$. Even a scale-invariant GW spectrum as small as $\Omega_\text{GW} \simeq 10^{-15}$ implies $f_g > f_\rwallace@example.net. $\omega/T \simeq 10^{-3}$.

With $\omega\ll T$ and $f_g\gg f_\gamma$, Eq.  reads $$\frac{\delta f_\gamma }{f_\gamma} \left(\omega_0,T_0\right)= \frac{\pi^4}{15}\left(\frac{T}{\omega}\right)^3{{\cal P}}\, \frac{\Omega_\text{GW}}{\Omega_\gamma}\,. 
 \label{eq:pred}$$

For a given detector sensitivity $\delta f_\gamma/f_\gamma$ and a given value of the conversion probability ${{\cal P}}$, relation  sets stringent bounds on the GW spectrum, which can be expressed in terms of the characteristic strain by means of @Maggiore:1900zz $$h_c = \left(\frac{3 H_0^2}{4 \pi^2} \Omega_\text{GW} f^{-2} \right)^{1/2}\,.$$ This is related to the one-sided power spectral density $S_h$ as $h_c = \sqrt{f S_h(f)}$. Fig. \[fig:hc\] contrasts the resulting constraints with existing bounds in the literature.

#### $N_\text{eff}$ bound.

GWs contribute to the energy budget of the Universe in the form of radiation and are as such constrained by the BBN and CMB bounds on the effective number of massless degrees of freedom $N_\text{eff}$ @Pagano:2015hma, $$\begin{aligned}
 \rho_{g}(T)   \leq \frac{7}{8} \left( \frac{4}{11} \right)^{4/3} \Delta N_\text{eff} \, \rho_\gamma(T) \,,
 \label{eq:Neffbound}\end{aligned}$$ with $\Delta N_\text{eff} \lesssim 0.1$ @Cyburt:2015mya [@Aghanim:2018eyx]. For a spectrum $\Omega_\text{GW}$ which is approximately scale invariant between $f_\text{min}$ and $f_\text{max}$ with $\ln(f_\text{max}/f_\text{min}) \sim {\cal O}(1)$, this implies $$\frac{\Omega_\text{GW}}{\Omega_\gamma} \lesssim \frac{7}{8} \left( \frac{4}{11} \right)^{4/3} \Delta N_\text{eff}\,,
 \label{eq:Neff}$$ whereas for a narrow spectrum peaked at $\tilde \omega$ with width $\Delta \tilde \omega \lesssim \tilde \omega$ this bound is relaxed by a factor $(\tilde \omega/\Delta \tilde \omega)$. Note that this bound applies only to GWs present already at CMB decoupling.

#### Probing the Rayleigh-Jeans tail of the CMB spectrum.

Below $\omega_0/T_0 \sim 10^{-2}$, galactic foregrounds dominate the radio sky. Here we focus on the results reported by ARCADE2 @Fixsen_2011 which covers the sweet spot of the low-frequency Rayleigh-Jeans spectrum before galactic foregrounds become important ($f = \omega_0/(2 \pi) =  3$, 8, 10, 30 and 90 GHz) and by EDGES @Bowman:2018yin, which is a recent measurement of the global 21cm absorption signal at 78 MHz.

ARCADE2 was a balloon experiment equipped with a radio receiver measuring the black body temperature of sky @Fixsen_2011. The cleanest frequency band is around 10 GHz enabling a mK resolution, $\delta f_\gamma/f_\gamma = \delta T/T_\text{CMB} \lesssim 4 \times 10^{-4}$ at $\omega_0/T_0  \simeq 0.18$. At smaller frequencies, ARCADE2 observed a significant radio excess beyond the expected galactic foreground whose origin remains an open question (see e.g. @Seiffert_2011 [@Feng:2018rje]). Assuming that this excess is entirely astrophysical, we can impose an upper bound on an additional contribution from a stochastic GW background using the 3, 8, 10 and 30 GHz frequency bands. In Fig \[fig:hc\], these frequencies are marked by crosses, the solid lines connecting them serve only to guide the eye.

Recently, the first observation of the global (i.e. sky-averaged) 21cm absorption signal was reported by the EDGES collaboration @Bowman:2018yin. The absorption feature was found to be roughly twice as strong as previously expected, which if true, would indicate that either the primordial gas was significantly colder or the radiation background was significantly hotter than expected. Conservatively, we may assume that the deviation from the expected value is due to foreground contamination, and place a bound on any stochastic GW background by using $\delta f_\gamma / f_\gamma \lesssim 1$ at $\omega/T = 1.4 \times 10^{-3}$ (78 MHz). The width of the observed absorption feature (19 MHz) determines the width of the frequency coverage.

