Symmetries and selection rules: optimising axion haloscopes for Gravitational Wave searches

Abstract

In the presence of electromagnetic fields, both axions and gravitational waves (GWs) induce oscillating magnetic fields: a potentially detectable fingerprint of their presence. We demonstrate that the response is largely dictated by the symmetries of the instruments used to search for it. Focussing on low mass axion haloscopes, we derive selection rules that determine the parametric sensitivity of different detector geometries to axions and GWs, and which further reveal how to optimise the experimental geometry to maximise both signals. The formalism allows us to forecast the optimal sensitivity to GWs in the range of 100 kHz to 100 MHz for instruments such as ABRACADABRA, BASE, ADMX SLIC, SHAFT, WISPLC, and DMRadio.

Keywords Axions and ALPs · Early Universe Particle Physics

1 Introduction

Gravitational wave (GW) experiments have begun to probe the GW spectrum over a vast range, from the Gigaparsec wavelengths probed by the CMB to thousands of kilometers, covered by current ground-based interferometers which operate in the 100 Hz range, yielding fundamental insights into cosmology, astrophysics, and particle physics. Reaching even higher frequencies poses a significant experimental challenge, but would represent a unique opportunity to probe possible extensions of the Standard Models of particle physics and cosmology. A cosmological source of GWs produced at a temperature T∗ could generate a stochastic GW background at frequencies of f above roughly 1 kHz times the ratio of T∗ to 10¹⁰ GeV, and leave a signature from modifications to the Standard Model at the highest temperatures. (An explicit example of such a source would be cosmological phase transitions.)

Unfortunately, probing relics of a possible high-temperature phase of the early Universe is fundamentally challenging. Experimental sensitivity to GWs can be expressed in terms of the strain h. As the energy density in GWs scales as the square of the strain times the square of the frequency times the square of the Planck mass, at higher frequencies even greater reach in terms of h is required to reach energy densities below the current bounds on the total energy in radiation in the early Universe derived from BBN and CMB observations. Instead, exotic astrophysical events sourcing transient signals appear to be a more promising medium-term target. For example, the merger of two equal mass objects yields GWs at a frequency of about 1 kHz times the ratio of a solar mass to the object mass, so that sources such as primordial black holes with mass well below a solar mass could populate the high frequency landscape.

In recent years, significant progress has been made in understanding the sensitivity of electromagnetic GW detectors in this frequency regime. In a flat spacetime perturbed by a gravitational wave, the metric being the flat metric plus a perturbation, the usual expressions for electrodynamics in flat spacetime receive corrections of the schematic form of the perturbation multiplied by the square of the field strength, yielding a graviton-two-photon vertex. As long appreciated, this interaction can lead to photon-GW mixing. More generally, however, a GW in the presence of an electromagnetic background will induce an electromagnetic response, in close analogy to the signal from axion (scalar) dark-matter arising from the axion-photon coupling. Exploiting the considerable experimental efforts to search for an electromagnetic response from wave-like dark matter, it has been shown that these same instruments can be used as GW telescopes. Largely motivated by the QCD axion, dark matter searches focus on signals of a MHz or above, and are therefore naturally suited to look for high-frequency GWs.

In this paper, we will continue the study of the sensitivity of axion haloscopes to GWs, with a particular focus on instruments operating in the "low-mass" magnetoquasistatic regime, and sensitivity in the MHz-GHz window; experiments already operating in this range include ABRACADABRA, ADMX SLIC, BASE, SHAFT, and WISPLC. These devices feature a strong static magnetic field which in the presence of an axion — or a GW — sources a small, oscillating induced magnetic field which is captured by a suitably placed pickup loop. Many of the existing instruments are effectively prototypes, with a sensitivity that can be improved significantly by increasing the volume of the magnetic field, and by reading out the magnetic flux induced in the pickup loop resonantly. By combining both of these improvements, the goal of the DMRadio collaboration is to reach the QCD axion prediction for axion masses between about a neV and a µeV. In view of the expected progress, it is timely to consider how synergies in axion and GW searches can be optimally exploited, in particular in view of different detector geometries currently proposed for axion searches.

Earlier work first proposed the use of low-mass axion haloscopes as GW detectors, and demonstrated that a toroidal magnetic field — as employed by ABRACADABRA, SHAFT, and the upcoming DMRadio-50L — could detect a passing GW. Here, we generalise that analysis to additional detector geometries, with a particular focus on the solenoidal magnetic field used by ADMX SLIC, BASE, WISPLC and which has been moreover proposed for DMRadio-m3. We provide analytical expressions for the effective current which the GW sources, the resulting induced magnetic field, as well as for resulting magnetic flux for various pickup loop geometries. Armed with these results, we will bootstrap the expected sensitivities to GW signals from axion searches. A further improvement over the earlier work is a careful treatment of the different timescales involved, in particular the potentially short duration of the GW signal.

