Challenges and opportunities of gravitational-wave searches at MHz to GHz frequencies

Challenges and opportunities of gravitational-wave searches at MHz to GHz frequencies

Nancy Aggarwal, Odylio D. Aguiar, Andreas Bauswein, Giancarlo Cella, Sebastian Clesse, Adrian Michael Cruise, Valerie Domcke, Daniel G. Figueroa, Andrew Geraci, Maxim Goryachev, Hartmut Grote, Mark Hindmarsh, Francesco Muia, Nikhil Mukund, David Ottaway, Marco Peloso, Fernando Quevedo, Angelo Ricciardone, Jessica Steinlechner, Sebastian Steinlechner, Sichun Sun, Michael E. Tobar, Francisco Torrenti, Caner Ünal and Graham White.

Living Reviews in Relativity (2021) 24:4. Review article. Received 6 April 2021 / Accepted 15 September 2021 / Published online 6 December 2021.

Abstract

The first direct measurement of gravitational waves by the LIGO and Virgo collaborations has opened up new avenues to explore our Universe. This white paper outlines the challenges and gains expected in gravitational-wave searches at frequencies above the LIGO/Virgo band, with a particular focus on Ultra High-Frequency Gravitational Waves (UHF-GWs), covering the MHz to GHz range. The absence of known astrophysical sources in this frequency range provides a unique opportunity to discover physics beyond the Standard Model operating both in the early and late Universe, and we highlight some of the most promising gravitational sources. We review several detector concepts that have been proposed to take up this challenge, and compare their expected sensitivity with the signal strength predicted in various models. This report is the summary of the workshop “Challenges and opportunities of high-frequency gravitational wave detection” held at ICTP Trieste, Italy in October 2019, that set up the stage for the recently launched Ultra-High-Frequency Gravitational Wave (UHF-GW) initiative.

Keywords: Ultra-high-frequency gravitational waves · Cosmological gravitational waves · Gravitational wave detectors · Fundamental physics with gravitational waves.

1 Introduction

Gravity and electromagnetism are the only two long range interactions in nature, but over the centuries we have explored the Universe only through electromagnetic waves, covering more than 20 orders of magnitude in frequencies, from radio to gamma rays. The discovery of gravitational waves in 2015 has opened a totally new window to observe our Universe.

Judging by what happens with electromagnetic waves, there should be interesting physics to be discovered at every scale of gravitational wave frequencies. Current and planned projects such as pulsar timing arrays, as well as ground- and space-based interferometers will explore gravitational waves in the well-motivated range of frequencies between the nHz and kHz range. However, both from the experimental and the theoretical point of view it is worth to consider the possibility to search for gravitational waves of much higher frequencies, covering regimes such as the MHz and GHz.

A strong motivation to explore higher frequencies from the theoretical perspective is that there are no known astrophysical objects which are small and dense enough to emit at frequencies beyond 10 kHz. Any discovery of gravitational waves at higher frequencies would thus indicate new physics beyond the Standard Model of particle physics, linked e.g., to exotic astrophysical objects (such as primordial black holes or boson stars) or to cosmological events in the early Universe such as phase transitions, preheating after inflation, oscillons, cosmic strings, thermal fluctuations after reheating, etc.

For early Universe cosmology, gravitational waves may be the only way to observe various events. In particular for the time between the Big Bang and the emission of the cosmic microwave background radiation, electromagnetic waves cannot propagate freely, whereas, due to the weakness of gravity, gravitational waves decouple essentially immediately after being produced and travel undisturbed throughout the Universe forming a stochastic background that could eventually be detected. Even though it may not be easy to unambiguously determine the concrete cosmological source of a gravitational-wave signal, its cosmological nature of the spectrum may be identified, similar to what happened with the original discovery of the cosmic microwave background.

In this context, the existence of a stochastic spectrum in the range from kHz to GHz is well-motivated: causality restricts the gravitational wave wavelength to be smaller than the cosmological horizon size at the time of gravitational wave production. This roughly implies a gravitational-wave frequency above the frequency range of the existing laser interferometers Virgo, LIGO and KAGRA for any gravitational-wave production mechanism that happens at temperatures larger than 10 to the power 10 GeV, assuming radiation domination all the way to matter-radiation equality. (Cosmological events occurring at lower temperatures can also source such high-frequency gravitational waves if the typical scale of the source is hierarchically smaller than the horizon at that time.) In particular, GHz frequencies correspond to the horizon size at the highest energies conceivable in particle physics (such as the Grand Unification or string scale) and phenomena like phase transitions and preheating after inflation would naturally produce gravitational waves with frequencies around the GHz range.

Established gravitational-wave detector designs are limited to frequencies up to the kHz range. In particular, resonant mass detectors, going back to the original bar design of Weber (1967), focused on isolated high frequencies, often targeting known millisecond-pulsar frequencies. Similarly, the well-established interferometric gravitational-wave detectors LIGO, Virgo and KAGRA cover parts of the high-frequency band up to a few kHz. For the purposes of this white paper, we shall therefore use the expression high-frequency gravitational waves to refer to frequencies that are above the LIGO detection band, i.e., starting from around 10 kHz. In particular, taking inspiration from the electromagnetic spectrum, we denote the MHz to GHz range by Ultra High-Frequency Gravitational Waves (UHF-GWs). Several proposals have been made for pushing the high-frequency end of interferometric detectors into this region, however, detectors for the MHz, GHz and THz frequency bands require radically different experimental approaches.

Over the years, there have been isolated attempts to search for gravitational waves of very high frequencies and a few proposals have been put forward. These new concepts have largely been suggested in the form of theoretical papers with no serious discussion of the potential experimental noise sources that might limit their performance, or occasionally, bench tests of early prototypes. The current status of many of these ideas must be regarded as highly preliminary. The published concepts span a wide range of technologies with no real consensus yet as to where to concentrate the community effort. In addition to the selection of suitable technological pathways towards a serious attempt at a detection at high frequencies, there needs to be an identification of the most realistic sources and thereby the waveforms and spectra for which such detectors should be optimised. This process demands a close collaboration of theorists and experimentalists.

The goal of this report is to summarise and start a dialogue among the specialised community regarding the importance and feasibility to explore searches for high-frequency gravitational waves. We are aware that this may be a long term goal but are convinced that the physics motivation is strong enough to start a systematic study of the different sources of high-frequency gravitational waves and their potential detectability. It is the purpose of this white paper to put together the different ideas both from theory and experiment to explore the importance of searching for high-frequency gravitational waves. The origin of this initiative was a workshop organised at ICTP in October 2019, “Challenges and Opportunities of High-Frequency Gravitational Wave Detection”, where members of the theoretical and experimental communities interested on high-frequency gravitational waves got together to explore the motivations and challenges towards this search. This workshop and the present white paper set the stage for the launch of the Ultra-High-Frequency Gravitational Wave (UHF-GW) initiative, whose goals include supporting the testing phase of currently existing detector proposals and stimulating the technological developments necessary to come up with new schemes for gravitational-wave detectors at high frequencies.

