Inverse Gertsenshtein effect as a probe of high-frequency gravitational waves

We apply the inverse Gertsenshtein effect, i.e., the graviton-photon conversion in the presence of a magnetic field, to constrain high-frequency gravitational waves (HFGWs). Using existing astrophysical measurements, we compute upper limits on the GW energy densities ΩGW at 16 different frequency bands. Given the observed magnetisation of galaxy clusters with field strength B ∼ μG correlated on 𝒪(10) kpc scales, we estimate HFGW constraints in the 𝒪(102) GHz regime to be ΩGW ≲ 1016 with the temperature measurements of the Atacama Cosmology Telescope (ACT). Similarly, we conservatively obtain ΩGW ≲ 1013 (1011) in the 𝒪(102) MHz (𝒪(10) GHz) regime by assuming uniform magnetic field with strength B ∼ 0.1 nG and saturating the excess signal over the Cosmic Microwave Background (CMB) reported by radio telescopes such as the Experiment to Detect the Global EoR Signature (EDGES), LOw Frequency ARray (LOFAR), and Murchison Widefield Array (MWA), and the balloon-borne second generation Absolute Radiometer for Cosmology, Astrophysics, and Diffuse Emission (ARCADE2) with graviton-induced photons. The upcoming Square Kilometer Array (SKA) can tighten these constraints by roughly 10 orders of magnitude, which will be a step closer to reaching the critical value of ΩGW = 1 or the Big Bang Nucleosynthesis (BBN) bound of ΩGW ≃ 1.2 × 10-6. We point to future improvement of the SKA forecast and estimate that proposed CMB measurement at the level of 𝒪(100-2) nK, such as Primordial Inflation Explorer (PIXIE) and Voyage 2050, are needed to viably detect stochastic backgrounds of HFGWs.

properties of reionisation [78], and cosmological aspects [79,80].We forecast the potential of the SKA in indirectly constraining the HFGWs.By comparing the upper limit constraints derived using existing measurements, forecast using SKA, and estimated using proposed future CMB surveys, we note the general status of HFGW detection proposals and future work to improve them.
The paper is structured as follows.In Section 2, we present the result of the gravitonphoton conversion in the classical limit, given the context of the large-scale magnetisation of the Universe.The detailed formulation of the effect can be found in appendix A. In Section 3, we present conservative upper bounds on HFGWs obtained from the kSZ observations with the ACT (Section 3.1), and from the reported excess radio background by EDGES, LOFAR, MWA, and ARCADE2 (Section 3.2).In Section 4, we present forecast constraints from SKA (Section 4.1) and future CMB surveys (Section 4.2).We discuss the findings in Section 5 and conclude in Section 6.

Graviton-photon conversion
We will describe the theory behind the conversion of gravitons to photons at galaxy cluster and cosmological scales in Section 2.1 and 2.2 respectively.In Section 2.3, we will discuss the implications of such conversions on the CMB.

Galaxy clusters
We briefly discuss here the framework in which gravitons are converted into photons.GWs with frequency f traversing through a galaxy-cluster magnetic field with amplitude B and coherent scale (or equivalently correlation length) l cor 1 can be converted into photons of the same frequency with a small but nonzero probability P g→γ (see appendix A for the detailed derivation) where n e is the electron density of the medium, and D is the total travel distance of GWs.We assume D > l cor and a constant field strength within the coherence scale of the magnetic field.Equation (2.1) shows that the conversion probability depends on the strength and structure of the magnetic field.Both strong (B > 1 µG [81][82][83][84]) and weaker fields (B ≲ 0.1 − 1 µG [85,86]) have been observed to be correlated over scales of O(10) kpc in galaxy clusters and in the dilute plasma between galaxies, known as the intracluster medium (ICM) (see Refs. [87,88] for reviews).These are obtained using various methods such as Faraday rotation measurements [89,90] or the equipartition hypothesis for the observed cluster-scale, 1 Note that we directly take the observationally inferred values of lcor here.In numerical simulations with a known magnetic field energy spectrum EB(k) in wavenumber k space, the correlation length lcor is instead computed via the expression lcor diffuse synchrotron emission [91,92] 2 .The origin of such large-scale correlated magnetic fields is debated.A commonly accepted hypothesis is that they result from the amplification of weak seed fields, which could originate either in the early Universe, referred to as PMFs, or at later epochs, e.g., during reionisation and structure formation.
The cosmological magnetohydrodynamic (MHD) simulations, which account for the amplification of weak seed magnetic fields through the combined effects of gravitational collapse and small-scale dynamos [97][98][99][100][101], reproduce ICM magnetic fields with B ∼ 1 µG strengths (see Ref. [102] for a review).However, the obtained strength as well as the coherence scale of the magnetic field at the current epoch depend on the magnetogenesis scenarios, and on the evolution of the field in the pre-and post-recombination epochs.For instance, in the case of PMFs, the correlation lengths of the field can be as large as 230 − 400 kpc at z = 0 [101].The observational constraints are also affected by the assumption of the field topology [82,[103][104][105].Therefore, the precise calculation of the inverse Gertsenshtein effect depends on our understanding of the magnetic field structure and strength on the relevant scales.Having this caveat in mind, we explore in our work the conversion of gravitons into photons on cluster scales (Section 3.1), setting B ∼ 0.1 µG over l cor ∼ 10 kpc.