Whilst the GW sensitivity for a solenoidal magnetic field is a practical result, as for the toroidal magnetic field, the calculation remains involved, and ultimately it becomes inefficient to compute the GW interaction with all possible magnetic field geometries. Motivated by this, we derive a series of symmetry based selection rules, which determine the parametric sensitivity to a GW signal depending upon the symmetries of the experimental magnetic field and the pickup loop used to read out the signal. From these results, we will demonstrate that configurations with a high degree of symmetry can kill the leading order sensitivity to a GW, even though they may be desirable to maximise the axion sensitivity. An analogue of this was already observed in the earlier work, where it was shown that if the flux from a toroidal magnetic field is read out through a circular pickup loop, then the leading order GW sensitivity, expected at second order in the product of frequency and length, vanishes, while sensitivity at third order remains. Here the frequency is the angular frequency of the GW, the length is a characteristic length scale for the experiment, and in the magnetoquasistatic regime of interest for low-mass axion haloscopes their product is much smaller than one.

We show that if both the external magnetic field and the pickup loop have cylindrical symmetry, that is, if they are invariant under azimuthal rotations and reflections in the z coordinate, any orientation of the pickup loop which is sensitive to the axion suffers from a cancellation of the leading order term for the GW signal. This symmetry is commonly exhibited by axion haloscopes, which make use of solenoidal or toroidal magnetic fields. To recover the dominant scaling, the cylindrical symmetry must be broken, for instance through the placement or geometry of the pickup loop. The latter can be most easily achieved by modifying the pickup loop to span only a fraction of the azimuthal angle, with the optimal GW sensitivity obtained when the cylindrical symmetry for the pickup loop is maximally broken (for instance, with a figure-8 configuration).

For existing experiments, as the largest axion signal is obtained for detectors with full cylindrical symmetry, this explains the earlier observation that the optimal axion and GW sensitivities cannot be simultaneously obtained for a haloscope based on a toroidal magnetic field, and furthermore demonstrates that this conclusion is generic. Nevertheless, we find that modifying the pickup loop geometry (or including several different pickup loops) allows one to obtain sensitivity to both the axion and GW signal, in a manner that at worst reduces the axion sensitivity by an order-one amount. (Footnote in the original: this same approach would also allow for discrimination between a GW and axion signal. Of course, we note that there are many ways to distinguish these signals, the most important being that in the accessible parameter space the GW signal will be transient, whereas that from dark matter is persistent.) We illustrate the power of symmetry arguments by determining the leading power sensitivity for a range of different detector geometries without explicit computation, in view of determining the optimal geometries for GW searches. For the most relevant cases, we provide the computation to confirm our results.

At the outset, we can already provide an intuitive argument as to why cancellations in highly symmetric detectors might be expected. To do so, rather than contrasting GW and axion electrodynamics, as we will in the remainder of the paper, let us consider a simpler comparison: a scalar versus a pseudoscalar. In particular, consider first the induced magnetic field arising from the interaction of a toroidal magnet with a pseudoscalar. If we consider the induced magnetic field in the z direction at the center of the toroid, as measured by the ABRACADABRA collaboration, we find a non-zero result. The consistent transformation of this result under parity, which can be confirmed directly, is critically reliant on the pseudoscalar nature of the axion. Indeed, if we ask what the induced field would be for a scalar interaction, there is no expression we can write consistent with parity and the cylindrical symmetry of the instrument. An explicit computation confirms that the scalar-induced field vanishes. This argument can be formalised into symmetry based selection rules which determine the geometries that are sensitive to scalar versus pseudoscalar coupling — indeed, there are configurations where the axion-induced field vanishes whilst the scalar-induced field does not — and we undertake that exercise in the appendix. The general lesson, however, is that highly symmetric detectors impose symmetry constraints on the induced fields that can be so restrictive that the measurable signal vanishes. This is true also for GWs, and we will determine an appropriate set of selection rules to determine the interplay between signals and geometry.

We can actually determine an additional general lesson by comparing the scalar and pseudoscalar interaction. As is well known, the axion interaction generates an effective current proportional to the derivative of the axion field contracted with the dual field strength, so that the interaction depends only on a derivative of the axion, as expected for a pseudo-goldstone boson. The equivalent expression for a scalar contains an extra term, proportional to the scalar multiplied by the current that generates the leading order fields in the laboratory. Accordingly, for the scalar there is an additional contribution to the effective current localised at the boundary of the magnetic volume, which turns out to be generic: it will be present also for the GW, although it has so far been overlooked in the literature. This contribution can be interpreted as an effective current at the boundary of the magnetic volume, determined by the component of the effective magnetisation vector parallel to the boundary surface.

In the remainder of this paper we will flesh out these ideas for the GW signal, and we organise our discussion as follows. Section 2 lays out the theoretical framework for our work, reviewing the relevant aspects of electrodynamics in a spacetime perturbed by a GW. Several points, such as a discussion of the symmetry properties of the induced magnetic field and response matrix formalism are presented here for the first time in this context. This sets the stage for deriving the GW sensitivity of axion haloscopes with solenoidal magnetic field configurations in section 3. The results are then generalised in section 4 where we derive symmetry principles which allow us to determine the parametric scaling of the GW sensitivity for various detector geometries without explicit computation. The symmetry arguments will then enable us to draw general conclusions about the optimal strategy for axion and GW searches in axion haloscopes.