The remainder of this report is organized as follows: Sect. 2 introduces some basic concepts and notation to discuss different types of gravitational-wave sources and to relate them to experimental sensitivities. An overview over gravitational-wave sources in the late and early Universe is given in Sect. 3, followed by a discussion of different detector concepts in Sect. 4. We conclude in Sect. 5.

We collect here a few acronyms that will be used throughout the paper: Gravitational Wave (GW), Ultra High-Frequency Gravitational Waves (UHF-GWs), Cosmic Microwave Background (CMB), Black Hole (BH), Innermost Stable Circular Orbit (ISCO), Big Bang Nucleosynthesis (BBN).

2 Setting up the notation: comparing different GW sources and detectors

Depending on the source and detector, the strength of GWs, detector noise, and signal-to-noise ratio are described using various different metrics. In general, before using any given metric, it is important to make sure that it is appropriately defined for the scenario under consideration. In this section we summarize the relevant quantities and notation. We follow the definition in Allen and Romano (1999) for stochastic strain sources, and definitions in Moore et al. (2015) for time-dependent strain sources.

2.1 Gravitational-wave sources at high frequencies

1. For stochastic GWs, for example those coming from cosmological sources, a spectral density prescription is most suitable. The most common models assume that they are approximately isotropic, unpolarized, stationary, and have a Gaussian distribution with zero mean. They can thus be fully defined by the second moment, Eq. (1), written in terms of the Fourier transform of the time-dependent strain in a given GW polarization and solid angle, evaluated at a frequency f, and the one-sided power spectral density. The energy-density in GWs per logarithmic frequency interval is represented by the density parameter of Eq. (2), conventionally normalized by the critical energy density, with Newton’s constant and the Hubble parameter today. The power-spectral density can be directly related to the 00-component of the stress energy tensor, Eq. (3). Often, a dimensionless characteristic strain is assigned to the normalized energy density for stochastic GWs, defined in Eq. (4a) as the square root of the frequency times the power spectral density, with the corresponding density parameter given in Eq. (4b).

2. For inspiral sources, such as BH mergers, a time-dependent strain can be obtained directly from Einstein’s equations. Inspirals have an evolving frequency evolution, so usually the stationary phase approximation is used to obtain an analytical form for the Fourier transform. The characteristic strain for such sources with inspiralling frequency can be defined so as to take the frequency evolution into account in the GW strength, Eq. (5), which for a wave of amplitude h0 gives the characteristic strain of Eq. (6), proportional to the amplitude times the square root of twice the frequency squared divided by the frequency drift rate.

2.2 Detectors

Each detector has a different way of searching for GWs, with different antenna patterns, frequency bands, binning, etc. This should be taken into account when defining the appropriate noise and signal-to-noise metrics. For interferometers (such as LIGO) the impact of spatial antenna patterns is of the order of unity. For simplicity, the detector noise floor is usually specified assuming the noise is stationary and Gaussian (even though in reality it is usually neither). Similar to the discussion of stochastic GWs, this noise floor is specified by using a power spectral density, Eq. (7). The angular brackets denote an average over multiple realizations of the system, which is obtained repeating the measure of the noise over several well separated time intervals of the same length. (This assumes that the system is ergodic, hence it is possible to trade an ensemble average with a time average.) In order to measure this noise, a fast Fourier transform of the detector noise in the absence of the signal is performed. This measured noise is compared to a numerical model, comprising of the sum of all the noises in the detector. An analysis showing each individual noise source (measured or modeled) summing up to the total measured noise is called the noise budget.

Unless otherwise specified, if a detector noise is specified in terms of spectral density, it should be treated as the power spectral density if it is in inverse hertz, or as its square root if it is specified in one over root hertz. For a visual comparison of signal strengths of inspirals and stochastic signals against detector sensitivities, conventionally a dimensionless noise amplitude has been introduced, Eq. (8), the square root of the frequency times the noise power spectral density.

Some experiments looking for long-lived sources (e.g. monochromatic sources or stochastic sources) can choose to average the noise over a long time. This gives an additional boost in signal-to-noise ratio, which is sometimes reported as an enhanced sensitivity, Eq. (9), the noise power spectral density divided by the number of averages. If the detector is operating at a centre frequency and it is integrating for an observation time, then the number of averages is the observation time divided by the duration of each spectrum, assuming the segments can be coherently averaged over the observation.

2.3 Signal-to-noise ratio

Understanding whether a signal is detectable using a particular detector requires development of a metric for the signal-to-noise ratio.

• The most efficient signal-to-noise ratio metric for broadband detection of transient sources uses matched-filtering, Eq. (10). If the frequency ranges of both the signal and the detector are sufficiently broad, so that the logarithmic frequency interval is of order one, then a characteristic inspiral strain comparable to the dimensionless noise amplitude roughly corresponds to a signal-to-noise ratio of order one. This explains why those two quantities are useful in assessing the reach of a particular broadband instrument looking for an inspiralling source.
• For a resonant detector with no sensitivity outside a small bandwidth, this signal-to-noise ratio simply collapses to the single frequency band of detection, Eq. (11), indicating that a correspondingly larger threshold value of the Fourier-transformed strain is required to yield a detectable signal at fixed dimensionless noise amplitude.
• For detecting approximately monochromatic sources, the signal-to-noise ratio similarly collapses to a single frequency. In this case, the bandwidth in Eq. (11) is given by the frequency resolution, i.e., either the width of the signal or the detector resolution, whatever is the relevant limiting factor. For searches of monochromatic GWs that last over long times, various astrophysical effects like the Earth’s motion need to be taken into account.
• Detecting stochastic sources usually requires utilizing cross-correlation between two or more GW experiments to distinguish the GW background from the experiment’s noise (see also Sect. 4.4). Therefore, defining a meaningful signal-to-noise ratio for detection of stochastic sources requires careful consideration of the noise, location, and alignment of each individual experiment. The signal-to-noise ratio can be increased by using more independent experiments, observing for longer times, and optimizing the size of frequency bins. Usually, the strength of the signal will be much less than the detector noise, so cross-correlation can provide a signal-to-noise ratio greater than 1. For the simple case of several colocated detectors measuring in a frequency band for an observation time, the signal-to-noise ratio can be written as in Eq. (12).

2.4 Comparison of signal strength and noise for narrowband detectors

Since most high-frequency detectors are narrowband, here we provide some handy expressions to compare signal strength and detector sensitivity for narrowband detectors. (We define a detector to be narrowband if its bandwidth is small enough such that the data is analyzed in a single bin, i.e., the noise is assumed to be frequency independent over the bandwidth and infinite outside the bandwidth.) The most natural way to express a detector’s sensitivity is in power or amplitude spectral density. On the other hand for signal strengths, the most natural units can depend on the type of source — dimensionless characteristic strain for inspirals, amplitude or power spectral density for stochastic sources, and wave amplitude for long-lived monochromatic sources. In order to compare the signal strength and detector sensitivity, we often strive to convey them in the same units. Here we will provide two ways to achieve this for narrowband detectors. The underlying principle for both methods is to first write down a reasonable signal-to-noise ratio metric, and use that to derive the appropriate comparable quantity. The signal-to-noise ratio for stochastic and long-lived monochromatic sources will be enhanced due to the integration over the observation time, in contrast with signal-to-noise ratio for transient sources, which will depend on just a single observation.