Cosmological scales
Beyond the scales of galaxies and clusters, redshift dependence of the electron distribution becomes important when considering inverse Gertsenshtein effect across cosmological epochs.The graviton-photon conversion probability modifies to [52] where f eq = k B T CMB /(2πℏ) ≈ 56.79 GHz is the characteristic frequency of CMB assumed to be in equilibrium, ∆l 0 = min(l eq , l 0 cor ) with l eq ≈ 95 Mpc being the comoving scale of the scalar mode entering the horizon at radiation-matter equality, and l 0 cor being the present-day coherence length of the magnetic field, and I(z ini ) is an integral determined by the ionisation fraction X e (z) [106] where z obs is the redshift at which observations are made.In line with the assumptions made in Ref. [52], we consider graviton-photon conversion taking place post-CMB and therefore take z ini ∼ 1100 in equation (2.3) (see Ref. [107] for a study of pre-CMB conversion).We note that contrary to the case discussed in Section 2.1, here we specifically assume the primordial origin of the observed large-scale magnetic fields.Although the current radio telescopes detect the diffuse radio emission only up to a redshift of z ∼ 0.9 [108][109][110], the analysis of the distant blazar spectra by the Fermi-LAT and High Energy Stereoscopic System (H.E.S.S.) collaborations [111] hints at the existence of Mpc-correlated, volume-filling magnetic fields in the IGM.This favours primordial magnetogenesis scenarios in the inflationary [112][113][114][115][116] or phase-transitional [117][118][119][120] epochs.
Uncertainties, again, remain in the understanding of the evolution of PMFs across different cosmological epochs (see Ref. [121] for a review).In this work, to analyse HFGWs on cosmological scales (Section 3.2), we choose the field properties normalised in equation (2.2), i.e., B ∼ 0.1 nG and l cor ∼ 1 Mpc, which yield considerably higher conversion probability than in individual local structures such as galaxies or clusters shown in equation (2.1).We justify the choice of the field strength B ∼ 0.1 nG by noting that although it is orders of magnitude higher than the lower bounds derived from the blazar spectra observations, i.e., B > 10 −14 G in voids [111], it is well within the derived values of B < 40 nG [93] and B < 4 nG [94] in the IGM, and 30 ≤ B ≤ 60 nG for filaments [122].In our future work we will take into account magnetic field characteristics in different epochs predicted or modelled by MHD simulations, i.e., weighting the integral in equation (2.2) by the corresponding field strength and coherence scales.

Impact on the blackbody CMB radiation
With a small but non-vanishing conversion probability P g→γ , the inverse Gertsenshtein photons introduce a small distortion ∆F γ to the CMB photon distribution which is otherwise assumed to be in equilibrium, i.e., where F γ is the overall photon distribution, and F eq describes the blackbody distribution of CMB photons We analogously define a graviton distribution F g (f, T ) to be such that the total GW energy is obtained as where Ω GW (f ) is the GW energy density in units of the critical density.The distribution functions F γ,g satisfy the Boltzmann equation, where is the Liouville operator with H being the Hubble parameter, and ∂ f F γ,g = 0 is assumed as we focus on conversions occurring at a fixed frequency.Here ⟨Γ γg ⟩ is the conversion rate, such that it integrates to the total probability along the line-of-sight (l.o.s.) P g→γ = l.o.s.⟨Γ γg ⟩dt.The solution of (2.8) can be obtained as [52] F γ (f, T ) F g (f, T ) = e −Pg→γ cosh P g→γ sinh P g→γ sinh P g→γ cosh P g→γ which, to the leading order in O(P g→γ ), yields the expression for ∆F γ as ∆F γ (f 0 , f eq ) = F g (f ini , T ini ) − F eq P g→γ . (2.10) Therefore, the fractional distortion to the photon distribution at a given frequency is where we assumed F γ = F eq at the initial time.Inverting equation (2.11) yields a constraint on the GW energy density Ω GW given by ∆F γ /F γ as . (2.12) Note that in similar works, sometimes the constraints are instead placed on the characteristic GW strain h c , related to Ω GW via In further sections, we will analyse the measurements by directly associating the distortion of the blackbody temperature to that of the photon distribution in equation (2.11), where is the characteristic frequency of a black body with temperature T γ and an overall distribution F γ .Explicitly in the Rayleigh-Jeans limit (f ≪ f eq ≈ 56.79 GHz) and the high-frequency limit (f ≫ f eq ≈ 56.79 GHz), equation (2.14) takes the forms where ∆T denotes the excess temperature on top of the CMB blackbody spectrum.Assuming that the inverse Gertsenshtein effect is the only mechanism causing such distortion, then δT g = ∆T , where δT g is the temperature of graviton-induced photons.In reality, many mechanisms can potentially contribute and therefore δT g < ∆T (see Section 3.1 for one such example and Sections 5.1 and 5.2 for discussions).