Many details of our analyses are deferred to appendices. Appendix A reviews Maxwell's equations in curved space time. Within it, we provide a careful derivation of the main equations governing the interaction of a GW with a background electromagnetism field, the derivation of the effective surface current, and an explanation of why the GW effects we consider scale at lowest order as the square of the frequency-length product. In appendix B we study scalar and axion electrodynamics, with a focus on sharpening an analogy to the GW case. We will explain how our GW selection rules extend to these spin-0 waves, and the consequences for various detector geometries. In appendix C we summarise the symmetry properties of the cylindrical magnetic field configurations employed by axion haloscopes, and demonstrate that they can be decomposed into a solenoidal and toroidal component. Appendix D expands our discussion of the response matrix formalism used to describe the detector response to a passing GW. In appendix E we summarise the explicit analytical expressions for all components of the effective current induced by a GW, up to third order and for both toroidal and solenoidal external magnetic field configurations. These expressions may be used as input for full detector simulations, or for detailed numerical calculations of the relevant GW effects. In appendix F we discuss in detail the bootstrapping of axion search results to establish GW sensitivity, carefully taking into account the different time scales involved in the possible signals and detectors. The appendix further discusses details of several possible sources for high-frequency GWs. Finally, appendix G is dedicated to the possibility of using an external electric instead of magnetic field for GW detection, and demonstrates how our symmetry arguments extend to this case.

2 Gravitational wave electrodynamics

To begin with, we review the general formalism used to compute the magnetic flux induced by a GW passing through a lumped-element circuit axion haloscope. We will review the discussion of the earlier work, pointing out an additional contribution to the induced magnetic flux due to effective surface currents at the boundary of the magnetic volume, which was previously overlooked. We then extend this approach to account for the transformation properties and symmetries of the various quantities, in particular the induced magnetic field, under rotations and reflections. The axion haloscopes targeting the magneto-quasistatic regime generally have a high degree of cylindrical symmetry, and we will study the impact of this on the GW signal systematically. Doing so will allow us to develop a systematic approach to the geometries of the external background magnetic field and pickup loop, and resolve fundamental questions such as determining the optimal geometry for GW and axion searches.

2.1 Proper detector frame

Throughout this paper we will work in the proper detector frame, in which coordinate distances to the origin match the proper distance, and thus coincide with those measured by ideal rigid rulers. (This is in contrast to the transverse traceless frame, in which coordinate distances are set by the geodesics of free-falling test masses, and a rigid instrument and experimental magnetic field no longer have a simple description.) As a consequence of this, up to non-inertial forces such as those associated with the rotation of the Earth (which can be neglected at high frequencies), the effect of GWs is simply given by a small Newtonian force proportional to their amplitude.

Throughout this paper, we will assume that these GW forces do not mechanically deform the experimental setup, in particular the static electromagnetic fields applied in the experiment remain static in the presence of a GW. Critically, this implies that in the proper detector frame the experimentally generated magnetic field coincides with that of flat spacetime. This assumption is in particular valid for GW frequencies below the mechanical resonance frequencies of the setup. At frequencies around and above the lowest mechanical resonance — set by the speed of sound in the material divided by the instrument size — the Newtonian GW force is no longer negligible. We expect this to impact part of the parameter space relevant for the experimental setups discussed here, and we leave a quantitative analysis to future work. Interestingly, in the case of microwave cavities, it was demonstrated that this effect can enhance the GW sensitivity.

Expanding the metric as the flat metric plus a perturbation, in the proper detector frame the GW at a given position can be expressed through Equation 2.1, which gives the time-time, time-space and space-space components of the perturbation in terms of a universal function of the phase, the two polarisation amplitudes, and the polarisation tensors defined in Equation 2.2. (The displayed tensor expressions are given here as named results; the explicit forms are in the source.) The polarisation tensors are built from two unit vectors transverse to the propagation direction, and the propagation direction itself is fixed by a polar and an azimuthal angle. In particular, the perturbation contracted with the position vector vanishes, and consequently the line element reduces to the flat one for purely radial displacements. From this we see that coordinate distances to the origin coincide with the corresponding proper distance, a defining characteristic of the proper detector frame.

In this work, we will limit ourselves to the regime where the product of frequency and instrument length is much smaller than one, as appropriate over most of the range covered by lumped-element circuit instruments. We can therefore treat that product as a perturbative parameter, and will do so often, for instance it will be implicit in our use of the Biot-Savart law and used throughout our discussion of the implications of the symmetry transformations. Further, since the universal phase function tends to minus one half at small argument, it follows that the metric perturbation in the proper detector frame has a leading order contribution at second order in the frequency-length product. The absence of any contribution at first order is a consequence of working in a freely falling reference frame assumed to be rigid. An immediate implication of this scaling is that for a GW incident on an electromagnetic field that is static in the proper detector frame, the leading order electromagnetic response induced will scale at second order. This demonstrates that the optimal observables for the GW one can construct will also be at second order.

As outlined in the introduction, one of the primary goals of the present work is to understand the role symmetry plays in the GW interactions. In particular, we will be studying the interaction between a GW and detectors with a high degree of symmetry. Existing axion instruments tend to have full cylindrical symmetry, that is, invariance under rotations about the vertical axis and arbitrary reflections. Therefore, it is worthwhile already to characterise the transformation of the GW polarisations and proper detector frame components when these transformations are applied to the position and incident direction at which we evaluate these quantities. Equations 2.3 and 2.4 record those transformations, introducing a sign that is plus one for the plus polarisation and minus one for the cross polarisation, which keeps track of their different behaviour under reflections; the perturbation transforms as a regular tensor under rotations about the vertical axis.