If we are interested in assessing the utility of a given detector to search for GWs from various types of sources, it would be natural to include the integration-time information in the signal depending on the source. On the other hand, if we wish to compare various detectors’ suitability to a given source, it is convenient to include the time and bandwidth information to convert the detector sensitivity to the source units. Here we provide ways to do both, for inspirals in Eqs. (13) and (14), for stochastic sources in Eqs. (15) and (16), and for monochromatic sources in Eqs. (17) and (18). Note that for monochromatic sources, there is no need to invent a characteristic strain, the most characteristic strain is the amplitude itself.

(The displayed equations (1) to (18) are described rather than reproduced here; the complete expressions are at the source.)

3 Sources

This section reviews various production mechanisms for GW signals in the high-frequency regime, typically in the range kHz to GHz, that fall into two broad classes. In Sect. 3.2 we discuss sources in our cosmological neighbourhood, which emit coherent transient and/or monochromatic GW signals. In Sect. 3.3 we turn to sources at cosmological distances which typically lead to a stochastic background of GWs. We emphasize that all proposed sources, with the notable exceptions of the neutron star mergers discussed in Sect. 3.2.1 (kHz range) and the cosmic gravitational microwave background discussed in Sect. 3.3.3, require new physics beyond the Standard Model of particle physics to produce an observable GW signal. Thus, while being admittedly somewhat speculative, these proposals provide unique opportunities to shed light on the fundamental laws of nature, even by only setting an upper bound on the existence of GWs in the corresponding frequency range.

3.1 Overview

Figures 1 and 2 summarize a representative selection of the sources which are discussed in more detail in the following subsections. The regions bounded by the colored curves illustrate the region of parameter space which may be covered by the corresponding source for appropriate parameter choices as specified below. Except for the cases of inflation with broken spatial reparametrization symmetry and the cosmic gravitational microwave background they should not be mistaken for GW spectra obtained for a fixed model parameter choice.

In the same figures, we also indicate the demonstrated (filled boxes) or expected (empty boxes) sensitivity of the detector concepts discussed in Sect. 4.1. In some cases we report two sensitivities for a single detector, using two different intensities of the same color (see for instance the case of levitated sensors), if the sensitivity depends on the details of the future implementation of the detector or on some assumptions needed to place the constraint. In the case of the levitated sensors the two colors refer to two different versions of the same detector concept: a 1 meter and a 100 meter implementation, see Sect. 4.2.1, the latter giving a better sensitivity, and in the case of the radiotelescopes EDGES and ARCADE, the two sensitivities refer to the weakest and strongest possible cosmic magnetic field, whose value is needed in order to place the constraint, see Sect. 4.2.2.

The comparison of signal strength and detector sensitivity in these figures should be taken with great caution, and serves as a very rough illustration only. In particular, the signals in Fig. 1 are coherent (and partially transient) signals whereas the signals depicted in Fig. 2 are stationary isotropic stochastic signals. A given detector concept will be more or less suitable for these different types of signals, which is not accounted for in this illustration. Further restriction may apply. For example, the quoted sensitivity for the radio telescopes ARCADE and EDGES assumes a cosmological distance between source and observer. Regarding the possible signals, we have aimed to make realistic estimates of the largest possible signals in different models. This does however not factor in the likelihood of such a signal occurring in the detector lifespan. This is in particular true for the coherent sources, see the respective subsections for details.

Figure 1 shows representative examples of coherent sources: the ringdown signal of neutron star mergers, mergers of compact objects such as primordial black holes and exotic compact objects, and signals from axion superradiance in both the axion annihilation and the axion decay channel. Figure 2 shows models producing a stochastic GW signal: interestingly, most of them are concentrated in the UHF band. These are produced in the early Universe and are thus subject to the cosmological constraint on the number of effective degrees of freedom during BBN and at CMB decoupling, see Sect. 3.3. They include inflation, preheating and oscillons, the cosmic gravitational microwave background, phase transitions, and topological defects such as cosmic strings and gauge textures.

(The parameter choices behind each curve in Figs. 1 and 2 are omitted for length; they are given in full at the source.)

3.2 Late Universe

In this section we revise various sources that are relevant for high-frequency GW production and are active in the late Universe.

3.2.1 Neutron star mergers

For not too high binary masses the merger of two neutron stars avoids the prompt collapse to a BH and leads to the formation of a massive rapidly rotating and oscillating neutron star remnant. The oscillations of this remnant are very characteristic of the incompletely known equation of state of high-density matter and generate GW emission in the kHz range. For instance, the dominant oscillation frequency of the post-merger phase scales tightly with the radii of non-rotating neutron stars. These radii are uniquely determined by the equation of state of neutron stars, and are therefore particularly valuable messengers of the underlying high-density matter physics. Simulation results show a tight correlation between the dominant GW frequency and neutron star radii; a fit to the data for fixed binary masses describes the relation with a maximum residual of only a few hundred meters, allowing for accurate radius measurements.

Subdominant features in the GW spectrum contain additional information about the equation of state and may also reveal the dynamics of the remnant, which is indispensable for a complete multi-messenger interpretation of neutron star mergers. The presence or absence of post-merger GW emission from a neutron star remnant on its own informs about the outcome of the merger (neutron star or BH). In combination with the measured binary masses, this information allows to constrain the threshold binary mass for prompt BH collapse, which is somewhere in the range 2.9 to 3.8 solar masses, depending on the equation of state. This threshold depends sensitively on the maximum mass of non-rotating neutron stars. Obtaining the threshold mass for prompt BH formation through post-merger GW emission will yield a robust determination of that unknown maximum mass, which is another important equation of state property that probes the very high-density regime. A robust measurement of it is also relevant for stellar astrophysics since it, for instance, affects the outcome of core-collapse supernovae. Pulsar observations only yield accurate lower bounds on it.

Generally, equation of state inference from the post-merger stage is complementary to other constraints, e.g., from the inspiral phase. The complementarity concerns the probed density regime, which is generally higher in the post-merger phase, and methodological aspects. Hence, the detection of post-merger GW emission is of highest importance to understand properties of high-density matter including the opportunity to probe the presence of a phase transition to deconfined quark matter.

The different features of the post-merger GW emission have frequencies in the range 1 to 5 kHz, with the dominant peak between 2 and 4 kHz. Simulated injections show that at a distance of 40 Mpc (comparable to that of GW170817) a strain sensitivity of roughly 3 times 10 to the minus 24 per root hertz is required for a detection of the main features. Hence, measurements can be anticipated with a small sensitivity improvement either of Advanced LIGO, Virgo and KAGRA or with a dedicated high-frequency instrument like NEMO (see Sect. 4.1.1).

(Sections 3.2.2 on mergers of light primordial black holes, 3.2.3 on exotic compact objects and 3.2.4 on black-hole superradiance are omitted for length; the complete text is at the source.)

