Gravitational wave probes of axion-like particles

We have recently shown that axions and axion-like particles (ALPs) may emit an observable stochastic gravitational wave (GW) background when they begin to oscillate in the early universe. In this note, we identify the regions of ALP parameter space which may be probed by future GW detectors, including ground- and space-based interferometers and pulsar timing arrays. Interestingly, these experiments have the ability to probe axions from the bottom up, i.e. in the very weakly coupled regime which is otherwise unconstrained. Furthermore, we discuss the effects of finite dark photon mass and kinetic mixing on the mechanism, as well as the (in)sensitivity to couplings of the axion to Standard Model fields. We conclude that realistic axion and ALP scenarios may indeed be probed by GW experiments in the future, and provide signal templates for further studies.


I. INTRODUCTION
The direct detection of gravitational waves (GWs) initiated a new era in cosmology and astrophysics while also opening up new avenues to explore fundamental physics in the early universe. In particular, axions or axionlike particles (ALPs) are a well-motivated extension of the Standard Model (SM), e.g. to solve the strong CP problem [1], to provide a dynamical solution to the electroweak hierarchy problem [2], to provide suitable inflaton [3] or dark matter candidates [4], or in the context of string theory [5].
Experimental searches for these particles are covering an increasing part of the parameter space. Several searches rely on the axion-photon coupling which is generically proportional to the inverse of the axion decay constant. This means the region corresponding to smaller decay constants (larger couplings) is more constrained, whereas larger values of the decay constant are usually difficult to probe.
In Ref. [6], we showed that axions or axion-like particles coupled to a light dark photon can produce a stochastic gravitational wave (GW) background when the axion field begins to oscillate in the early universe, allowing exploration of the parameter space inaccessible by experiments that rely on the axion-photon coupling. The rolling axion induces a tachyonic instability which amplifies vacuum fluctuations of a single gauge boson helicity, sourcing chiral gravitational waves. The energy transfer from the axion into light vectors also widens the viable parameter space for axion dark matter.
The goal of this paper is to explore the phenomenological impact of our findings. First, we show that gravitational waves can be produced in realistic axion and ALP scenarios where other couplings such as kinetic mixing a camachad@uni-mainz.de b w.ratzinger@uni-mainz.de c pedro.schwaller@uni-mainz.de d bstefan@uni-mainz.de of the visible and dark photon, couplings of the axion to SM fields, or finite dark photon masses are also present. Furthermore, we provide a simple analytic fit to the gravitational wave spectrum extracted from our numerical simulation which may be used for further studies or comparison with GW signals from other sources. We present the main result of our paper in Fig. 2, where we identify the regions of ALP parameter space that will be probed by future gravitational wave experiments. Since a strong polarization of the GW signal peak is a firm prediction of our scenario, in Fig. 3 we also indicate the region where this feature may be probed following the recent results of Ref. [7]. It is striking that gravitational waves may be able to provide evidence for axions with very large decay constants, which are otherwise inaccessible.

II. THE AUDIBLE AXION MODEL
Here we give a brief overview of the model presented in Ref. [6]. The original simplified model consisted of an axion field φ and a massless dark photon X µ of an unbroken U (1) X Abelian gauge group where the parameter f is the scale at which the global symmetry corresponding to the Nambu-Goldstone field φ is broken 1 . We assume this global symmetry is also explicitly broken at the scale Λ ∼ √ m f , which generates the potential V (φ) and a mass m for the axion.
While the expansion rate of the universe H = a /a 2 is greater than the axion mass m, the axion field is overdamped and does not roll 2 . In a radiation-dominated universe, H becomes of order m at the temperature T osc ≈ √ mM P , at which point the axion will begin to oscillate in its potential. At this time, we assume initial conditions expected from misalignment arguments where θ is the initial misalignment angle. The φX µν X µν coupling results in a non-trivial dispersion relation for the gauge field helicities that depends explicitly on the velocity φ of the axion field. As the axion field oscillates, one of the gauge field helicities will have a range of modes with imaginary frequencies (negative ω 2 ), resulting in a tachyonic instability that drives exponential growth. This process transfers energy from the axion field into dark gauge bosons and amplifies vacuum fluctuations of the tachyonic gauge boson modes into a rapidly time-varying, anisotropic energy distribution that sources gravitational waves. Because the first helicity to become tachyonic has the largest value of φ , it receives and keeps an exponentially enhanced advantage compared to the second helicity, resulting in a highly chiral dark photon population. We showed in Ref. [6] that this parity violation is also imprinted onto the gravitational wave spectrum. For an in-depth review of the particle production process and its applications, see Refs. . We will now briefly discuss some possible extensions to the original simplified model.