2.2 Effective current induced by GWs, and 2.3 the induced magnetic field

(Sections 2.2 and 2.3 are largely a chain of tensor equations that did not survive text extraction cleanly. Their content is given here as named results; the complete derivations are at the source.)

The interaction of GWs with electromagnetic fields can be effectively described as an additional current augmenting Maxwell's equations in a flat spacetime — Equations 2.5 and 2.6. Here the ordinary current is the one present in the absence of the GW, whereas the effective current is a divergence of terms built from the metric perturbation contracted with the background field strength. The homogeneous Maxwell equations are unaffected by the presence of the GW. Throughout this paper we work to linear order in the perturbation, so the field strength appearing on the right-hand side contains only the background fields.

In further analogy to ordinary electromagnetism in a medium, one can define an effective polarisation vector and an effective magnetisation vector — Equation 2.7 — such that the effective current is the pair consisting of minus the divergence of the polarisation and the curl of the magnetisation plus the time derivative of the polarisation, Equation 2.8. This final formulation is reminiscent of polarisation and magnetisation vectors for electromagnetism in a medium. Hence, the task of calculating the electromagnetic fields induced by a GW is equivalent to performing standard electromagnetic calculations in such media.

The effective current induced by the GW sources an induced magnetic field determined by the Biot-Savart law — Equation 2.10 — integrated over the detector volume filled by the external magnetic field. Under the assumption that this integration region is invariant under reflections, which it is for the cylindrically symmetric detectors we consider, the Biot-Savart law implies a definite transformation law for the induced magnetic field under reflections, Equation 2.11, and simple covariance under rotations about the vertical axis. That transformation law is a key tool in studying the implications of detector symmetry for the associated GW signal.

The observable used in low-mass axion haloscopes is the induced magnetic flux through a suitably placed pickup loop — Equation 2.12 — the surface integral of the induced field against the unit normal of the loop. Our symmetry arguments also depend on the transformation of the pickup loop normal under reflections, Equation 2.13. A response matrix formalism, developed further in the appendix, encodes the detector response to a passing GW and proves to be what reveals the symmetry structure most clearly.

3 The GW sensitivity of solenoidal detector geometries

Axion haloscopes in the magnetoquasistatic regime use one of two magnetic field geometries. The first is a toroidal field, confined to an annular region, as used by ABRACADABRA, SHAFT and the upcoming DMRadio-50L. The second, treated here, is a solenoidal field: a uniform vertical field inside a cylinder of radius R, and zero outside it.

Among the detectors making use of a solenoidal field, we will first focus on instruments where the induced flux is read out through a vertical pickup loop, as implemented in the ADMX SLIC and BASE experiments. Moreover, the planned WISPLC and DMRadio-m3 experiments are also planning to implement a related configuration. (Although other future instruments will use a toroidal magnetic field, for instance DMRadio-50L.) Therefore, as a straightforward generalisation of the earlier work, we first calculate the expected magnetic flux from the incoming GW for a solenoidal magnetic field and different locations of the pickup loop. Armed with an understanding of how the GW interacts with a detector for two explicit cases, in the next section we will then generalise our discussion for general geometries.

(Footnote in the original, on the differences between these experiments: ADMX SLIC has a single rectangular pickup loop at a fixed polar angle. The BASE experiment relies instead on many such pickup loops placed symmetrically in the horizontal plane, whereas DMRadio-m3 will use a full toroidal sheath. For the WISPLC experiment, the current design features a pickup loop at fixed polar angle, but which is located outside the region of the external magnetic fields.)

Before proceeding, we note that in the main text we will only consider the interaction between GW and laboratory magnetic fields. The rationale for this is that axion haloscopes exclusively make use of magnetic fields, as larger energy densities can be built up in magnetic than electric fields. Further, the axion interaction with a magnetic field is controlled by the time derivative of the axion field, which for dark matter is much larger than its gradient, which the electric field couples to. As the GW is both relativistic and couples differently than the axion, this final consideration does not apply, and therefore for completeness we briefly discuss the interaction with an electric field in the appendix.

3.1 The GW signal for a solenoidal magnetic field

(This subsection derives the induced flux explicitly for a rectangular pickup loop at fixed polar angle, for a pair of such loops, and for a figure-8 configuration, in each case expanding in the frequency-length product and in the inverse height of the instrument. The results are Equations 3.3 to 3.9. The derivation is a sequence of displayed expressions that did not survive text extraction cleanly and is omitted here for length; the complete calculation is at the source. Its two load-bearing outcomes are used below: for the azimuthally symmetric arrangements the second-order term cancels and the leading flux appears only at third order, while for a figure-8 readout, which maximally breaks the azimuthal symmetry, the second-order term survives.)

3.2 Strain sensitivity from recasting axion limits and projections

Combining the above results, we now determine the GW sensitivity of axion haloscopes making use of a solenoidal external magnetic field. In detail, we recast the constraints on the axion photon coupling from BASE and ADMX SLIC, as well as the future instruments WISPLC and DMRadio-m3, as expected sensitivities to the amplitude of GWs. All of these instruments are designed with the goal of detecting a coherently oscillating axion dark matter background field, which takes a simple sinusoidal form with an amplitude fixed by the local dark-matter density and the unknown axion mass — Equation 3.10.