3.3 Early Universe

We now turn to cosmological sources emitting GWs at cosmological distances, i.e., in the early Universe. In this case, the source is associated to an event in our cosmological history, triggered e.g., by the decreasing temperature of the thermal bath, and typically occurs everywhere in the Universe at approximately the same time. This results in a stochastic background of GWs which is a superposition of GWs with different wave vectors. The total energy density of a GW background, with characteristic wavelengths well inside the horizon, decays as relativistic degrees of freedom with the expansion of the Universe, that is as the fourth inverse power of the scale factor. This implies that a GW background acts as an additional radiation field contributing to the background expansion rate of the Universe. Observables that can probe the background evolution of the Universe at some particular moment of its history can therefore be used to constrain that energy density at such moments. In particular, two events in cosmic history yield a precise measurement of the expansion rate of the Universe: BBN and photon decoupling of the CMB. An upper bound on the total energy density of a GW background present at the time of BBN and CMB decoupling can be therefore derived from the constraint on the amount of radiation tolerated at those cosmic epochs, when the Universe had a temperature of about 0.1 MeV and about 0.3 eV, respectively.

A constraint on the presence of extra radiation is usually expressed in terms of an effective number of neutrino species after electron-positron annihilation. Since the energy density in GWs must not exceed the allowed extra radiation, one obtains a constraint on the redshifted GW energy density today in terms of the number of extra neutrino species, Eq. (36). We recall that the above bound applies only to the total GW energy density, integrated over wavelengths way inside the Hubble radius (for super-horizon wavelengths, tensor modes do not propagate as a wave, and hence they do not affect the expansion rate of the Universe). Except for GW spectra with a very narrow peak, the bound can be interpreted as a bound on the amplitude of a GW spectrum over a wide frequency range. The bound obviously applies only to GW backgrounds that are present before the physical mechanism (BBN or CMB decoupling) considered to infer the constraint takes place.

Constraints can be placed by BBN alone, and/or in combination with CMB data, giving Eq. (37) for a stochastic GW background produced before BBN, with wavelengths inside the Hubble radius at the onset of BBN, corresponding to present-day frequencies above about 1.5 times 10 to the minus 12 hertz. A similar bound can also be obtained from constraints on the Hubble rate at CMB decoupling. This translates into an upper bound on the amount of GWs, which extends to a greater frequency range than the BBN bound. Since high-frequency GWs carry a lot of energy, these bounds pose severe constraints on possible cosmological sources of high-frequency GWs.

(Sections 3.3.1 on inflation, 3.3.2 on preheating and oscillons, 3.3.3 on the cosmic gravitational microwave background, 3.3.4 on phase transitions, 3.3.5 on topological defects, 3.3.6 on evaporating primordial black holes and 3.4 on miscellaneous sources are omitted for length; the complete text is at the source.)

4 Detection of gravitational waves at high frequencies

After the first detection of GWs at frequencies in the range 0.1 to 2.0 kHz, the expansion into other frequency bands is a natural next step — as it was in the 1950s when radio, X-ray and UV observations became possible with new technology. As detailed in the previous section, many exciting questions in astronomy, cosmology and fundamental physics are tied to GW signals with frequencies far above the capabilities of current detectors or their upgrades. Figures 1 and 2 give an impression of the range of GW amplitudes expected for various coherent and stochastic sources. Even GW upper limits with no known source targets at the time of publication of this white paper may be valuable in restricting physical theories.

In this section, we will investigate the experimental possibilities for the detection of high-frequency GWs. First, we will give an overview of current GW detectors and their limitations, followed by the introduction of several concepts for the detection of high-frequency signals. Depending on the detector concept and the targeted sources, we quote detector sensitivities in terms of strain amplitude spectral density or in terms of the dimensionless quantities defined for stochastic and for monochromatic signals, see Sect. 2 for details. Careful consideration of operation and bandwidth is needed to convert between these quantities.

4.1 Laser interferometers and resonant mass detectors and their limitations

The first GWs were detected by the Advanced LIGO detectors in the US and the Advanced Virgo detector in Italy. In early 2020, the Japanese KAGRA detector joined LIGO’s third observing run. These detectors are all based on the principle of a Michelson interferometer, using large suspended mirrors with several kilometers distance between them. Several other detectors are in the design phase. These detectors typically have their peak sensitivity at frequencies of a few hundred Hz.

However, some future detectors are designed to particularly expand the detection band towards either low or high frequencies. To expand the detection band of Earth-bound interferometers to frequencies below 10 Hz, cryogenically cooled mirrors, large beam diameters and operation underground are considered. LISA, also based on laser interferometry, is a planned, satellite-based detector to increase the arm length beyond the possibilities on Earth and to reduce environmental noise sources such as seismic noise. LISA will have its peak-sensitivity in the mHz range. To increase interferometer sensitivity towards higher frequencies, options are an increase of laser power and/or resonant operation. The planned Australian NEMO detector will be targeting frequencies of up to several kHz, see Sect. 4.1.1 below. (We note that through the GW memory effect, these interferometers are sensitive to high-frequency GW bursts far beyond their nominal frequency band.)

While increasing the arm-length of an interferometer increases strain signal in some frequency band, longer arms are only really beneficial as long as the GW wavelength is longer than the interferometer arms. For significantly higher frequencies (MHz) interferometers with arm-lengths of meters are more suitable, but are of course at the same time limited by the smaller strain sensitivity achievable with shorter arms. This constitutes the main limitation of laser interferometers, used as direct strain meters, towards higher GW signal frequencies.

A concept to detect GWs which existed prior to the interferometers are resonant bar detectors, initially proposed and built by Joseph Weber in the 1960s. Their modern successors, resonant spheres, have peak sensitivities at several kHz. In Sect. 4.1.3, we will give a summary of these resonant spheres.

4.1.1 Laser interferometers: Neutron Star Extreme Matter Observatory (NEMO)

The first detection of a binary neutron star merger in 2017 has increased the interest in the development of GW detectors with sensitivity in the few kHz regime which will be capable of detecting the merger and ringdown part of the waveform. It is expected that such detectors will need to have strain sensitivities approaching 10 to the minus 24 per root hertz in the range 1 to 4 kHz for events that are likely to occur a few times per year. This sensitivity should be achieved by the third generation terrestrial GW detectors that are anticipated to come online in the later half of the 2030s. The Australian GW community is currently exploring the feasibility of a new detector, NEMO, dedicated to detecting this merger phase and the following ringdown as well as testing third generation technology on a smaller scale. The planned sensitivity of this detector would reach 10 to the minus 24 per root hertz in the range 1 to 2.5 kHz. This detector will work in collaboration with the existing second generation GW detector network that will provide sky localization for electromagnetic follow-up.

The dominant high-frequency noise source for interferometric GW detectors is quantum phase noise or shot noise as it is otherwise called. The magnitude of this noise source is inversely proportional to the square of the product of the circulating power incident on the test masses and the length of the arms of the detector. This generally necessitates extremely high powers in the arms of the interferometers (about 5 MW in the case of NEMO). Such high circulating powers lead to technical issues such as parametric and tilt instabilities and thermal induced distortions. These issues can be challenging to deal with, however a dedicated high-frequency detector promises to make this easier. This is because low-frequency sensitivity limits the actuation that can be applied to the test masses to correct instabilities and distortions. Further, relaxing the low-frequency sensitivity relaxes the requirements on seismic isolation and test mass suspension systems that can significantly reduce the cost.