Existing measurements
We now present the constraints on the HFGWs from currently available measurements.

Kinematic Sunyaev-Zel'dovich effect
The kSZ effect has been conventionally used as a method to observe and study galaxy clusters since the induced CMB distortions act as markers for the underlying electron distribution in these clusters.We provide a brief description of the kSZ modelling and refer the interested readers to Ref. [123] for more detail.The temperature contrast observed against the CMB can be characterised as [66] δT kSZ  where σ T is the Thomson cross-section, v p is the peculiar velocity of the observed galaxies, and τ (θ) is the optical depth along the line of sight (l.o.s.).The kSZ temperature in the above equation can be simplified in the observed redshift range of ACT, 0.4 < z < 0.7 [65], by approximating e −τ ≈ 1.In addition, the root-mean-square (rms) peculiar velocity along the l.o.s. is v rms p /c ∼ 1.06 × 10 −3 .Therefore, equation (3.1) is determined essentially by the electron density n e alone where n e , in turn, depends on the gas density ρ gas .Modelling the baryonic physics that affects the gas distribution requires detailed hydrodynamical simulations [e.g.124,125].However, we use an approximate method, called baryonic correction model (BCM), that are computationally less expensive and validated against advanced hydrodynamical simulations [126,127].We briefly describe this model in Appendix B. Reproducing the measurements provided by ACT requires convolving our model with the telescope filters, which are described in Ref. [65].We refer the interested readers to Refs.[66,123] for detailed modelling of this ACT observation.
Along with the kSZ effect, the temperature excess over the CMB can also receive a contribution from the inverse Gertsenshtein effect as HFGWs propagate through galactic and cluster magnetic fields.We assume that the total measured temperature contrast ∆T = δT kSZ + δT g with δT g being the temperature of graviton-induced photons.The effect of varying the baryon distribution inside clusters by incorporating δT g can be seen in Figure 1.The black line corresponds to a BCM with ∆T g = 0, aligning closely with the measurements within the 95% credible region sourced from Ref. [123].The combined temperature profile ∆T is tuned to stay within the error margins of ACT datasets.We note that δT g enhances the overall temperature profile but the exact features depend on the magnetic field properties.Assuming that magnetic fields exist only inside the clusters and become absent in the ICM (left panel of Figure 1), the enhancement diminishes for the corresponding scales outside individual clusters.In this case, the strongest constraint, h c ≥ 8 × 10 −22 , takes place at the smallest angular scale R ≈ 1, i.e., closest to the center of the cluster, corresponding to the largest GW propagation distance.Assuming uniform magnetic field B 0 = 1 µG throughout the entire field of observation of R ∼ 6 arcmin, corresponding to the scale ∼ 3.7 Mpc at the mean redshift z ∼ 0.55 (right panel of Figure 1), the temperature profile amplifies more with increasing angular scales.In this case, GW strains are upper bounded at the largest observed scales R ≈ 6, giving h c ≥ 4 × 10 −22 .Therefore, detailed knowledge of magnetic fields within clusters and in the ICM is needed to determine precisely the effects of δT g , although the results of the two assumptions shown in the left and right panels of Figure 1 differ within a factor of two.Using equations (2.12) and (2.15), and assuming the scenario of uniform magnetic fields throughout the field of observation, we note that the kSZ observations of ACT provide upper limits on the HFGW energy density Ω GW (f ) and strains h c (f ), i.e., Ω GW ≲ 3.5×10 15 and h c ≲ 8×10 −22 at f = 90 GHz, and Ω GW ≲ 9.9×10 15 and h c ≲ 4×10 −22 at f = 150 GHz.We find that the inverse Gertsenshtein effect is a subdominant correction on top of the more significant kSZ effect, i.e., δT g ≪ δT kSZ for all observed angular scales, even for the unrealistically large HFGW amplitudes considered here.