In the presence of a magnetic field, the axion background generates an effective current proportional to the time derivative of the axion field multiplied by the magnetic field. For the configuration considered, that current induces a magnetic flux proportional to the axion-photon coupling, the time derivative of the axion, the field strength, the loop height and the difference of the squares of the two loop radii — Equation 3.11 — with the result for a finite instrument height determined numerically.

Our goal is to derive sensitivity to the GW strain by reinterpreting results on the axion-photon coupling, established assuming an axion signal flux of that form (modified for the specific experimental configuration). To do so, we compare the axion flux and the GW flux, but further we account for the fact that in all expressions derived so far the axion and GW are treated as persistent monochromatic waves, when this is not the case for either one. The dark-matter axion is indeed persistent, but has a coherence time set by a quality factor of about a million, indicating a highly coherent signal. The GW signal on the other hand is model dependent (specific examples of superradiance and primordial black hole mergers are discussed in the appendix), but can be described as lasting for a finite duration with a finite coherence time, centered at a frequency, so that the signal has its own quality factor. For resonant instruments, one must also account for the quality factor and coherence time of the instrument. The ultimate strain sensitivity is determined from considering the interplay of each of these scales, together with the experimental run time. The end result is that rather than simply matching the GW and axion fluxes, we instead equate the GW flux to a coherence ratio multiplied by the axion flux — Equation 3.12 — where the coherence ratio accounts for the difference in coherence between the signals. As defined, a coherence ratio greater than one implies the GW signal is harder to detect than a naive matching of the flux would suggest.

Here, we will restrict our attention to a single case, where we take the GW duration equal to its coherence time, and imagine a resonant instrument that spends a time scanning each axion mass that is long compared with every other scale. In this case the coherence ratio takes a four-branch form — Equation 3.13 — depending on how the instrument quality factor compares with those of the axion and the GW. Let us briefly describe the physical origin of each term. The overall quarter-power scaling of run time to coherence time encodes the suppression that arises as the GW signal does not persist for the full time the instrument scans this frequency; the quarter scaling follows our assumption that the scan time is the largest scale, implying that the sensitivities have entered the asymptotic scaling regime consistent with the Dicke radiometer equation. The ratio of axion to GW quality factors to the quarter power arises as signals that are more coherent are easier to discover. More coherent signals are narrower in the frequency domain, and therefore can generally be teased out over a smaller amount of background. In addition, when the axion quality factor is below the instrument's there is a slight penalty to the axion signal as the full axion signal is not resolved by the instrument. Finally, there is a strong penalty applying to the GW signal whenever the instrument quality factor exceeds the GW quality factor. When this occurs, the GW fails to fully ring up the resonance response of the instrument, which strongly decreases the power it deposits, explaining the linear scaling of this factor, as opposed to the quarter scaling of all others.

Using the coherence ratio for any given instrument, we can translate GW signals defined by their amplitude, duration, and coherence time to an effective signal strength. We can then compare that effective strength to the equivalent plane wave sensitivity shown in figure 3, which is derived simply from matching the flux, assuming a coherence ratio of one.

For the results in figure 3, the frequency range is fixed by the corresponding axion mass range, and therefore falls into the MHz band. The left figure demonstrates the estimated sensitivity of three existing instruments: ABRACADABRA, BASE, and ADMX SLIC (the reach for SHAFT is comparable to ABRACADABRA). Note that BASE and ADMX SLIC perform a resonant search strategy for the axion, and therefore at present have a deeper sensitivity, but narrower frequency coverage than ABRACADABRA which completed a broadband search. For ABRACADABRA and BASE, the cylindrical symmetry of these instruments suppresses the leading order flux, for reasons we demonstrate in the next section. For this reason we also show the sensitivity the instruments could obtain if they implemented a figure-8 style geometry. On the right we show the sensitivity for future instruments, in particular DMRadio and WISPLC. For DMRadio, we show the projected reach of the 50L, m3, and GUT variants of these instruments, in each case assuming they have adopted a figure-8 style readout. We assumed a solenoidal magnet for m3, but toroidal for 50L and GUT. For WISPLC, we have repurposed their anticipated axion sensitivity with a resonant detection strategy, and assumed a single pickup loop (more precisely, WISPLC has two individual pickup loops). For these cases, we chose the angular direction of GW and relative pickup loop direction which maximise the GW sensitivity. Further, as the GW signal depends on the incident direction, in each case we have simply taken one of the two polarisations, and chosen the incident direction to maximise the signal, although performing an angular average instead would only minimally impact the results.

For comparison, on the right we also draw the expected effective signals coming from superradiance or primordial black hole binaries as benchmarks for various parameter choices. For drawing the signal curves in the right panel of figure 3 we adopt the following simplified experimental setups: an instrument quality factor of ten thousand and a measurement time of a thousand seconds for superradiance, and a quality factor of ten thousand with a measurement time of one millisecond for primordial black holes. Note that in the latter case, the signal depends on the black hole masses and the curve presented in the figure corresponds to the maximal signal for a given frequency.