4.1.2 Interferometers up to 100 MHz

As was first pointed out by Mizuno (1995), in laser interferometers the overall stored energy in the form of circulating laser power sets a limit on the achievable sensitivity and bandwidth, which is a consequence of the quantum Cramér-Rao bound. For a given laser power, higher bandwidth needs to be traded in for an increase in sensitivity. While opto-mechanical resonances can be introduced in the signal response of interferometers to shape the sensitivity curve for specific frequencies, it appears unlikely that the stored laser power can be further increased by several orders of magnitude. Therefore, broadband interferometric detectors reaching into the MHz detection range (while maintaining LIGO or Virgo-level strain sensitivity) seem not to be a viable option when taking also the arm-length argument from above into account.

Nevertheless there are three notable efforts (two existing and one under construction) of laser interferometers in the MHz range, which currently set the best experimental upper limits on GWs in their respective frequency bands.

One option is to build kHz-bandwidth interferometric detectors that are centered around much higher frequencies. Akutsu et al. (2008) have published upper limits from such a system working at 100 MHz. The detector used a synchronous recycling architecture based on a resonant recycling cavity of dimension 75 cm and a Nd:YAG laser with a power output of 0.5 W. The limit on stochastic GW signals was reported to be about 10 to the minus 16 per root hertz, placing a bound on the characteristic stochastic strain of about 7 times 10 to the minus 14. A study of the potential of this technique showed that a sensitivity of 10 to the minus 20 per root hertz is possible at 100 MHz with a bandwidth of 2 kHz, but the sensitivity decreases with increasing frequency and is not competitive above 1 GHz.

The sensitivity of a single instrument can be surpassed by correlating two co-located instruments in the case of searching for stochastic signals from GWs or other sources. The Holometer experiment at Fermilab consists of two co-located power recycled Michelson interferometers with 40-meter long arms. While their primary research target has been signatures of quantization of spacetime, they reach a sensitivity of 10 to the minus 21 per root hertz approximately in the band 1 to 13 MHz. Using a 704-hour dataset from the Holometer experiment, the authors of Martinez and Kamai (2020) concluded that there are no identifiable harmonic sources such as cosmic string loops and eccentric BH binaries emitting in the frequency range 1 to 25 MHz.

The experimental GW group at Cardiff University is planning a set of two wide-band table-top interferometers sensitive in the band 1 to 100 MHz. These will be able to set new upper limits on a stochastic GW background in this frequency band.

4.1.3 Spherical resonant masses

The principle of a resonant mass detector is that its vibrational eigenmodes can get excited by a GW. These mechanical oscillations are transformed into electrical signals, using electromechanical transducers, and amplified by electrical amplifiers. These resonant detectors have a relatively small bandwidth, usually of less than 100 Hz. Thermal noise, Johnson–Nyquist noise, pump phase noise (if the transducer is parametric), back-action noise, and amplifier noise are the internal noises of this kind of detector. Therefore, the resonant mass antenna and transducers are made of high-quality factor materials in order to decrease thermal (mechanical) and Johnson–Nyquist (electrical) noises.

The idea of a spherical resonant mass antenna for GW detection has a long history and was first proposed by Forward (1971) followed by several decades of exploration and proposals. In 1991, Aguiar proposed a large spherical antenna project in Brazil. This detector, Mario Schenberg, in São Paulo, Brazil, was started to be built in 2000, around the same time as Mini-GRAIL, in Leiden, Netherlands. These two spherical detectors were active for about 15 years. At present, they are decommissioned, but Schenberg is planned to be reassembled at INPE, in São José dos Campos, about 100 km from its initial site at the University of São Paulo. Such detectors have a bandwidth of 50 to 100 Hz with peak frequencies around 3 kHz for the quadrupole modes. To increase the frequency range, a xylophone configuration of several spheres has been proposed.

Spherical antennas provide more information, compared to the classical bar antennas, because of their quadrupole modes, while also being significantly more sensitive due to their favorable geometry of having a larger cross-section at identical mass. From the output of six transducers tuned to the quadrupole modes of the sphere, a single sphere can obtain complete information about the polarization and direction of the incoming wave.

In 2004, Mini-GRAIL reached a peak strain sensitivity of about 1.5 times 10 to the minus 20 per root hertz at a frequency of 2942.9 Hz at temperatures of 5 K. Over a bandwidth of 30 Hz, the strain sensitivity was about 5 times 10 to the minus 20 per root hertz. Schenberg, operating also at 5 K, reached strain sensitivities of about 1.1 times 10 to the minus 19 per root hertz for its quadrupolar modes (around 3.2 kHz) and about 1.2 times 10 to the minus 20 per root hertz for its monopolar mode (around 6.5 kHz), in 2015. Both antennas could reach sensitivities around 10 to the minus 22 per root hertz when operating at 15 mK. Schenberg, because it uses parametric transducers, can reach higher sensitivities if it implements squeezing of the signal. In this case, it would have similar sensitivities as the ultimate sensitivities of Advanced LIGO and Virgo around 3.2 kHz.

The conceptual difficulties in pushing this technology to higher frequencies are similar to the issues discussed for laser interferometers. Searching for GWs at higher frequencies requires smaller resonating spheres and consequently requires measuring smaller absolute displacements to achieve the same strain sensitivity. Note also that contrary to laser interferometers, resonant mass detectors have not yet reached the standard quantum limit yet. It thus seems unlikely that this technology can be pushed significantly beyond the kHz region.

4.2 Detection at frequencies beyond current detectors

In this section we will introduce several ideas and concepts for the detection of GWs at high frequencies beyond the capabilities of currently existing GW detectors.

4.2.1 Optically levitated sensors

Optically levitated dielectric sensors have been identified as a promising technique for resonant GW searches spanning a wide frequency band from a few kHz to about 300 kHz. A dielectric nano-particle suspended appropriately at the anti-node of a laser standing wave within an optical cavity will experience a force when a passing GW causes a time-varying strain of the physical length of the cavity. The particle will be displaced from the location of the trapping light anti-node, resulting in a kick on the particle at the frequency of the GW space-time disturbance. The trapping frequency and mechanical resonance linewidth are widely tunable based on the laser intensity and laser cooling parameters chosen.

When detecting the resulting displacement of the particle at the trapping resonance frequency, the sensitivity is limited by Brownian thermal noise in the particle itself rather than the displacement detection of the particle. This results in improved sensitivity at higher frequency (unlike traditional interferometer style detectors which decrease sensitivity at high frequency due to laser shot noise). The low-friction environment made possible by optical levitation in ultra-high vacuum enables extremely sensitive force detection, which becomes ultimately quantum-limited by photon-recoil heating from discrete scattering events of individual trap laser photons.

A 1-meter prototype Michelson-interferometer configuration detector called the Levitated Sensor Detector is under construction at Northwestern University in the US, with a target sensitivity of better than about 10 to the minus 19 per root hertz at 10 kHz and about 10 to the minus 21 per root hertz at 100 kHz. With Barker and coworkers at partner institution University College London, fiber-based approaches are being investigated to permit longer cavities without the need for expensive optics. The ultimate strain sensitivity of a 10-meter room-temperature instrument is estimated to be better than approximately 10 to the minus 20 per root hertz at 10 kHz and 10 to the minus 22 per root hertz at 100 kHz. For a cryogenic 100-meter instrument this can be improved by more than an order of magnitude across much of the frequency band. A detailed analysis of the search reach for GWs produced by axions via the BH superradiance process is provided in Aggarwal et al. (2020).

