Novel Search for High-Frequency Gravitational Waves with Low-Mass Axion Haloscopes

Abstract

Gravitational waves (GWs) generate oscillating electromagnetic effects in the vicinity of external electric and magnetic fields. We discuss this phenomenon with a particular focus on reinterpreting the results of axion haloscopes based on lumped-element detectors, which probe GWs in the 100 kHz to 100 MHz range. Measurements from ABRACADABRA and SHAFT already place bounds on GWs, although the present strain sensitivity is weak. However, we demonstrate that the sensitivity scaling with the volume of such instruments is significant – faster than for axions – and so rapid progress will be made in the future. With no modifications, DMRadio-m³ will have a GW strain sensitivity of about ten to the minus twenty at 200 MHz. A simple modification of the pickup loop used to readout the induced magnetic flux can parametrically enhance the GW sensitivity, particularly at lower frequencies.

Introduction

The present gravitational wave (GW) program is focussed on the nHz to kHz frequency range, motivated by the signals expected from the merging of known compact astrophysical objects. This focus leaves the ultra-high frequency (UHF) range, above a kHz, largely unexplored, despite its unique opportunity to probe the physics of the very early Universe.

In this work we propose a novel strategy for the UHF range based on GW electrodynamics: the modified version of electromagnetism applicable in the spacetime metric of a GW. We show there exists a close analogy to axion electrodynamics – the appropriate formalism when working in the background of an ultralight axion – and exploit this connection to convert axion haloscopes into GW telescopes. In the vicinity of static electric and magnetic fields, a GW sources a small electromagnetic signal oscillating at the GW frequency. This opens up the possibility of using low-mass axion haloscopes with a frequency range of 100 kHz to 100 MHz as detectors for UHF GWs. Our proposed search strategy will utilize the anticipated rapid progress being made in this field by lumped-element axion detectors. We will show how the existing results of ABRACADABRA and SHAFT can already be recast as novel limits on GWs. Going forward, DMRadio will dramatically extend both the axion and GW reach at these frequencies. As we will demonstrate, the reach of DMRadio can be enhanced with a simple modification to the signal readout, by using a semicircular “figure-8” loop to measure the magnetic flux. With this adjustment, the future reach of DMRadio will represent both a competitive and complementary approach to UHF GWs. Looking even further into the future, the scaling of the GW reach with the instrument volume is particularly advantageous, which we highlight with the larger instrument we label DMR-100.

There are existing proposals for the UHF band, including optically levitated sensors, bulk acoustic wave (BAW) devices, interferometers such as the holometer, current and future microwave cavity instruments such as the axion haloscope approach introduced for ADMX and SQMS, as well as cosmological probes of GWs based on observations by the radio telescopes EDGES and ARCADE. We emphasize that the different proposals we show are at varied levels of maturity, and therefore caution against overly quantitative comparisons. Instead, we refer to the specific references for details. Our proposal is related to the (inverse) Gertsenshtein effect, which describes the conversion of GWs into photons, a concept that has been proposed as method to search for GWs, in different frequency regimes, in the laboratory and cosmology.

We organize the discussion as follows. We begin with several general comments on the modification to electrodynamics induced by a passing GW, before specializing to the case of interest: the sensitivity of instruments that use a toroidal magnetic field. We then outline how we can exploit this to recast existing ABRA and SHAFT results, and future DMRadio searches. In the Supplementary Material we provide the full details of our calculations and a brief discussion of GW sources in the UHF band.

Gravitational Wave Electrodynamics

We describe the spacetime in the presence of a GW by the linearized metric: the flat metric plus a perturbation whose components are all far smaller than one. The perturbation to the flat-space metric generates a correction to the kinetic term of electromagnetism, which can be written as an effective current — Equation 1: the divergence of the field-strength tensor equals an effective current whose time component is minus the divergence of an effective polarisation vector and whose spatial part is the curl of an effective magnetisation vector plus the time derivative of the polarisation.

Equation 2 gives those two vectors. The effective polarisation is built from the metric perturbation contracted with the electric field, plus half the trace of the perturbation times the electric field, plus the time-time component times the electric field, minus the cross product of the time-space components with the magnetic field. The effective magnetisation is the corresponding combination with the electric and magnetic fields exchanged and the signs and the spatial trace term adjusted. Manifestly, near large external electric or magnetic fields, GWs will source oscillating fields, the detection of which is the focus of this work.

The above formalism facilitates a comparison with axion electrodynamics. In particular, the coupling between the axion and electromagnetism is also described by Equation 1, but with the polarisation given by the axion-photon coupling times the axion field times the magnetic field, and the magnetisation given by the same coupling times the axion field times the electric field, as follows from the axion-photon interaction Lagrangian.