Further example values of the coherence ratio for different instruments and two different benchmark signals are provided in table 2. The first benchmark is a GW signal of duration one second, with a coherence limited only by its finite duration. The second is an example of a persistent highly coherent signal. In both cases, we assume the frequency spectrum of the signal to be centered around the detector's resonant frequency. These two benchmarks are loosely inspired by the properties of GWs from primordial black hole mergers and from superradiance, respectively. The advantage of these simplified signals is to facilitate comparisons across different detector concepts for astrophysical signals. This is an alternative to the comparison of sensitivity curves in terms of noise spectral densities, which requires detailed knowledge of the relevant noise sources. A further alternative is the use of the characteristic strain. However, in realistic cases the timescales mentioned above enter into the estimation of the characteristic strain, and hence again detailed knowledge of signal and detector properties becomes necessary for comparisons. Of course, the method used here of defining simplified benchmarks for comparisons also has its drawbacks.

Table 2 Coherence ratio for the different experiments considered in figure 3, for two benchmark signals: benchmark one has duration equal to coherence time of one second; benchmark two is persistent and highly coherent. As an example, the best estimated sensitivity of ADMX SLIC when taking a coherence ratio of one is a strain of about 1.7 times 10 to the minus 16, so that the sensitivity to benchmark one would be about 2.8 times 10 to the minus 16, and to benchmark two about 1.7 times 10 to the minus 17.

| Instrument | Instrument quality factor | Measurement time | Reference frequency | Coherence ratio, benchmark 1 | Coherence ratio, benchmark 2 | |---|---|---|---|---|---| | ADMX SLIC | 3 × 10³ | 320 s | 50 MHz | 1.6 | 0.1 | | BASE | 4 × 10⁴ | 1 min | 0.7 MHz | 3.0 | 0.39 | | WISPLC | 10⁴ | 1 min | 30 kHz and 5 MHz | 6.7 and 1.9 | 0.86 and 0.24 | | DMRadio | 2 × 10⁷ | 8 mins and 60 ns | 100 kHz and 30 MHz | 787 and 1 | 0.18 and 1 |

We can also use the formalism introduced above to estimate the sensitivity to stochastic gravitational backgrounds. Their energy is constrained by BBN and CMB observations. Comparing the spectrum expressed in terms of the characteristic strain with the energy in a plane gravitational wave, we identify the strain with the characteristic strain, so that the BBN and CMB bounds indicate the maximal GW strain achievable from cosmological stochastic backgrounds. The remaining task is to estimate the coherence ratio in this case. Setting the coherence time and duration equal to the measurement time and the GW quality factor of order one, well below those of the axion and the instrument, we obtain a coherence ratio much greater than one. In other words the low coherence further suppresses the effective signal strength, which, together with the bound on the effective number of relativistic species, implies that these signals are unfortunately currently out of range by several orders of magnitude. Of course, recasting axion searches for highly coherent signals is not the optimal strategy to search for stochastic backgrounds. Nevertheless, this simple estimate illustrates the challenges that such a dedicated search will be facing.

Figure 3 demonstrates that axion haloscopes can place competitive bounds on GWs in this frequency range. Due to the challenges mentioned above in comparing different sensitivity estimates, we refrain here from including other detector concepts in this figure. This is, however, a very active field and other concepts such as bulk acoustic wave devices, levitated sensors, interferometers, other electromagnetic GW detectors and indirect detection methods have reached, or are expected to reach, similar sensitivities. As evident from figure 3, a further increase in sensitivity is needed to reach possible astrophysical (or cosmological) signals. Our work should be seen as part of the quest of paving a possible path towards this.

4 Selection rules for general detector geometries

In section 3 we studied in detail the interaction of a GW with a solenoidal magnetic field, adding to the existing results where the wave interacts with a toroidal field. In this section we seek to generalise these results with a symmetry based study of a broader class of magnetic fields and pickup loops. We will consider detectors with both solenoidal and toroidal magnetic fields, but in each case we will consider all possible directions for the pickup loops: designed to measure the induced magnetic field in the vertical, azimuthal, and radial directions.

A central goal of our analysis is to identify, by symmetry alone, what is the leading power of the GW flux for a given detector. To do so, we will derive three selection rules that hold for the interaction of a GW with a cylindrical instrument, that allow us to study the more general case and identify promising detector geometries without explicit calculation. The results are catalogued in table 3 supplemented by the outcome of explicit computations, but let us briefly summarise the key findings. The leading contribution we expect to the flux is at second order in the frequency-length product; however, we have already seen for the BASE experiment and for ABRACADABRA that the flux at this order vanishes. These are two examples of a general result: instruments with full cylindrical symmetry (of both the magnetic field and pickup loop) designed to search for axions have a leading power sensitivity of at most third order.

This is a consequence of two observations. Firstly, as we will demonstrate, detectors with azimuthal symmetry are only sensitive to one of the two GW polarisations. They are also only sensitive to either a scalar or an axion, as under parity the scalar transforms as the plus polarisation, whereas the axion transforms as the cross polarisation. Secondly, azimuthal symmetry enforces that only the plus contribution can enter at second order. Consequently, instruments with full cylindrical symmetry which can detect scalars coupled to electromagnetism may also detect plus-polarisation GW flux at second order, but no leading order contribution can appear for axion experiments which employ full cylindrical symmetry to enhance the axion signal. If the cylindrical symmetry of the pickup loop is broken, the leading power sensitivity to the GW can be restored, at the cost of an order-one factor to the axion signal.