Excess radio background
Previous studies have reported an excess of radio background through various methods, including direct measurements from ARCADE2 [128] and LWA1 [129], as well as indirect observations through the 21-cm signal during cosmic dawn.This 21-cm signal can be observed by radio experiments as the differential brightness temperature at position x and redshift z, which is given as [67,75] where x HI , δ b and T s are neutral hydrogen fraction, baryon overdensity and spin temperature, respectively.This signal is observed against a radio background, which follows a blackbody spectrum of temperature T radio .
The previously assumed radio background to be the CMB has been challenged by measurements from ARCADE2 [128] and LWA1 [129].These measurements indicate the presence of excess radio signal over the CMB, denoted as T radio ̸ = T CMB (1+z) [128,129].Additionally, the unconventional sky-averaged 21-cm signal evolution detected by EDGES [70] at z ≈ 17 further supports the plausibility of this excess radio signal [69].This radio background can be effectively described by the model: where A r represents the frequency-independent amplitude coefficient, and β ≈ −2.55 stands for the spectral index.Note that the bounds on Ω GW (f ) in equation (2.12) can be rewritten in terms of the A r parameter as where the temperature distortion is given by ∆T /T CMB = A r (ν obs /78 MHz) β , and we have taken the Rayleigh-Jeans limit f ≪ f eq applicable to the radio frequency regime.Figure 2 demonstrates the upper bounds of Ω GW (f ) in equation (3.5), given certain values of A r and a cosmological magnetic field with properties normalised in equation (2.2).To give an intuition of the constraining power of A r , we compare to two theoretical expectations, namely the critical energy density Ω GW = Ω crit = 1, and the more realistic ∆N eff bound given by where the variation of the effective number of relativistic degrees of freedom is measured to be ∆N eff ≲ 0.1 [10,130,131], and the radiation energy density is Ω γ ≃ 5.4 × 10 −5 [10].
We can also use the measurements from radio telescopes such as LOFAR and MWA to constrain A r .These telescopes are attempting to statistically measure the spatial distribution of δT b with the 21-cm power spectrum during the epoch of reionisation [e.g.73].The field of view of these telescope are large enough for the signal to contain cosmological information, which has been used by previous authors to study cosmology [80,134,135].To provide a conservative limit, we assume that the entire excess radio signals were created by Instrument  [69], LO-FAR [132], and MWA [133] in the O( 102 ) MHz regime.We also show the corresponding redshift z along with the A r values constrained at 68% confidence level.
inverse Gertsenshtein effect.We then use equations (3.4) and (2.15) to convert the values of A r ̸ = 0 reported by authors interpreting observations from EDGES [69], LOFAR [132], and MWA [133] to constraints on the GW energy density Ω GW (f ) and characteristic strain h c (f ) at the corresponding frequency f3 .Note that this is done assuming magnetic field properties normalised in equation (2.2).The resulting constraints on GWs at MHz frequencies are shown in Table 1.Though several orders of magnitude weaker than the ∆N eff bound, these constraints have been derived from independent measurements.Previous studies have explored several phenomena, including emission from high redshift radio galaxies [72] and superconducting cosmic strings [137], to explain the excess radio background.If such phenomena are confirmed to contribute to this excess signal, then the constraints on GWs can be improved further.
In Table 2, we have listed measurements from ARCADE2 [138] that directly measure the photon background at GHz frequencies, encompassing both the CMB and GWs converted into photons.∆T 0 gives the measured excess temperature over the CMB at different frequencies (f ).This table also includes constraints on GHz frequency GWs via equation (2.15), albeit being relatively weak.We anticipate improvements in these constraints with upcoming CMB experiments, which will be elaborated upon in Section 4.2.

Future surveys
We will now discuss the future surveys that are capable of improving the constraints on HFGWs.First, we present a forecast study for the upcoming radio telescope, the Square Kilometre Array (SKA).Later, we will discuss the potential of future CMB experiments.