4.2.2 Inverse Gertsenshtein effect

The Gertsenshtein effect describes the conversion of photons to GWs in the presence of a magnetic field and was considered already decades ago as a source of GWs (Gertsenshtein 1962). While the coupling constant for this process is too small to be of interest for experiments in the near to medium future, the inverse Gertsenshtein effect, frequently referred to as magnetic conversion, can indeed be used to search for GWs. While such dedicated instruments do not exist yet (apart from small prototypes), a first step in this direction has been done by using existing data from axion-search experiments. In these experiments, typically a strong static magnetic field of several Tesla is set up with field lines perpendicular to some interaction region, through which a beam line passes. In their nominal usage these experiments would search for axion-like particles, which can convert to photons in the presence of the magnetic field. These photons would be detected at the end of the beam line by electromagnetic detectors within the frequency band of interest (e.g. photodetectors for optical radiation). The very same experimental arrangements can also be used to search for GWs, by re-interpreting the acquired data, as has been pointed out and performed by the work in Ejlli et al. (2019). This work could set first upper limits for GWs at optical and X-ray frequencies (i.e., around 500 THz and 10 to the sixth THz respectively).

In the future this class of experiment is expected to continue with larger detectors of this sort being constructed. Given the motivation in the context of searches for high-frequency GWs, a dual usage of these detectors could be imagined with dedicated instruments and operational modes to search for GWs. For example, the planned IAXO detector aims at searching for axions produced in the core of the sun. If it were fitted with different electromagnetic receivers, from radio to optical frequencies, GW searches could be facilitated in these bands. A particular advantage of searches with IAXO would be that the device can be pointed (within some limits) to different points in the sky. It could be of particular interest to point to patches in the sky where, for example, a binary BH merger is predicted to happen. The latter is a real prospect once the LISA space interferometer will be operational, which can detect inspiralling BHs long before their merger.

Note that, in principle, the inverse Gertsenshtein effect might be exploited at all frequencies, and that one further advantage of this concept is the tunability of the solid angle, which changes according to the direction of the magnetic field, see Sect. 4.4.5. In particular, the inverse Gertsenshtein effect has substantial room for development especially at GHz frequencies where many of the early Universe signals converge. Using magnets developed for particle accelerators a conversion path length of 100 metres with uniform magnetic field of 5.6 T is quite realistic, and being implemented for the ALPS experiment searching for axion-like particles. Electromagnetic detectors having a thermal noise equivalent to 0.1 Kelvin would lead to a sensitivity around a dimensionless noise amplitude of 10 to the minus 26. This sensitivity could be further enhanced by the inclusion of a Fabry Perot cavity in the conversion volume and a factor of 100 improvement might be possible. Unfortunately, this improved sensitivity would be gained at the expense of the wide bandwidth of the technique and this would limit the applicability to stochastic signals. Ringwald et al. (2021) estimates the sensitivity that can be reached in the GHz region by using Single Photon Detectors (SPD) and Heterodyne radio receivers (HET). The corresponding limits are reported in Figs. 1 and 2.

The conversion of GWs to photons by the inverse Gertsenshtein effect cannot only be exploited in a laboratory setting but also by considering astrophysical or even cosmological detectors. In this case, the magnetic field is weaker and the background is much harder to control, but cosmic magnetic fields can extend coherently over kpc or even Mpc, implying an enormous detector volume. The frequency range of MHz to GHz coincides with the Rayleigh-Jeans tail of the cosmic microwave background, a target of existing and upcoming radio telescopes. For example, the data of ARCADE 2 and EDGES can be recast to respectively constrain the characteristic stochastic strain below 10 to the minus 24 (or 10 to the minus 14) in the range 3 GHz to about 30 GHz, and below 10 to the minus 12 (or 10 to the minus 21) at about 78 GHz, for the strongest (weakest) cosmic magnetic fields in accordance with current astrophysical data.

4.2.3 GW to electromagnetic-wave conversion in a static electric field

Lupanov (1966) considered the inverse Gertsenshtein effect but using a static electric field rather than a static magnetic field. The physics is essentially the same in the two cases but since the intensity of electric fields in laboratory settings is limited by the tendency to pull electrons from nearby conductors (or dielectrics) and thereby cause local short circuits, the available energy densities in electric fields are about one millionth of that created by magnetic fields in the several Tesla range. Hence the use of electric fields seems not to offer any advantages.

4.2.4 Resonant polarisation rotation

Cruise (1983) showed that a GW could induce a rotation of the plane of polarisation in electromagnetic waves in certain geometries, some of which might be relevant astronomically. In 2000 the idea of resonant polarization rotation was extended to a situation in which the electromagnetic wave was a circulating wave in a microwave waveguide ring. The original effect was amplified by the (potentially significant) quality factor of the waveguide ring. A proof of concept apparatus was constructed by Cruise and Ingley (2005, 2006). Such a device would be narrowband with a sensitivity of about 10 to the minus 14 per root hertz at frequencies of 100 MHz. It is difficult to see the sensitivity of this scheme for GW detection increasing very far beyond the published value.

4.2.5 Heterodyne enhancement of magnetic conversion

Li and Yang (2004), Li et al. (2006), Baker et al. (2008) and Li et al. (2009) have suggested enhancing the conversion efficiency of magnetic conversion detectors such as those discussed in Sect. 4.2.2. This proposal has been specifically aimed at the detection of cosmological (relic) signals of a stochastic nature and with dimensionless amplitudes in the range 10 to the minus 30 up to 10 to the minus 26 at 5 GHz, about the highest signal consistent with the BBN limit. The conversion from GW to electromagnetic wave is enhanced by seeding the conversion volume with a locally generated electromagnetic wave at the same frequency as that being searched for. In conditions in which the gaussian local oscillator beam is parallel to the incoming signal and at right angles to the static magnetic field, an additional beam of electromagnetic waves is generated by the conversion process, travelling at right angles to the incoming beam and the locally generated beam. The technical challenge is then to distinguish this perpendicular beam of, say, 800 photons per second from the locally generated beam at the same frequency and carrying 10 to the power 24 photons per second at frequencies of several GHz. This demands a geometric purity in the Gaussian beam of better than 10 to the minus 21, far beyond the current state of the art. The authors have proposed this interesting idea over many years but the lack of laboratory results on the performance of the necessary subsystems leaves the feasibility of this concept an open question.

4.2.6 Bulk acoustic-wave devices

Bulk acoustic-wave devices are one of the pillars of frequency control and frequency metrology. In its simplest form, a piece of piezoelectric material is sandwiched between two electrodes, converting the acoustic waves inside the material into electrical signals. With its relatively compact size and robustness, this technology gives one of the best levels of frequency stability near one second of integration time. More recently, it was demonstrated that quartz bulk acoustic-wave devices exhibit extremely high-quality factors (up to 8 times 10 to the ninth) at cryogenic temperatures for various overtones of the longitudinal mode covering the frequency range 5 to 700 MHz. For this reason it was proposed to use the technology for various tests of fundamental physics such as Lorentz invariance tests, quantum gravity research and search for high-frequency GWs.