4.1 Three selection rules for the interaction of a GW with a cylindrical detector

(The three rules are stated here as the paper states them; their proofs, which turn on the transformation properties of the induced field and the pickup loop normal under reflections, are omitted for length. The complete proofs are at the source.)

Selection rule 1. For an instrument with azimuthal symmetry, the flux is proportional to the plus polarisation at second order in the frequency-length product. (The paper emphasises that this statement is coordinate dependent, and holds for its own definition of the plus polarisation. In general one can convert plus into cross by a rotation of forty-five degrees around the propagation direction. The coordinate independent statement is that for geometries with azimuthal symmetry, only a single polarisation appears at leading order.) As a corollary, at leading order an azimuthally symmetric detector can only depend on the incident GW direction through the square of the sine of the polar angle.

Selection rule 2. For an instrument with azimuthal symmetry, the flux is proportional to either the plus or the cross polarisation, but not both. This holds to all orders in the frequency-length product.

Selection rule 3. For an instrument with full cylindrical symmetry, the flux will be either an even or an odd function of frequency.

Let us work through several explicit examples, each assumed to be azimuthally symmetric.

• Toroidal magnet with a horizontal pickup loop. Consider first a configuration as used by ABRACADABRA. By selection rule 2 the plus polarisation cannot contribute, only the cross polarisation will. Concretely, for ABRACADABRA this implies that any azimuthally symmetric pickup loop will receive no contribution proportional to the plus polarisation, for all possible vertical positions of the pickup loop. Selection rule 1 further implies that the leading order contribution can only occur at third order. Both of these results were observed by explicit calculation in the earlier work.

• Solenoidal magnet with an array of vertical pickup loops. Inspired by BASE, we next consider a setup where only the cross polarisation can contribute. Explicitly, the flux generated by the plus polarisation from a pickup loop at one polar angle will exactly cancel the flux the plus polarisation generates in a pickup loop at the opposite angle. Thus, even though the geometry has changed significantly, our polarisation selection rule applies identically to the ABRACADABRA-type configuration. This explains the cancellation for azimuthally symmetric solenoidal detectors observed in section 3.

• Solenoidal magnet with a horizontal pickup loop. Here the cross contributions to the flux vanish to all orders whereas the plus contribution survives, and with it a contribution to the flux of second order. In this case, one might worry about the feasibility of separating the tiny induced field from the large background magnetic field. Any detection strategy would exploit the alternating nature of the GW flux, as opposed to the ideally static background field, and potentially also the angular dependence of the GW.

The two selection rules derived so far allow us to understand an important consequence for the detection of GWs with axion haloscopes. In particular, by selection rule 2, the flux in an azimuthally symmetric detector sensitive to the cross component has no dependence on the plus component. But then by selection rule 1, such an instrument will not have the optimal sensitivity to the GW, since the second-order contribution will vanish. This is an important observation for axion haloscopes, because the pseudoscalar axion field transforms in the cross-polarisation way, and therefore to be sensitive to the axion one is forced into a configuration where the leading GW flux vanishes. The only way of evading this conclusion is to break the azimuthal symmetry. One could break this maximally by introducing a figure-8 configuration. This would revive a plus contribution at second order; however, as the axion induced magnetic field has no angular dependence, its contribution will vanish. Hence, to detect both axion and GW, one could use a pickup loop with an opening angle smaller than a full turn, which avoids a complete cancellation.

The combination of our three selection rules shows that instruments with full cylindrical symmetry have a highly restricted form of the induced GW flux. In table 3, we apply these selection rules for all possible pickup loop orientations, and for both solenoidal and toroidal external fields, always assuming full cylindrical symmetry. In the first line of each cell, we denote the surviving polarisation, whether even or odd powers contribute, and what is the leading order contribution. The second line provides the explicit leading order flux.

The result for a horizontal pickup loop in a toroidal magnetic field was previously presented in the earlier work; our new result corrects this expression by a factor of one third, which is due to taking into account the contribution from the effective surface current previously overlooked. For the second-order results in the earlier work, obtained from the use of a figure-8 pickup loop to break the azimuthal symmetry, the surface current contribution vanishes, leaving them unchanged.

For each case in the table, the selection rules determine the leading order contribution to the flux without any explicit calculation being required, which achieves one of the central goals of this work. For example, take a solenoidal magnetic field with a radial pickup loop. Only the plus polarisation can contribute from selection rule 2. However, only odd powers of frequency contribute by selection rule 3, and therefore the leading order contribution is at third order. If, however, the loop was moved up or down in the vertical direction, breaking the cylindrical symmetry, selection rule 3 would no longer hold, and we would have a contribution at second order as allowed by selection rule 1 — Equation 4.6 — so that a leading order contribution has been resurrected. Consider also the case of a toroidal magnet with a radial pickup loop. Now only the cross polarisation and even orders will contribute. But by selection rule 1, for such a configuration the leading order contribution cannot occur until fourth order, where we already expect corrections from our use of the Biot-Savart law. Again, placing the loop at the vertical bottom or top of the instrument would parametrically enhance the flux — Equation 4.7.