We can extend this analogy in order to estimate the expected GW sensitivity of axion haloscopes. For both the axion and GW, the magnitude of the induced fields is controlled by a dimensionless combination, either the strain or the axion coupling times the axion field. The axion dark-matter program aims to probe the QCD axion, for which the axion mass times the decay constant is of the order of the pion mass times the pion decay constant, and the axion-photon coupling is the fine structure constant divided by two pi times the decay constant. Matching the strain sensitivity to the average axion signal for the QCD axion, we find a strain of about ten to the minus twenty-two. This estimate sets the scale for the GWs that can be potentially detected. In this argument we introduced the electromagnetic fine structure constant, the local dark-matter density, as well as the pion mass and decay constant.

Cosmological GW sources, which are typically isotropic and incoherent, are bounded by constraints on the total amount of radiation in the Universe to satisfy a strain less than about ten to the minus twenty-nine times the ratio of 100 MHz to the frequency, times the square root of the effective number of extra neutrino species — well below our estimated reach. More promising search targets are rare exotic astrophysical GW sources. As a concrete example, if primordial black holes (PBH) with masses well below a solar mass contribute to the dark-matter density, then some fraction of these will exist in binaries and emit high frequency GWs through their inspiral and eventual merger phase. In order to estimate the size of this signal, we take the most up-to-date estimates of the fraction of PBHs in binaries and their expected merger rate, although we emphasize that uncertainties remain such as the impact of accretion on the merger rate. Combining the merger rate with the assumption that PBHs saturate the relevant microlensing constraints, and accounting for the local overdensity of binaries given by the Milky Way halo, we arrive at the estimated sensitivity required in order to see one event per year at that frequency, marginalizing over the primordial black hole mass. We thus primarily focus on localized, approximately coherent GW signals in the following.

In the transverse-traceless (TT) gauge, the non-vanishing components of a plane GW take the standard form of Equation 3, built from the two polarisation amplitudes, the polarisation tensors constructed from two unit vectors transverse to the propagation direction, and the azimuthal and inclination angles of the wave. The choice of the transverse unit vector is a convention, any unit vector perpendicular to the propagation direction can be adopted; different choices will mix the definitions of the plus and cross amplitudes.

In the TT gauge, the form of Equation 3 appears to considerably simplify Equation 2, although for our purposes this can be deceptive because the experimentally generated electric and magnetic fields in the equation are not naturally defined in the TT frame. Instead, throughout this work we will operate exclusively in the frame where all detector quantities are defined. This is the proper detector frame, in which we find Equation 4: the time-time, time-space and space-space components of the perturbation, each written as the square of the frequency multiplied by a universal function of the phase and by combinations of the position vector with the transverse-traceless tensor evaluated at the origin. That universal function is the exponential of the phase minus one minus the phase, all divided by the square of the phase, and it tends to minus one half at small argument. On very general grounds, Equation 4 shows that the effective current and therefore the fields it induces are rapidly suppressed for frequencies below the inverse length scale of the instrument. As we show, there are further suppressions if the experiment is highly symmetric.

Application to a Toroidal Magnetic Field

We consider now an explicit experimental configuration to detect the oscillating fields sourced by a passing GW. In particular, we follow the original ABRA proposal of establishing a DC toroidal magnetic field — Equation 5: the field points along the azimuthal direction with magnitude equal to the maximum field times the ratio of the toroid inner radius to the radial coordinate, between the inner radius and the inner radius plus the width, and zero otherwise. The toroid has inner radius R, width a and height H, and the pickup loop of radius r sits at the centre of the toroid, where the background field vanishes.

The combined effect of the magnetic field and a GW is the effective current in Equation 1 that, according to the Biot-Savart law, sources a magnetic field in the region inside the inner radius — Equation 6, the standard Biot-Savart integral over the toroid volume of the radial and azimuthal components of the effective current. Corrections to the Biot-Savart law from the displacement current enter only at fourth order in the frequency. To detect this magnetic field, a pickup loop is placed at the center of the toroid, which will be sensitive to a magnetic flux equal to Equation 6 integrated over the area of the loop. We next consider two different loop geometries, beginning with the approach used in existing axion instruments.

Circular pickup loop. For a complete circle, the radial current contribution vanishes — independent of the form of the effective current — leaving Equation 7, the flux from the azimuthal current alone. A shift of the azimuthal angle removes the angular dependence in the integrand in all terms except the azimuthal current. Using the results of the previous section, the azimuthal integral of that current is proportional to the cross polarisation amplitude, the frequency, the maximum field and a second-order Bessel function of the frequency times the radial coordinate times the sine of the inclination angle — Equation 8.