For the latter purpose, a bulk acoustic-wave device represents a resonant mass detector whose vibration could be read through the piezoelectric effect and Superconducting Quantum Interference Devices (SQUIDs). The approach has the following advantages: highest quality factor (high-sensitivity), internal (piezoelectric) coupling to SQUIDs, allows parametric detection methods, large number of sensitive modes (more than 100) in a single device, modes scattered over wide frequency range (1 to 700 MHz), well-established and relatively inexpensive technology (mass production), high-precision (insensitive to external influences such as seismic vibration and temperature fluctuations). On the other hand, it is shown that at low temperatures identical devices demonstrate significant dispersion in mode frequencies, thus, showing low accuracy. The level of sensitivity of bulk acoustic-wave detectors is estimated at the level of about 2 times 10 to the minus 22 per root hertz subject to the mode geometry. With additional investment into research and development, this level can be improved and the frequency range extended down to hundreds of kHz range.

A search for high-frequency GWs with a single bulk acoustic-wave device and two modes at 4 K has been running in the University of Western Australia since November 2018.

4.2.7 Superconducting rings

The quantum properties of vortices in superfluids may interact with spin components of the GWs. In addition, an extension of the electromagnetic impedance at a boundary in the case of GWs in a superconducting fluid suggests that the impedance mismatch well-known in classical bar detector theory is much reduced for a GW arriving at a boundary in a superconductor, essentially creating a very efficient mirror that could be used as a building block for an interferometer or Sagnac ring. Anandan and Chiao (1982) and Chiao (2002) proposed a new detector format which utilises these putative principles, reaching a sensitivity of about 10 to the minus 31 in monochromatic strain amplitude. Resonant operation will restrict the bandwidth in the GHz range. A good review of the issues surrounding the interaction of mesoscopic quantum systems with gravity was prepared on an European Space Agency contract by Kiefer and Weber (2005). This review casts doubt on some of the assumptions made by Anandan and Chiao.

4.2.8 GW deformation of microwave cavities

Caves (1979) published a theoretical study of a microwave cavity with a high mechanical quality factor. Mechanical deformation of the cavity by a GW coupled two of the cavity’s resonant microwave modes and transferred the electromagnetic excitation to a previously unexcited cavity mode. Reece et al. (1984) built a similar system with a higher resonant frequency of 1 MHz and one operating at 10 GHz, while Pegoraro et al. (1978) designed a system with a sharp resonance at about 1 GHz. These schemes certainly offer some sensitivity in the frequency range above 1 GHz but that is limited to around a strain of 10 to the minus 21 by the thermal noise in the microwave sensors. Even at cryogenic temperatures the sensitivity will be many orders of magnitude away from the level required for detecting cosmological sources. The bandwidth of this scheme is nominally limited by the bandwidth of the cavity, up to effects like detection or electronics gains and noises.

4.2.9 Graviton-magnon resonance

As pointed out in Ito et al. (2020), a GW passing through a ferromagnetic insulator can resonantly excite magnons (collective excitations of electron spins), similar to the excitations of phonons in resonant bar detectors. The readout is achieved by placing the magnetic sample inside a microwave cavity, coupling the magnon to a photon mode. This idea builds on the technique of ferromagnetic haloscopes proposed for axion searches. The sensitivity of such detector reaches a strain of about 7.6 times 10 to the minus 22 per root hertz at 14 GHz and about 1.2 times 10 to the minus 20 per root hertz at 8.2 GHz. The sensitivity of this approach can be greatly improved by incorporating single frequency counters that are already available. A few orders of magnitude in sensitivity have been shown for axion detection.

4.3 Summary of detector sensitivities

In Table 1 we summarize the existing and proposed technologies for high-frequency GW detection, reporting the corresponding sensitivities. For all experiments that quote their sensitivity in terms of a power spectral density noise, a conversion to dimensionless stochastic strain has been performed using Eq. (16b) with 1 year as the observation time and the specified detector bandwidth. For the other detectors, we specify the dimensionless strain variable on a case by case basis: we use the stochastic form for detectors that look for a stochastic signal, while we use the monochromatic form for detectors that look for a monochromatic GW. In the case of microwave cavities (see Sect. 4.2.8) we denote the sensitivity simply by h as it can refer either to bursts or to long duration signals, and we refer the reader to the original papers for the details, while we report the best sensitivity estimates. We also specify whether each experiment has already been built, is under construction, is only devised or only the physical mechanism has been identified (theory). The sensitivity values labeled by an asterisk refer to the planned sensitivities that will be achieved by the proposed future improvements of the currently built setups.

Note that a square bracket in the frequency column refers to the bandwidth of the detector, while a round bracket refers to the range of frequencies that can be covered by the detector itself. We report the bandwidths used in Table 1 to obtain dimensionless strain from power spectral density: Mini-GRAIL had a bandwidth of about 30 Hz; the Schenberg antenna had a bandwidth of about 50 Hz; the 0.75-m interferometer has a bandwidth of about 2 kHz; the optically levitated sensors have a bandwidth of one tenth of the frequency; about 10 to 50 kHz for Cruise’s and Ingley’s detector; for enhanced magnetic conversion the bandwidth is about 1 Hz; bulk acoustic wave resonators have a bandwidth of the frequency divided by 10 to the eighth, where the frequency is in the ranges reported in Table 1; and the bandwidth for Pegoraro’s detector is about 1 Hz. For references that specify their detector sensitivity in dimensionless strain without specifying the exact form of the dimensionless strain, the sensitivities are labeled as just h.

(Table 1 itself, listing every concept with its operational frequency and proposed sensitivity, is omitted here because the extracted table cannot be rendered faithfully; it is at the source.)

4.4 Cross-correlation detectors

For a coalescing binary the information about the waveform of GWs is available. The best approach to the detection problem is to use this information by projecting the observed data over the set of expected signals. Usually this set can be parameterized by a small number of parameters, varying in some allowed range. A scalar product between data and the set of templates is evaluated, using its maximum as a detection statistic. It should be noted that the set of expected signals does not have a linear space structure, in the sense that the linear combination of two possible signals is not generally speaking a possible signal. For this reason the computational cost of a search over a template bank grows very fast with the number of free parameters.

When the number of parameters is large, or when a parameterization of the waveform of the expected signal is not possible at all, other detection methods must be used. In the present context this is the case for several cosmological processes, which are expected to produce a GW signal that can be described as an overlap of a very large number of contributions. There is not a waveform here, the expected signal is a stochastic process and the best approach for the detection is the cross correlation one.

4.4.1 Relic gravitational radiation

The detection of the CMB in 1965 by Penzias and Wilson gave the first experimental insight into the properties of relic radiation, the remnants currently observable of the Big Bang. The CMB is a stationary, stochastic radiation field, basically isotropic down to levels as low as 10 to the minus 5, with a Gaussian distribution.