Discussion
==========

Cosmological sources of GWs typically produce stochastic GW backgrounds with a frequency roughly related to the comoving Hubble horizon at the time of production. Processes in the very early Universe at energy scales far beyond the reach of colliders thus generically produce GWs in the MHz and GHz regime. Despite the large amount of redshift, these violent processes can produce sizeable GW signals, saturating the $N_\text{eff}$ bound . Some examples are axion inflation @Barnaby:2010vf[^7], metastable cosmic strings @Buchmuller:2019gfy and evaporating light primordial black holes @Anantua:2008am [@Dong:2015yjs]. Further significant contributions may be expected from preheating @Dolgov:1989us [@Traschen:1990sw; @Kofman:1994rk; @Kofman:1997yn; @Figueroa:2017vfa] and first order phase transitions occurring above $10^7$ GeV @Witten:1984rs [@Hogan:1986qda; @Punturo:2010zz; @Hild:2010id; @Evans:2016mbw]. The sensitivity of radio telescopes can however not yet compete with the cosmological $N_\text{eff}$ bound, unless one considers essentially monochromatic signals (which may arise e.g. from large monochromatic scalar perturbations @Bugaev_2010 [@Saito_2010]).

Since the dominant contribution to ${{\cal P}}$ arises around reionization, a particularly interesting target are GW sources active around $10 \lesssim z \lesssim  10^3$, which would not be constrained by the $N_\text{eff}$ bound. During the dark ages, there is no generic reason to expect GW production in the GHz regime but there are models which predict such a signal for suitable parameter choices. For example, mergers of light primordial black holes in this epoch (with masses of about $10^{-9..-7}~M_\odot$) would result in GHz GW signals today @Maggiore:1900zz, see @Raidal_2019 [@Gow_2020] for a discussion of possible rates. Superradiant axion clouds around spinning black holes yield an essentially monochromatic GW signal with $f \lesssim$ MHz @Arvanitaki:2009fg [@Arvanitaki:2010sy; @Arvanitaki:2012cn], with higher frequency possible when considering primordial black holes with masses below the Chandrasekhar limit.

We emphasize that the use of radio telescopes allows to search for GWs in a wide frequency regime. While the absence of any excess radiation can already constrain some models under the assumption of strong cosmic magnetic fields, the potential of this method will truly unfold with further improvements in the sensitivity of radio telescopes –driven in particular by the advances in 21cm cosmology– or in the case of a positive detection of excess radiation.

An example of future advances in radio astronomy is the case of the Square Kilometer Array (SKA). Assuming an effective area per antenna temperature of at least $\unit[10^2]{m^2/K}$ @Braun:2019gdo[^8] in the $0.1-10$ GHz range, a few hours of observation will lead to sensitivities in the ballpark of $\unit[]{\mu Jy}$, which must be compared against CMB fluxes of at least $\unit[10^3]{Jy}$. SKA measurements are thus very promising although sufficient foreground subtraction will be extremely challenging.

#### Acknowledgments.

It is a pleasure to thank Nancy Aggarwal, Sebastien Clesse, Mike Cruise, Hartmut Grote and Francesco Muia for insightful discussions on the Gertsenshtein effect and high-frequency GW sources at the ICTP workshop "Challenges and opportunities of high-frequency gravitational wave detection". Likewise, we would also like to thank Torsten Bringmann, Damian Ejlli, Kohei Kamada and Kai Schmidt-Hoberg. This work was partially funded by the Deutsche Forschungsgemeinschaft under Germany's Excellence Strategy - EXC 2121 "Quantum Universe" - 390833306. C.G.C. is supported by the Alexander von Humboldt Foundation.

[^1]: The first relation in Eqs.  differs from that of @Fujita:2020rdx in the negative sign.

[^2]: See the discussion in the Supplemental Material which includes Refs. @Venhlovska:2008uc [@Dvorkin:2013cga; @2012PhRvD..85d3522F; @Carlson:1994yqa]

[^3]: See the discussion in the Supplemental Material showing that the effect of the magnetic field on the wave velocity is negligible, which includes a reference to @Monitor:2017mdv

[^4]: In Eq. , $B^2 \to 2 B^2/3$ extracts the transverse component.

[^5]: For numerical computations, we use the values reported in Ref. @Kunze:2015noa

[^6]: See Ref. @Pospelov:2018kdh for a related argument for decaying dark matter.

[^7]: The dominant GW signal of axion inflation may also be related to its subsequent preheating phase @Adshead:2019lbr, also leading to MHz to GHz signal which can easily saturate the $N_\text{eff}$ bound.

[^8]: See also <https://www.skatelescope.org/wp-content/uploads/2014/03/SKA-TEL_SCI-SKO-SRQ-001-1_Level_0_Requirements-1.pdf>

{'source': 'arXiv_out/2006'}