4.2 Increased GW sensitivity for pickup loops that break azimuthal symmetry

All three selection rules above followed from the full azimuthal symmetry of the detector. When broken, the restrictions the rules impose are lifted, and in many geometries this allows for a parametric improvement in the GW flux. While the breaking can occur at the level of the magnetic field or pickup loop, the latter is far more practical. For instance, an instrument could use multiple pickup loops, one for the axion, and another for the GW.

With such a possibility in mind, the paper computes the leading second-order flux for various geometries in the case where the detector has a pickup loop that spans a partial opening angle for a horizontal or radial readout, or a set of loops that span a fraction of a toroid in the case of the vertical loop. Those results, for both solenoidal and toroidal fields and for vertical, azimuthal and radial readouts, are Equations 4.8 and 4.9; they are stated to second order and to leading order in the inverse instrument height. (The displayed expressions are omitted here for length; the complete forms are at the source.) Observe that when the opening angle is a full turn, only the result for the vertical readout survives, consistent with table 3.

5 Discussion

Both axions and GWs induce effective polarisation and magnetisation terms in Maxwell's equations. While the formalism for exploiting this effect to search for the axion has been in place for four decades, the GW analogue and its synergies with axion searches remains nascent. Our work expands our understanding of this latter case, and by focussing on lumped-element circuits for axion detection (such as ABRACADABRA, SHAFT, BASE, ADMX SLIC, WISPLC and the DMRadio program), we estimate their sensitivity to current and future high-frequency GW searches, as shown in figure 3.

We also expand the theoretical foundations of the interaction of GWs with instruments operating in the magnetoquasistatic regime, extending the earlier results in a number of ways. Most importantly, we have developed a symmetry based formalism that largely fixes the form of the leading GW signal in various instruments. We considered external magnetic fields with a cylindrical symmetry — toroidal or solenoidal fields — as used in all ongoing and planned axion haloscopes. We derived selection rules for the signal strength, which, based on symmetry alone, fix the leading power sensitivity, and hence parametrically determine the GW strain sensitivity without calculation. This allows one to immediately determine the impact of different geometries for the external magnetic (or electric) field and the pickup loop on the achievable GW strain sensitivity. As summarised in table 3, highly symmetric detectors place strong restrictions on the form of the induced flux as a direct consequence of the tensor nature of the GW. These arguments can be extended to a scalar or pseudoscalar (axion) coupled to electromagnetism, as we show in the appendix. Taken together, we observe that in optimising the sensitivity to axions, existing instruments can often parametrically suppress the GW signal. Fortunately, however, the observed cancellation can quite easily be remedied by minimally breaking the instrument's cylindrical symmetry, for instance by changing the position or shape of the pickup loop. We demonstrated this for different detector geometries, obtaining a parametric increase for the GW sensitivity.

Our work provides several technical improvements on the formalism and initial studies of the earlier work. First, we include the contribution from effective surface currents induced by the GW, arising due to the change in effective magnetisation at the boundary of the static magnetic fields. These effects are generically of the same order as the effects obtained from the interaction of the GW with the magnetic field itself, and for instance modify some of the toroidal results relevant for ABRACADABRA, SHAFT, and DMRadio-50L. Second, we give a thorough discussion and prescription of how to compute sensitivities for transient signals, focussing on resonant detectors. This is particularly important for high-frequency GWs, since the duration of the expected signals can be much shorter than the observation time. We include the effect of finite coherence time and finite duration of the signal, the scanning strategy and the quality factor of the instrument. Our prescription is based on bootstrapping the axion search results, allowing an immediate recasting of existing and upcoming axion searches in terms of GW searches. Third, we introduce linear response matrices describing the detector response to the GW signal. With this new formalism, we recover the earlier results, but the alternative approach played a key role in revealing the symmetry properties of the detectors, facilitating the derivation of the selection rules mentioned above. With all this at hand, we provide analytical results for the effective current induced by a GW up to third order for a solenoidal magnetic field configuration.

Much work remains. The symmetry based arguments introduced here can be deployed for the full set of signals axion haloscopes could detect, including, for instance, dark photons. Such arguments can help determine the full physics reach of the future axion dark-matter program. A dedicated GW search will require a targeted data analysis strategy as well as a detailed detector simulation. While this is beyond the scope of the current paper, any such analysis can draw on the tools we have provided here. A further open question relates to the impact of the mechanical response of the detector to GWs, which may become relevant once the GW frequency lies above the lowest mechanical resonance mode. We leave this to future work. The achievable strain sensitivities we obtain by bootstrapping the axion searches still lie above any expected signals from astrophysical or cosmological sources. Nevertheless, the sensitivities are competitive with other experiments and proposals in this frequency regime. We aim with this work to join the worldwide effort of paving the way towards high-frequency GW detection.

(Acknowledgements, funding statements and the reference list are omitted here, as are appendices A through G — additional details of GW electrodynamics, scalar and axion electrodynamics, parity properties of external magnetic fields, the response matrix, explicit expressions for the effective current, recasting dark matter sensitivity to GW strain sensitivity, and GW detection with an electric field. Sections omitted for length; the complete text is at the source. Reference numerals have been removed from the body for readability.)