It is natural to suppose that a useful starting point in planning GW observations of a relic radiation would be to assume that cosmologically-sourced GWs can be modeled by a stochastic field with relatively simple properties. In the spacetime volume of a given experiment, we can describe our background as a superposition of plane waves, Eq. (52). Here the polarization degrees of freedom of the field are labelled by an index whose number can depend on the considered theory of gravitation. The parameters of the signal are the amplitudes introduced in Sect. 3, one for each mode of the field. When these must be considered stochastic variables, the relic gravitational radiation field is completely described by their joint probability distribution, and is called GW stochastic background.

For a Gaussian stochastic background this joint probability distribution is Gaussian, and is completely determined by the second order expectation value, Eq. (53). Further assumptions lead to the simplification of this general expression. For example, stationarity in a given reference frame requires the two-point expectation value to be a function of the time difference only, and as a consequence the correlation between two amplitudes can be non-zero only when their frequencies coincide. Stationarity in every reference frame implies homogeneity, and only correlations with opposite wave vectors are allowed. Each stochastic model can have its peculiar signature: in the simplest stationary, isotropic and Gaussian model a parameterization of the model can be given in terms of an array of functions, Eq. (54), which is a generalization of Eq. (1) and allows for a non trivial polarization structure.

If the radiation is stationary, a temporal signature which could be exploited by a single-instrument detection procedure is not available. Moreover, isotropy and homogeneity do not allow for a signal modulation which could be obtained in principle by changing the orientation or the position of the detector. It could still be possible to detect the stochastic background as an excess noise in the apparatus. However in order to do that the amplitude of the signal must be large enough to make it evident given a theoretical estimate of the noise budget, which is always uncertain. This means that the strategy for detection will necessarily be different from the strategy for discrete source detection.

The most obvious approach is the use of spatial correlations. If a detector is to be developed for the detection of GW relic radiation then a decision to operate it as a correlation detector will have a far reaching influence on many aspects of its design. An excellent review of GW relic radiation and appropriate methods of detection, in the context of the high-frequency band, has been published by Allen (1997) and important properties of correlation detectors have been explored by Michelson (1987). The basic principle is to compare the signal from two detectors. This is comparing a random signal with another stationary, stochastic, isotropic, gaussian signal from the same source. Similar to template matching as a means of detecting discrete sources, in this case the template itself is random and that affects the statistical gain from performing a cross correlation between two detectors.

(Sections 4.4.2 on the properties of correlation detectors, 4.4.3 on the overlap function, 4.4.4 on exploiting the overlap function at very high frequencies, 4.4.5 on signal switching and 4.4.6 on data acquisition and long term storage are omitted for length; the complete text is at the source.)

5 Discussion and conclusions

The search for high-frequency gravitational waves is a promising and challenging search for new physics. It provides a unique opportunity to test many theories beyond the Standard Model that could not be tested otherwise.

Various models proposed to address open questions in particle physics and cosmology predict a gravitational-wave signal in the frequency range from about 10 to the third to 10 to the tenth hertz, which could be a coherent signal — e.g., from mergers of compact objects or from axion superradiance around black holes — or stochastic — e.g., originating from certain models of cosmic inflation, from a phase transition in the very early Universe or from oscillons, evaporating primordial black holes, etc. Many of these models can lead to relatively large signals, corresponding to an order-one fraction of energy density in the early Universe converted to gravitational waves. This energy is red-shifted in the expanding Universe, rendering even these strong signals challenging to detect today. Moreover, in many cases the amplitude of the signal depends sensitively on the model parameters and may be significantly lower in large parts of the model parameter space.

The high-frequency band comes with particular challenges and opportunities. High-frequency gravitational waves carry a high-energy density, implying that bounds provided by cosmology on the fraction of energy contained in gravitational waves translate to stringent bounds on the characteristic gravitational-wave strain. This poses a severe challenge for detection, since the magnitude of any observable effects is typically governed by the strain and not by the energy density. This renders the detection of cosmological sources of high-frequency gravitational waves much more challenging than comparable searches at lower frequencies. On the other hand, the lack of known astrophysical gravitational-wave sources in this frequency range poses a unique opportunity for foreground free searches of new physics.

At the moment, there is no general consensus on the most promising detection strategy in this frequency band, though many proposals have been put forward in the past decades. The proposals that we are aware of are summarized in Table 1, together with their frequency and sensitivity range. We emphasize that the same sensitivity (in terms of characteristic strain) at a higher frequency typically implies a reduced sensitivity to the viable parameter space of a given cosmological source, as discussed above. In this sense, detectors based on magnetic conversion or on the deformation of microwave cavities seem to be particularly promising avenues, though a more careful study of noise levels and of the margin on improvement with foreseeable technology development is needed in many cases.

We hope that this document will stimulate the necessary discussion and we strongly encourage feedback regarding further proposals or critical assessments which we may have missed. We have the ambition to consider this document as a first step towards a coherent international collaboration to seriously consider the search for high-frequency gravitational waves.

None of the proposals listed in this report currently reach the sensitivity needed to probe the new physics outlined above. At best, achievable proposals are still at least six orders of magnitude beyond the required sensitivity. However, we recall that, one hundred years ago, the technological gap in both the LIGO and LISA frequency ranges was of about 16 orders of magnitude. Also, less than 50 years ago, Misner, Thorne and Wheeler declared that “such detectors have so low sensitivity that they are of little experimental interest”, referring to laser interferometers. The first laser interferometer gravitational-wave detector, built at Hughes Research Laboratories in the 1970s, had a sensitivity which was eight orders of magnitudes below the design sensitivity of the currently operating LIGO, Virgo and KAGRA detectors. There are currently clear development paths leading to detectors operating at sensitivities of about 10 to the minus 26 using, e.g., magnetic conversion (see Sect. 4.2.2) and several new ideas such as magnon devices and superconducting systems which have received little detailed design development so far.

We therefore take the past history of laser interferometry as an encouraging lesson for the development of gravitational-wave detectors in the high-frequency band. The challenges are strong but the opportunities are unique. As a rule of thumb, probing very early epochs of our cosmic history and consequently particle physics at very high-energy scales requires searching for gravitational waves at high frequencies with correspondingly small experimental devices. As can be seen from Figs. 1 and 2, the Ultra High-Frequency band, ranging from the MHz to the GHz, is an exciting window to explore fundamental physics up to the grand unification or string theory scales of order 10 to the sixteenth to 10 to the seventeenth GeV, thereby probing our cosmological history right up to the Big Bang. A detection of a gravitational-wave signal in this frequency range would be a smoking gun signal for new physics, since no known astrophysical processes can generate sizable gravitational-wave signals at these frequencies. It would be remarkable if the experimental test of fundamental physics at the highest energies and earliest times in the history of the Universe could eventually be achieved not with huge particle accelerators nor satellite interferometry, but with small size table-top experiments.

This white paper set up the stage for the launch of the Ultra-High-Frequency Gravitational Wave (UHF-GW) initiative, that supports the creation of a network of researchers for the development of gravitational-wave science in the high-frequency range. One of the goals of the initiative is to stimulate the technological development that is needed to build successful gravitational-wave detectors at high frequency.

(The summary tables of sources, the appendix and the reference list are omitted here; the complete text is at the source.)