Forecast with the SKA
The observation of the 21-cm signal will improve substantially with the SKA that is currently under construction at two different sites.Here we will focus on the low-frequency component built in Western Australia that will cover frequencies f ∼ 30−300 MHz [75].In this study, we  2: Bounds of GW energy density Ω GW (f ), characteristic strain h c (f ), and the corresponding radio temperature excess ∆T 0 from ARCADE2 [138] in the O(10 0 ) GHz regime.Note that measurements of ARCADE2 at f ∈ {29.5, 31, 90} GHz are not shown here as they are consistent with CMB.
will use the power spectra expected from the SKA that will quantify the spatial fluctuation strength of the 21-cm signal at different redshifts.
In order to model the power spectra, we use the analytical framework initially proposed in Ref. [139].This framework has been actively improved to study the impact of cosmological structure formation on the 21-cm signal at high redshift (z ≳ 6) [79,80].We construct the mock observation for SKA using the open source package, Tools21cm [140].The error in the measurement accounts for the instrumental noise that increases with increasing wavenumber (k).This noise was estimated assuming an observation time of 1000 hours and the latest plan of SKA antenna distribution.For a detailed description of this calculation, we refer the readers to Ref. [141].We only consider the power spectra at k ≳ 0.1 Mpc −1 preventing the regime dominated by cosmic variance and foreground contamination [142].
We examine the identical mock observation illustrated in Figure 5, and the model parameters assumed during the observation listed under 'Mock Value' in Table I of Ref. [80].These observations are constructed at 12 redshifts covering frequency f ∼ 80 − 200 MHz.For this mock observation, the value of A r is zero, which corresponds to no excess radio background.To infer the constraints expected on A r , we performed an Monte Carlo Markov Chain (MCMC) analysis.We provide the full result in appendix C.This analysis constrained A r ≲ 10 −9 at 68% confidence level.With magnet field properties in equation (2.2), i.e., B 0 ∼ 0.1 nG and ∆l 0 ∼ 1 Mpc, this A r value corresponds to Ω GW ≳ 6.0 × 10 4 at f ∼ 30 MHz and Ω GW ≳ 4.7 × 10 2 at f ∼ 200 MHz.We show this constraint in Figure 3. Compared to the current limits at these MHz frequencies (Table 1), SKA will improve it by 7 to 10 orders of magnitude.
The reionisation and heating of the IGM by the first photon source or galaxies impact the spatial distribution of the signal at ionised bubble scales, which would have a scale dependent effect on the 21-cm signal [143][144][145].This scale-dependent information in the SKA data will help constrain the early galaxy properties and allow constraining A r with the amplitude of the spectrum [132].However, we assume that the 21-cm background is uniformly amplified by primordial HFGWs.Therefore, the improvement we predict for the SKA is not sensitive to the assumed properties of the first galaxies for our mock observation.We also want to mention a caveat of our forecast study, which is the assumption that the foreground signal is assumed to be perfectly removed.
In this study, we assumed the sensitivity of the first phase of SKA.The later stages are planned to be more sensitive that will allow the constraints on HFGWs to be even stronger.We should also note that the constraints could also be improved with longer observation time.and the corresponding frequency f min are also shown.The magnetic field properties are chosen to be in line with equation (2.2).
However, processing large radio data is challenging due to various complexities, including the intricate process of data calibration (see Refs. [146,147] and references therein).Therefore we have considered 1000 hour observation time, which is the initial target [75].The SKA will be sensitive enough to give us measurements beyond the power spectrum, including the bispectrum [e.g.148,149] and image datasets [e.g.143,150,151].These measurements will have more constraining power, which we will explore in the future.

Forecast with future CMB surveys
In the future, a number of precision surveys of CMB and its spectral distortions could improve the constraints on HFGWs beyond the GHz regime.Here we consider the expected capabilities of proposed missions such as Polarized Radiation Interferometer for Spectral disTortions and INflation Exploration (PRISTINE) [152], the Primordial Inflation Explorer (PIXIE) [153,154] and its next-generation concept Super-PIXIE, as well as the scheme Voyage 2050 [155], assumed to be a few times more sensitive than Super-PIXIE.We take the foreground-marginalized error budget for temperature measurements of these missions [152] and compute the upper limits of HFGW energy density Ω GW (f ) within the corresponding detectors' frequency bands.The anticipated temperature signal errors are shown in Table 3 for PRISTINE (2 years), PIXIE (4 years), Super-PIXIE (8 years), and Voyage 2050 [152].Spectral distortions at O(10 0−2 ) nK precision, if achieved, would significantly tighten the HFGW constraints, even reaching below the critical value Ω GW = 1 and the current ∆N eff bound Ω GW ≃ 1.2 × 10 −6 at the THz frequency regime.On the other hand, the ∆N eff bound could also be tightened with these precision future surveys and a careful comparison between the expectations is needed.The forecast constraints from SKA and future CMB surveys are shown in Figure 3 together with the estimates from Sections 3.1 and 3.2.We show that PIXIE/super-PIXIE and, consequently, PRISTINE have the capability to significantly tighten constraints, bringing them into an intriguing range.Voyage 2050, which is at the conceptualisation phase, will substantially improve the constraints on HFGW probing Ω GW ≲ 1.2 × 10 −6 in the THz regime.