A. Finite dark photon mass
First, we consider the possibility of a non-zero mass for the dark photon which could arise through a dark Higgs or Stueckelberg mechanism. The main effect of m X is to modify the dark photon dispersion relation which can reduce the efficiency of or prevent tachyonic growth. To further quantify this statement, we go back to the analysis in Ref. [6], where we showed that the tachyonic growth of the mode functions becomes inefficient if they grow less than O(1) during one oscillation of the axion field. This happens when −ω 2 ± < (am) 2 is satisfied for all modes k. At this time, the scale factor has roughly increased by an amount (see Appendix A) a a osc ≈ αθ 2 From this expression and for αθ 10, we see that we require m X θαm/2 in order to have tachyonic production. Dark photons with masses well below this bound will not affect the success of our mechanism. If the produced dark photons redshift enough to become nonrelativistic, they might contribute to dark matter (see Section IV for details). Further discussion on the consequences of finite dark photon mass and its impact on the GW spectrum can be found in Appendix A.

B. Kinetic Mixing
Next, we examine whether the relevant photon-dark photon kinetic mixing operator affects our mechanism. Indeed, this operator will inevitably be generated by renormalization group flow if there exist states which carry both electromagnetic and U (1) X charge [36]. If kinetic mixing leads to an effective coupling of the dark photon to the SM radiation bath, one might worry that it induces a large thermal mass for the dark photon that prevents tachyonic growth.
In the case of an exactly massless dark photon m X = 0, the kinetic mixing term is unphysical as it can be removed via the field redefinition This redefinition leaves the coupling of the SM photon to the electromagnetic current unchanged, namely Thus, it is clear that only the field combination that couples to the SM plasma A develops a thermal mass. The orthogonal combination X is identified with the dark photon, which remains completely decoupled.
However, if the dark photon has a finite mass m X = 0, the mixing is physical. Diagonalizing the kinetic terms by performing the field redefinition Eq. (8) now leads to a non-diagonal mass matrix This mass term, as well as the thermal mass Π induced by the SM plasma for A must then be included in the dispersion relation, which reads The photon thermal mass is of order Π ≈ e 2 T 2 , which at the time when the axion begins to oscillate gives Π ≈ e 2 mM P . As discussed in Section II A, the existence of the tachyonic instability requires m X θαm/2. Futhermore, the momenta that experience tachyonic growth are those with k θαm, so we are deeply in the regime where m 2 X , k 2 Π. In this limit, the effective mass matrix in Eq. (10) always has a small eigenvalue m 2 X (1 + O( 2 )) which is independent of T 2 , despite the kinetic mixing [37]. This result, while perhaps surprising at first, becomes clear when we consider the limit m X → 0, where the dark photon must decouple. Thus, we conclude that the field combination associated with the dark photon X does not acquire a thermal mass via kinetic mixing, so we are subject only to the usual constraints on , see e.g. Refs. [38][39][40][41][42][43][44][45][46][47][48][49][50][51].