Independent of the incident GW direction, the plus component decouples. Furthermore, in contrast to what would be naively expected from Equation 4, the flux receives no contribution at second order in the frequency because the Bessel function of second order begins at the square of its argument. Instead, in the limit where the inner radius is much smaller than the height, which is in turn much smaller than the inverse frequency, the leading contribution to the flux is Equation 9: proportional to the cross polarisation amplitude, the cube of the frequency, the square of the pickup loop radius, the square of the toroid inner radius, the toroid height, the square of the sine of the inclination angle, and the logarithm of one plus the ratio of the toroid width to its inner radius.

For comparison, we note that in this same limit an axion induces a flux given by Equation 10: the axion-photon coupling times the square root of twice the local dark-matter density, times the maximum field, times pi times the square of the pickup loop radius, times the toroid inner radius, times the same logarithm. Nonetheless, a GW will generate a flux, and existing axion searches for such a flux can constrain UHF-GWs.

To isolate the fate of the expected second-order contributions, let us note that, at leading order in the frequency, the radial and azimuthal components of the effective current are given by Equation 11: each is the square of the frequency times the maximum field times the toroid inner radius, divided by the radial coordinate, multiplied by a combination of the cross and plus polarisation amplitudes with sines and cosines of the azimuthal and inclination angles and with the radial and vertical coordinates. All of these terms will vanish for a circular pickup loop geometry: as noted above the radial current cannot contribute in this case in general, and from the explicit expression we see that at second order the azimuthal integral of the azimuthal current vanishes.

Having identified this, however, we can see that with an alternative geometry, these leading terms will survive. To be explicit, using the above currents we can compute the leading contribution to the magnetic field in the plane at the vertical center of the toroid, from Equation 6 — Equation 12: the vertical field is proportional to the square of the frequency, the maximum field, the radial coordinate of the observation point, the toroid inner radius, the same logarithm and the sine of the inclination angle, multiplied by the cross amplitude times the cosine of the azimuthal difference minus the plus amplitude times the cosine of the inclination angle times the sine of that difference. The sinusoidal variation of both polarizations demonstrates that the maximum flux is achieved with a pickup loop with oppositely oriented semicircles for the first and second halves of the azimuthal range — the “figure-8”.

The “figure-8” configuration. Integrating Equation 12 over the figure-8 configuration yields Equation 13: the flux is proportional to the square of the frequency, the maximum field, the cube of the pickup loop radius, the toroid inner radius, the logarithm and the sine of the inclination angle, multiplied by a combination of the cross and plus amplitudes with the sine and cosine of the azimuthal angle of the incoming wave.

The result is now second order in the frequency and sensitive to both polarizations. While it will maximize the GW sensitivity, the figure-8 pickup loop is insensitive to the axion signal, as the vertical field generated by the latter is independent of the azimuthal angle. A single semicircle is sensitive to both, with fluxes given by half of Equations 10 and 13, respectively. Comparing the two expressions, we also see that to compensate the frequency-squared factor, the GW flux scales with an extra power of the loop radius, indicating the improved volume scaling over the axion flux. Accounting for the minimal inductance of the pickup loop, we see that the figure-8 flux scales as the seven-sixths power of the volume whereas the axion flux scales as the five-sixths power. Ultimately, the beneficial volume scaling can be traced back to the fact that our measurement is linear in the induced fields, which themselves are proportional to the strain for the GW or the axion field for the axion.

From Equation 13, we note that the response to the two polarizations differs and depends on the direction of the incoming GW. With two identical detectors angled appropriately, polarization measurements as well as sky localization become possible. For a sufficiently coherent source, one may hope to use the Earth’s rotation or even a mechanical rotation of the experimental setup to break these degeneracies with a single detector.

Gravitational Wave Sensitivity

Low-mass axion haloscopes perform a search for anomalous magnetic flux, and in the absence of a significant signal above background, interpret the results as limits on the axion-photon coupling through the use of Equation 10. With the same equation we can convert existing and projected limits on that coupling into limits on the axion flux, which we can recast as strain sensitivities when compared with our predictions for the GW flux. The procedure is not quite as simple as equating the latter to the axion flux, however. The sensitivity also depends on the relative coherence time of the two signals — a longer coherence time corresponds to a narrower signal in the frequency domain, which in general can be more sensitively detected over the background.

The coherence time of a signal with a given mean frequency is two pi times the quality factor divided by that frequency, where the quality is a dimensionless measure of the inverse width of the frequency distribution. The non-relativistic nature of axion dark-matter implies a highly coherent signal with a quality of about a million, giving a coherence time of about a microsecond for an axion mass of one nano-electron-volt. Considering experimental runtimes longer than the coherence time, the flux sensitivity scales as the fourth root of the quality, and so our actual limit on the GW flux is given by the axion flux limit times the fourth root of the ratio of the axion quality to the GW quality. Beyond this we assume the signal persists during the relevant experimental runtime, and fix the incident GW direction along one horizontal axis.