Discussions
In Section 5.1, we compare the sensitivity of ongoing and planned experiments to constrain HFGWs.Later in Section 5.2, we will briefly discuss a few more alternate methods.[69,70], LOFAR (green) [132], MWA (black) [133], ARCADE2 (orange) [138], and ACT (purple) [65].Forecast with the upcoming SKA-Low (red) is shaded in red .Forecast with proposed and conceptual PRISTINE (cyan) [152], PIXIE/Super-PIXIE ( blue ) [153,154] and Voyage 2050 (light blue) [155] are also shown in comparison with tilted hatches .The black lines in the lower part of the figure indicate the theoretical upper bounds of GWs: the solid line shows where Ω GW = 1, and the dash-dotted line indicates the ∆N eff bound from equation (3.6).

Sensitivity comparison
From Figure 3, we make the following observations regarding the constraints on HFGWs from existing and projected future detectors.
• Among the existing instruments, ARCADE2 provides the tightest constraint at Ω GW ≲ 10 11 , followed by radio telescopes EDGES, LOFAR, and MWA with similar constraints at Ω GW ≲ 10 13 .The least competitive constraints come from ACT at Ω GW ≲ 10 16 .
• The constraints from ACT are the least competitive due to the low conversion probability applicable to kSZ observations of individual galaxy clusters, i.e., P ∼ 10 −30 in equation (2.1).This is much lower than the conversion probability applicable for global radio excess signals, i.e., P ∼ 10 −20 in equation (2.2).
• Constraints obtained in this work using existing measurements are comparable to similar work in the literature [44,52,156], although a much tighter bound of Ω GW < 1 has been claimed in the X-ray frequency band [156].
• We found that all upcoming and future observations are significant improvements at their corresponding frequencies.Even with an realistic amount of observing time, i.e. 1000 hours, and the conservative assumption that graviton-induced photons saturate the entire excess signals, the SKA is expected to improve the existing constraints in the MHz frequency band by roughly 10 orders of magnitude to Ω GW ≲ 10 2−4 .
• Removing foreground contributions, proposed (PRISTINE, (Super-)PIXIE, Voyage 2050) CMB surveys can significantly tighten the Ω GW upper limits, with Voyage 2050 reaching below the BBN bound in the THz regime, if the anticipated precisions are achieved.However, note that the results are highly dependent on the foreground modelling and that, as of the time of writing, these surveys remain a concept.
• Similar to the projected CMB surveys, our conservative estimates in the MHz to GHz regimes can be further improved by combining foreground contribution from other mechanisms, such as decays of relic neutrinos [157,158], axions [159] and other dark matter candidates [160,161].We leave this to future studies.

Alternative methods to probe HFGW
Though this work focused on approach (i) introduced in Section 1, we want to discuss the ongoing efforts to explore approach (ii).Many laboratory proposals exploit the similarities between axion-photon and graviton-photon couplings to constrain HFGWs using axion detectors.The QCD axions are pseudoscalars initially proposed as a dynamical solution to the strong charge-parity (CP) problem [162][163][164][165] (see Ref. [166] for a review of axions and axion-like particles).Numerous experimental efforts are underway to survey the parameter space of axions, due to their desirable properties as a dark matter candidate [167][168][169] (see Refs. [170][171][172] for reviews).The coupling between axion and EM fields reads L ⊃ g aγ aF F , where a is the axion field and g aγ the coupling strength.This resembles the coupling between gravity and the EM sector via the term L ⊃ hF 2 .The similarity implies that both axions and gravitons can convert to photons in external magnetic fields and that data from existing axion haloscopes can be reinterpreted to constrain HFGWs [58,59].In general, approach (ii) has the advantage of having full knowledge and control of the magnetic field.The properties of large-scale magnetic fields, on the other hand, have not been precisely constrained.However, the future SKA surveys have the potential to substantially improve our understanding about their origin and structure on large-scales [173].Laboratory field strength (up to ∼ 10 T) can also be much larger than cosmological fields as well, although the latter compensate by having a much larger effective detector volume.Besides the two approaches of the inverse Gertsenshtein effect overall, other methods to detect HFGWs have been proposed, based on the couplings between gravitons and materials or mediums other than EM waves.These include, but not limited to, quantum sensors to detect graviton-phonon conversion [174,175], optically levitated sensors [176,177], microwave cavities [178][179][180][181], bulk acoustic wave devices [182], and graviton-magnon resonance detectors [183].Note that it is nontrivial to unify the sensitivity treatment and comparison across different detection proposals.Therefore, we leave the quantitative comparison of the advantages and disadvantages of the proposed methods to a future work.