C. QCD Axion
Finally, we examine the case where the ALP φ is taken to be the QCD axion itself, which is the focus of Ref. [26]. In this limit, m and f are not independent parameters but are instead related by m 2 f 2 = χ QCD , where χ QCD = (75.5 MeV) 4 is the QCD topological susceptibility. In particular, the QCD axion has the following couplings to SM gauge bosons where G a µν and F µν are the gluon and photon field strengths, respectively, and g φγγ is a model dependent coupling, e.g. g φγγ = −1.92 α EM /(2πf ) in the KSVZ model [52,53]. Here, we note that none of these couplings spoil the effectiveness of our mechanism because the tachyonic growth of these states are regulated by plasma effects. The photon acquires a Debye mass of order Π ∼ e 2 T 2 via hard thermal loops, preventing tachyonic growth [54,55]. Similarly, the gluon self-coupling induces a magnetic mass m(T ) ∼ g 2 T [56 -58]. As a final consideration, model dependent couplings of φ to SM fermions also exist. However, the production of fermions is not exponential due to Pauli-blocking. Thus, the exponential production of dark photons dominates over SM channels.

III. GRAVITATIONAL WAVE SPECTRUM
Here, we present an improved computation of the GW spectrum as compared to the results of Ref. [6]. The computation requires the discretization of a double integral over the tachyonic momenta, resulting in the simulation time growing as the square of the number of gauge modes. By switching to a more memory efficient code written in Python, we were able to solve the coupled axion-gauge boson equations of motion using N = 10 5 gauge modes. For the GW spectrum computation, an N GW = 500 subset of these modes were taken, which is an order of magnitude improvement over our previous simulation. We show the results of the improved numerical calculation in Fig. 1.
The spectrum is strongly polarized in the peak region, whereas the tail is unpolarized as shown in the  [6]. Black is the total spectrum, whereas red and blue are the individual polarizations. Green gives the "conservative spectrum" as described in Section III.
figure. Furthermore, the dashed green line indicates a "conservative spectrum", which is the part we expect to remain even if the process of gauge bosons backscattering into axions results in a strong back-reaction. This effect, which we neglect here, induces inhomogeneities in the axion field which can end the energy transfer from effects which rely on coherent resonance of gauge modes with the zero-momentum axion condensate. However, we expect the gravitational waves produced during the initial tachyonic instability phase to survive, and we use the estimate for the closure of the tachyonic band given in Ref. [6] to obtain the conservative spectrum shown in Fig. 1.

A. GW Spectrum Fit Template
To make connection with experimental searches for stochastic GW backgrounds, we use the improved numerical simulation to extract a GW signal template. Such a template enables simple estimates of the GW spectrum and signal-to-noise (SNR) calculations for a given set of model parameters, without having to run a complicated numerical simulation. We approach our GW signal template from the ansatz that the low frequency part of the GW spectrum is given by a power law while the high frequency part falls off exponentially, with some transition region that gives the peak. A reasonable ansatz of this form is whereΩ GW ≡ Ω GW (f )/Ω GW (f peak ) andf ≡ f /f peak . In Ref. [6], we derived simple analytic scaling relations for the peak amplitude and frequency of the GW spectrum, where these expressions hold for α ∼ 10 − 100. The parameters A s and f s are fit to the GW spectrum from our numerical simulation to correct for the O(1) factors by which the scaling relation is off. The parameter p specifies the power law index and γ controls how quickly the exponential behavior takes over at high frequencies.
Discussion of the fit to the simulation and the best fit values for the parameters A s , f s , γ, p can be found in Appendix C.
Together, Eqs. (12) and (13) allow one to go directly from the underlying fundamental model parameters α, m, f to the GW spectrum. We note that a nonzero mass for the dark vector has the effect of providing a low-frequency cutoff to the dark photon power spectrum. However, as discussed in Appendix A, there is no significant effect on the GW spectrum as long as m X m/2.

