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 $Omega_{rm GW}$ at 16

different frequency bands. Given the observed magnetisation of galaxy clusters

with field strength $Bsimmu{rm G}$ correlated on $mathcal{O}(10),{rm

kpc}$ scales, we estimate HFGW constraints in the $mathcal{O}(10^2),{rm

GHz}$ regime to be $Omega_{rm GW}lesssim10^{16}$ with the temperature

measurements of the Atacama Cosmology Telescope (ACT). Similarly, we

conservatively obtain $Omega_{rm GW}lesssim10^{13} (10^{11})$ in the

$mathcal{O}(10^2),{rm MHz}$ ($mathcal{O}(10),{rm GHz}$) regime by

assuming uniform magnetic field with strength $Bsim0.1,{rm nG}$ 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. Although none of these existing constraints fall below the critical

value of $Omega_{rm GW} = 1$ or reaches the Big Bang Nucleosynthesis (BBN)

bound of $Omega_{rm GW}simeq1.2times10^{-6}$, the upcoming Square Kilometer

Array (SKA) can improve the sensitivities by roughly 10 orders of magnitude and

potentially become realistic probes of HFGWs. We also explore several

next-generation CMB surveys, including Primordial Inflation Explorer (PIXIE),

Polarized Radiation Interferometer for Spectral disTortions and INflation

Exploration (PRISTINE) and Voyage 2050, that could potentially provide

constraints competitive to the current BBN bound.

## The Roadmap for High-Frequency Gravitational Wave Research

In this article, we have examined the constraints on high-frequency gravitational waves (HFGWs) and outlined a roadmap for future research in this field. Here, we summarize the key conclusions and highlight potential challenges and opportunities on the horizon.

### Conclusions

- We applied the inverse Gertsenshtein effect, which involves graviton-photon conversion in the presence of a magnetic field, to constrain HFGWs.
- Using existing astrophysical measurements, we computed upper limits on the GW energy densities at 16 different frequency bands.
- Our analysis revealed that HFGW constraints in the range of $mathcal{O}(10^2),{rm GHz}$ regime are estimated to be $Omega_{rm GW}lesssim10^{16}$, based on magnetization of galaxy clusters.
- Similarly, assuming a uniform magnetic field with strength $Bsim0.1,{rm nG}$, we obtain $Omega_{rm GW}lesssim10^{13} (10^{11})$ in the $mathcal{O}(10^2),{rm MHz}$ ($mathcal{O}(10),{rm GHz}$) regime by saturating the excess signal reported by various radio telescopes.
- Although none of the existing constraints reach the critical value of $Omega_{rm GW} = 1$ or the Big Bang Nucleosynthesis (BBN) bound of $Omega_{rm GW}simeq1.2times10^{-6}$, future observations hold promise for more significant discoveries.

### Future Roadmap

- Square Kilometer Array (SKA): The upcoming SKA has the potential to improve sensitivities by approximately 10 orders of magnitude. This significant boost in capabilities could make it a realistic probe for HFGWs.
- Next-generation CMB Surveys: Several next-generation surveys, including Primordial Inflation Explorer (PIXIE), Polarized Radiation Interferometer for Spectral disTortions and INflation Exploration (PRISTINE), and Voyage 2050, are being considered. These surveys have the potential to provide constraints that are competitive with the current BBN bound.

### Challenges and Opportunities

- Challenges:
- The development and implementation of cutting-edge technologies and instruments capable of detecting and analyzing HFGWs with high precision.
- The need for more accurate measurements of magnetization in galaxy clusters to improve constraints on HFGWs.
- Addressing potential sources of noise and interference in radio telescope observations to enhance sensitivity to HFGWs.

- Opportunities:
- Advancements in technology, such as the development of more sensitive detectors and improved data analysis techniques, can greatly enhance our ability to study HFGWs.
- The collaborative efforts between different research teams, observatories, and space agencies can lead to innovative solutions and new discoveries in the field.

Overall, the future of HFGW research looks promising. With the upcoming SKA and next-generation CMB surveys, we have the potential to make breakthroughs in our understanding of these elusive phenomena. Overcoming technical challenges and leveraging new opportunities will be key to unlocking the mysteries of high-frequency gravitational waves.