Conclusion
In this work we estimated the upper bounds on the stochastic background of HFGWs at 16 frequency bands in the O( 102 ) MHz and O( 102 ) GHz regimes by applying the inverse Gertsenshtein effect to large-scale magnetic fields.The bounds are obtained conservatively by saturating with induced Gertsenshtein photons: (i) the excess of radio background over CMB reported by EDGES, LOFAR, MWA, and ARCADE2, and (ii) the error margins of the kSZ observations made by ACT, assuming a fixed underlying model of baryonic physics inside galaxy clusters.We note that these constraints are comparable in competitiveness as similar works, and that they all lie many orders of magnitude larger than the ∆N eff bound.Therefore, probing a high-frequency SGWB using inverse Gertsenshtein effect might be challenging with the existing instruments.However, the ∆N eff bound as a benchmark only applies to SGWB produced before BBN at T ∼ 1 MeV, and late-Universe mechanisms might generate GWs that reach above the bound Ω GW ≃ 1.2 × 10 −6 at certain frequencies.In addition, for transient HFGW events, the theoretical bounds on a stochastic background also become invalid.Thus, albeit the challenges, inverse Gertsenshtein effect is worth careful further studies in different contexts as a means to probe HFGWs.It is especially so since significant improvements can be expected with future measurements.The upcoming SKA is expected to be much more sensitive than the current radio telescopes considered in this and other similar works.We forecast an approximately 10 orders of magnitude tighter constraint from SKA when applied in the context of excess radio background.Here we have only focused on the low frequency component of SKA that will observe the IGM during cosmic reionisation.The medium frequency component of SKA will observe the 21-cm signal produced by the neutral hydrogen inside galaxies [184] and help fill the gap seen in Figure 3 above f ∼ 200 MHz.We will explore this regime in the future.Finally, the future CMB surveys anticipated to detect spectral distortions at O(10 0−2 ) nK level could potentially reach the current ∆N eff bound.Finally, obtaining more realistic upper limits of HFGWs requires careful considerations of realistic magnetic fields, possibly using direct numerical simulations, as well as going beyond the conservative estimations by saturating Gertsenshtein photons.This implies a detailed understanding of other systematic and/or physical mechanisms contributing to the global foreground signals.
where B crit = m 2 e ∼ 10 13 G is a critical value.In the relevant regimes we consider, i.e., B ∼ µG and ω ≲ O(10 2 ) GHz, the condition (A.10) is clearly not satisfied.Therefore, we neglect the QED corrections and reduce the Maxwell equation (A.6) to where we have split the Faraday tensor F µν = Fµν + f µν into a quasi-static background Fµν and a small perturbation |f µν | ≪ Fµν induced by GWs.We also used the Lorenz gauge for both EM and GW quantities, i.e., ∂ µ A µ = ∂ µ h µν = 0.

Conversion probability
The GW equation (A.5) in terms of its components reads where the EM tensor (A.7) is assumed to be dominated by the magnetic fields As a result of the split of the Faraday tensor, the magnetic field (B = B + b) also decomposes into a homogeneous part B and a small induced part |b| ≪ | B|.To bilinear order in B and b, and let ∂ i = (0, 0, ∂ l ) in the longitudinal direction only, then the induced magnetic fields are Similarly the Maxwell equation (A.11) can be recast to 11) ∂ l h (12)  ∂ l h (21) ∂ l h (22)   B2 − B1 , (A. 16) which, in the same frame of (B 1 , B 2 ) → (B, 0) as above, leads to the simplified form of , where we assumed ∂ 0 ≃ −iω, −i∂ l ≃ k and ω + k ≃ 2ω.Note that we expressed the wavenumber k in terms of ω using dispersion relation k = µω, and used the fact that µ − 1 ≪1 in our case to arrive at ω + k ≃ 2ω.We also focus on monochromatic waves such that A λ (t, l) = e −iωt A λ (l) and h λ (t, l) = e −iωt h λ (l).Then the spatial solution A λ (l) and h λ (l) in the system of equations (A.17) and (A.15) can be written as Note that (A.28) is suitable for a propagation distance up to the coherence scale, l ≲ l cor , so that the uniformity of magnetic field can be assumed.For a larger distance D > l cor , the total conversion probability can be approximated by averaging out the sinusoidal part in equation (A.