IV. PROBING AUDIBLE AXION MODELS
With the results of the previous section, we can now identify the regions of parameter space that may be probed by future GW experiments. Detectability requires an SNR above a certain experiment dependent threshold. Using the signal template from Section III A, we obtain the SNR ρ for a given parameter point using where Ω eff is the effective noise energy density of a given experiment and T obs is the observation time in seconds. For more details on the computation and the recommended threshold values for each experiment, we refer to Ref. [59]. Our results are shown in Fig. 2, where the detectable regions lie below the curves labeled as SKA, LISA, BBO, DECIGO and ET, respectively. Interestingly, GW experiments are most sensitive for large values of f corresponding to very weakly coupled axions. These probes are therefore highly complementary to other existing limits (orange shaded) or planned searches (orange lines), which are typically more sensitive for larger couplings. An exception is the constraint coming from black hole superradiance, which is also most reliable for large f and also indirectly relies on GW observations [60,61]. It should also be emphasized that the GW signal regions do not depend on the axion relic abundance today, and therefore do not require the axion to account for all of dark matter. For experiments which probe the axion-photon coupling g φγγ , we assume the KSVZ relation g φγγ = −1.92 α EM /(2πf ) to convert between g φγγ and 1/f . The non-decoupling behavior of the GW signal is due to the fact that larger f corresponds to more energy in the axion field Ω osc φ ∝ m 2 θ 2 f 2 which is available to be converted into gravitational radiation. This holds as long as the initial misalignment angle θ takes on natural values of O(1). Additionally, we are always assuming m X m/2 and α ∼ 10 − 100 such that the particle production process is efficient, see e.g. Refs. [6,26].
In Fig. 3 we show a close up of the parameter space that leads to detectable signals, as well as bounds arising from cosmology. If the dark photon is massless or sufficiently light to stay relativistic until recombination, the number of effective relativistic degrees of freedom N eff sets an upper bound on f . A simple estimate can be done assuming that all the energy in the axion field is converted into dark gauge bosons. This leads to a bound of f (5 − 7) × 10 17 GeV shown in Fig. 3, depending on whether the axion starts oscillating before or after the QCD phase transition.
The dark photon might also become non-relativistic before recombination and therefore contribute to dark matter, as in Refs. [28][29][30][31]. The shaded green region of Fig. 3 shows the potential parameter space for vector dark matter, which is cut off at the lower bound of f ≈ 3 × 10 16 GeV where the dark photons are too hot to be compatible with structure formation. For additional information on the derivation of these bounds, see Appendix B. The diagonal gray lines indicate how much the axion abundance must be suppressed compared to the ordinary misalignment case to avoid over-production. While the exponential production of dark photons can lead to a very strong suppression [6,26], it is currently unclear whether a suppression by more than three orders of magnitude persists if backreaction effects are taken into account in the equations of motion [25]. Otherwise, other mechanisms for depleting the axion abundance might be needed in the bottom right region of parameter space.
A smoking gun for Audible Axion models is the completely chiral nature of the peak of the gravitational wave spectrum, inherited from the parity violation in the dark photon population. Specifically, the peak region is generated by the parallel addition of dark gauge bosons with the same helicity, resulting in a gravitational wave of the same polarization. This can provide powerful background rejection, since SGWBs from astrophysical sources are not expected to carry a net polarization.
Conventional wisdom has been that planar GW interferometers, such as LISA or Einstein Telescope (ET), cannot measure the polarization of an isotropic stochastic GW background [62][63][64]. However, it was pointed out that the dipolar anisotropy induced by the Doppler shift due to the relative motion of our solar system with respect to the cosmic reference frame can be exploited to allow planar detectors to detect net circular polarization [65,66]. In particular, LISA and ET would be able to detect net circular polarization with an SNR of O(1) for a SGWB with amplitude h 2 Ω GW ∼ 10 −11 [7]. In Fig. 3 we indicate the region in parameter space where the signal is strong enough such that LISA and ET can pick up on the polarization following the analysis of Ref. [7].