What remains is to determine a value for the GW quality. This will depend on the specific source. As mentioned already, the localized sources that are our focus can be coherent, but are not expected to be as extremely coherent as a dark-matter signal. As benchmarks, we therefore consider sensitivity to both a coherent signal with quality one thousand and an incoherent signal with quality one, to indicate the dependence upon this choice.

With these choices, we show the reach of existing and future haloscopes, taking the default circular pickup loop geometry. This roughly amounts to using Equations 9 and 10, except that in all figures we use the full flux expressions rather than these leading order results, which are provided in the Supplementary Material. For ABRACADABRA-10 cm and SHAFT, we recompute the GW flux for the explicit geometries specified in those works, as well as accounting for the ferromagnetic core adopted by SHAFT. The future projections of DMRadio-50L, m³ and GUT are obtained by scaling up the toroidal geometry of the ABRACADABRA-10 cm instrument to the volumes of these future experiments. As the exact parameters of these instruments have yet to be specified, these results should be interpreted as representative of the parametric reach — the sensitivities will change by order-one amounts once the geometry of the instruments is known. For instance, while the 50L instrument will adopt a toroidal geometry, m³ will be solenoidal, and the geometry of DMRadio-GUT has not yet been specified. In all cases, there is a suppression in the sensitivity at lower frequency, which is a result of the frequency-cubed factor in Equation 9.

To overcome this, we show the combined sensitivity if the DMRadio instruments adopted the figure-8 configuration, assuming a GW quality of one thousand. While such DMRadio results are still more than a decade away, in order to highlight the significant volume scaling of our approach, we also show the reach of a scaled up version of DMRadio-GUT. In particular, DMR-100 is an instrument with the same toroidal geometry as above, but scaled up to a magnetic field volume of 100 cubic metres, the largest instrument suggested in the original ABRA proposal. The magnetic field volume is also referred to as the effective volume, or written as the geometric coupling times the volume of the toroid.

Conclusions

We provide a formulation of GW electrodynamics which demonstrates that low-mass axion haloscopes are also UHF-GW telescopes. Through the use of an optimized figure-8 pickup loop geometry, the DMRadio program may discover not only the dark matter of our Universe, but also exotic sources of GWs.

Going forward, there are several questions opened by our results that warrant further study. Determining the optimal experimental configuration that allows DMRadio to unlock the frequency-squared GW scaling, while maintaining full axion sensitivity will be critical. To that end, including both a circular and figure-8 loop in the detector may be required, and we note that the mutual inductance between the two loops would vanish. Further, lumped-element instruments have at present only performed measurements for frequencies far below the inverse toroid radius. Given the frequency-squared scaling of the GW effective current, experimental results at higher frequencies would be particularly welcome. From a theoretical perspective, it would also be interesting to understand the limit where the frequency approaches the inverse radius, where a calculation based on the Biot-Savart law alone is insufficient. Nevertheless, the exact behavior has yet to be quantified even for axions, and would be particularly interesting to consider for GWs given the frequency-squared scaling of the effective current. Understanding the extension to other geometries will be important, particularly the solenoidal configuration already used by ADMX SLIC and that will be adopted by DMRadio-m³. Finally, the discussion we presented has been couched purely in terms of the raw GW strain sensitivity. However, as our understanding of the sources at UHF improves, it will be necessary to reinterpret these results in terms of the specific model parameters describing the source populations. At such a time, it would also be worthwhile to understand the additional information that multiple instruments could uncover, as we briefly discussed.

While these issues should be resolved, a larger question looms. A number of distinct experimental proposals have coalesced on a strain sensitivity of about ten to the minus twenty-two for megahertz GWs, a level that is still many orders of magnitude away from any signal of the early Universe. Whether we can hope to probe such strain sensitivities remains to be determined.

Note added

For an extended discussion of the GW signal from PBH binaries including also the stochastic component see the work of Franciolini, Maharana and Muia, which presents results obtained in parallel and independently of the results shown here. Our results are consistent with their findings.

Acknowledgements

We thank Diego Blas, Torsten Bringmann, Sebastian Ellis, Gabriele Franciolini, Yonatan Kahn, Joachim Kopp, Francesco Muia, Jonathan Ouellet, Andreas Ringwald, Chiara Salemi, and Jan Schuette-Engel for helpful discussions. Additionally, we thank Jonathan Ouellet and the authors of the microwave-cavity paper for comments on the manuscript. C.G.C is supported by the Deutsche Forschungsgemeinschaft under Germany’s Excellence Strategy EXC 2121 “Quantum Universe” - 390833306 and by the Alexander von Humboldt Foundation.

(The Supplementary Material and the sixty-five-item reference list are omitted for length; the complete text is at the source. The same team’s fuller treatment of the symmetry selection rules behind these results is on this site at /library/stm-259fcab99d.)