B Modelling baryonic physics inside galaxy clusters
We briefly elaborate on the BCM that yields the gas profile ρ gas and electron numder density n e used for the kSZ effect calculations in Section 3.1.We employed a baryon model, initially presented in Ref. [188], and further refined in subsequent works [126,127], to capture and characterise the processes inside clusters.The matter inside the galaxy clusters consists of collisionless dark matter (clm), gas and the central galaxy matter (cga).In BCM, the total matter (ρ dmb ) is given as, where the gas profile is given by with β being a mass-dependent slope, f cga the stellar-to-halo fractions of the central galaxy, and f star the total stellar content.These quantities are parameterised as, where i ∈ {cga, star}, M S = 2 × 10 11 M ⊙ /h and the power indices are η star = η and η cga = η +η δ .Here Ω b /Ω mat is the baryon fraction, r vir and M vir are the virial radius and mass.Note that in the large-cluster limit, equation (B.2) approaches the truncated Navarro-Frenk-White (NFW) profile, i.e., lim M vir ≫Mc β = 3.In essence, the electron number density depends on a 7-parameter model with five gas parameters θ θ θ gas ≡ (log M c , µ, θ ej , γ, δ), and two stellar parameters θ θ θ star ≡ (η, η δ ).In this study, we fixed the model with θ θ θ gas = (13.4,0.3, 4, 2, 7) and θ θ θ star = (0.32, 0.28), which is within the 95% credible region of the constraints in Ref. [123].The dark and light contours represent the 68% and 95% confidence level.We see that this measurement will not only constrain the astrophysical processes, it will provide strong upper limits on the excess radio background (log 10 A r ) that can be translated into constraints on the HFGWs.

C Full posterior distribution of the SKA forecast study
where L(d d d|θ θ θ) and π(θ θ θ) are the likelihood and prior distribution respectively.We assume a Gaussian likelihood that is given in equation ( 21) of Ref. [79].This likelihood incorporates the impact of both the cosmic variance and system noise expected from 1000 hour with SKA observation.
We use an MCMC sampler implemented in the publicly available emcee package 5 [189] for exploring model parameter space.Below we describe these model parameter along with the assumed prior range for each of them.
1. f * ,0 : The amplitude of the star formation rate in the dark matter haloes hosting photon sources during these early times.The number of photons emitted is proportional to the stellar mass.Therefore we can control the amount of emitted photons with this parameter.We consider a flat prior between 0.01 and 1 in log-space, which is consistent with models required to interpret the ultraviolet luminosity function measurements [e.g.190,191] .We assume f * ,0 = 0.1 for producing the mock observation.
2. M t : The minimum dark matter mass that can sustain star formation.We consider a flat prior between 3.2 × 106 and 3.2 × 10 9 M ⊙ /h in log-space.We assume M t = 10 8 M ⊙ /h, which is close to the threshold that can sustain source formation due to molecular cooling [192,193].
3. A r : The parameter describing the magnitude of excess radio background corresponding to the 21-cm signal.In order to explore interesting values shown in Figure 2, we consider a flat prior between −30 and 30 in log-space.We produced the mock observation assuming absence of excess radio background that is quantified as A r = 10 −30 .
As the goal of this study is to study the constraints in the HFGWs that depends only on A r , we have chosen a simple two parameter astrophysical model to describe the photon sources during cosmic dawn.In Figure 4, we show the full corner plot 6 for the posterior distribution from the MCMC run.We find that the ground truth for the two astrophysical parameters are predicted correctly at both the 68% and 95% confidence level.We estimated an upper limit on the excess radio background with A r ≲ 10 −9 at 68% confidence level from the 1D marginalised posterior distribution.

Figure 1 :
Figure1: In both panels, ACT observations[65] are indicated by the purple dots and their error bars, and the kSZ temperature spectrum δT kSZ from the baryon physics modelling[123] is shown in black solid lines.The effects of graviton-induced photons are indicated by blue shaded regions, assuming the absence of magnetic fields outside clusters and a background of HFGWs with strain h c = 8 × 10 −22 (left panel) and the presence of magnetic fields with strength B 0 = 1 µG and HFGWs with h c = 4 × 10 −22 (right panel).The main panels and the insets respectively correspond to the cases of f = 90 GHz and f = 150 GHz.

Figure 2 :
Figure 2: The upper limits of GW energy density Ω GW (f ) imposed by a range of corresponding values of the parameter A r in equation (3.4), assuming cosmological magnetic fields with properties discussed in Section 2.2.The two white lines respectively indicate the A r values necessary to reach the critical value of Ω GW = 1 and the ∆N eff limit of Ω GW ≃ 1.2 × 10 −6 .

Figure 4 :
Figure 4: Posterior distribution on the model parameters expected from upcoming SKA-Low measurements with 1000 hour observing time.The assumed parameter values for the mock observation is shown with blue lines.The dark and light contours represent the 68% and 95% confidence level.We see that this measurement will not only constrain the astrophysical processes, it will provide strong upper limits on the excess radio background (log 10 A r ) that can be translated into constraints on the HFGWs.