V. DISCUSSION AND CONCLUSIONS
In [6] we showed that a stochastic gravitational wave background can be produced by an axion coupled to a dark photon, specifically by the tachyonic instability induced in the dark photon by the axion dynamics. This instability leads to exponential growth of dark photon vacuum fluctuations which act as the GW source.
Here, we have shown that this GW signal is also produced for a broader class of models that allow for a massive dark photon and/or kinetic mixing with the SM photon. Furthermore, we argue that couplings of the axion to gluons and photons do not affect the success of the mechanism, which illustrates the viability of the QCD axion case.
The central results of our paper are Fig. 2 and Fig. 3, where we show the regions of axion parameter space that may be probed by future GW experiments. Since the GW signal is strongest for large f , GWs probe complementary regions of parameter space to most other experiments, which rely on sizeable couplings of the axion to the visible sector that are suppressed by 1/f . In Fig. 3, we zoom in on the GW signal region and show cosmological constraints as well as the region where the dark photon itself could be dark matter. For most of the parameter space relevant for GW detectors, the axion relic abundance needs to be strongly suppressed, which might require an extension of the model.
Since the shape of the GW signal is universal for dark photon masses less than roughly the axion mass, we provide a fit template function which parameterizes the dependence of the GW amplitude and peak frequency on the axion mass m and decay constant f . This fit, which is extracted from our new simulation with an order of magnitude higher mode density, provides a quick translation from the underlying model parameters to the detectability of the GW signal for all experiments and parameter points, providing a tool which can be directly used by experimental collaborations to probe our and similar models.
We call the minimum speed at which particle production becomes kinematically allowed |φ min | ≡ 2f am X /α. Tachyonic particle production will end when φ drops below this value, at which point the most tachyonic scale is Before the backreaction from particle production be- comes strong, φ scales roughly as φ ≈ θf (a osc /a) 3/2 am, so we can estimate the value of the scale factor at the time when kinematics shuts off the particle production process as at which time the most tachyonic scale is We expect this scale to provide a hard low-k cutoff in the gauge power spectrum. Indeed, we show in Fig. 4 the dark photon power spectrum for different values m X , where Eq. (A5) provides a good description of the low-k cutoff.

Tachyonic Band
Following the analysis of Ref. [6], the tachyonic band k − < k < k + is found by solving for the tachyonic modes with growth timescales less than the conformal oscillation time. Specifically, it is given by solving ω 2 = −(am) 2 , with the result which reproduces the massless result when m X → 0. The band closes once the scale factor has increased by an amount a a osc = αθ 2 which gives the result quoted in Section II A. Because the scale factor for tachyonic band closure approaches the kinematic closure value for m X /m 1, one can see that the tachyonic band always closes before kinematics shut off the tachyonic production. The most tachyonic scale at the time of tachyonic band closure is which explains why the spectrum starts to fall off before the hard kinematic cutoff given by Eq. (A5), since modes below this scale are not efficiently produced.

Gravitational Wave Spectrum
In the limit αθ 10, requiring a/a osc > 1 in both Eqs. (A5) and (A8) gives the condition that is required in order to have any tachyonic production. However, to keep the tachyonic band open until the scale factor has grown by an order of magnitude (as is typically required to produce an observable GW signal), we require which evaluates to m X m/2 for αθ ∼ 50. We show in Fig. 5 the effect of a massive dark photon on the GW spectrum. One sees that for m X m/2 that the GW spectrum is largely unaffected. For larger values of the dark photon mass, the tachyonic instability is not as efficient and less energy is transferred from the axion into dark photons and gravitational waves. Thus, the net effect is to damp the GW spectrum.
where the energy density of SM and dark photons are ρ γ and ρ X , respectively. Since both species scale as radiation, the fraction ρ X /ρ γ only changes when SM fields become non-relativistic and transmit their entropy to the photon bath. Before transferring its energy to dark radiation, the oscillating axion field redshifts as matter and therefore grows ∝ a compared to the SM radiation bath. Following the analysis of Ref. [6], we assume the majority of the axion energy is transmitted to dark radiation close to the closure of the tachyonic band. For the parameters in question (θ ≈ 1 and α ≈ 50), tachyonic band closure occurs when the scale factor has grown by an amount a * a osc = θα 2 Fits to the GW simulation data (magenta) using the template given in Eq. (C1) evaluated using the best fit parameters in Table I. The plots are as follows: Total spectrum (black, top), Dominant chirality spectrum (red, center), and Conservative spectrum (green, bottom).