Gravitational Waves and Possible Fast Radio Bursts from Axion Clumps

The axion objects such as axion mini-clusters and axion clouds around spinning black holes induce parametric resonances of electromagnetic waves through the axion-photon interaction. In particular, it has been known that the resonances from the axion with the mass around $10^{-6}$eV may explain the observed fast radio bursts (FRBs). Here we argue that similar bursts of high frequency gravitational waves, which we call the fast gravitational wave bursts (FGBs), are generated from axion clumps with the presence of gravitational Chern-Simons (CS) coupling. The typical frequency is half of the axion mass, which in general can range from kHz to GHz. We also discuss the secondary gravitational wave production associated with FRB, as well as the possible host objects of the axion clouds, such as primordial black holes with typical masses around $10^{-5}M_{\odot}$. Future detections of FGBs together with the observed FRBs are expected to provide more evidence for the axion.


I. INTRODUCTION
The first observation of the gravitational waves has started a new era for astrophysics, cosmology and even particle physics. Searching for gravitational wave signals in the universe at different frequencies from different sources has started, from the primordial gravitational waves at 10 −16 Hz to the pulsars and binary system signals up to 10 4 Hz. What's more, the higher frequency up to MHz or GHz gravitational wave searches have also been proposed [1,2], and those signals can be generated by the axion annihilation around the black holes [3][4][5][6]. The axion is a well-motivated dark matter candidate, though the distribution of such dark matter around the universe remains an unknown and interesting topic. Considering that in early universe evolution the density variations of both matter and dark matter occurred and acted as the seeds for cosmic structures today, there are great reasons to believe that axion distributes in cosmic space unevenly.
The axion was proposed as the pseudo-Nambu-Goldstone boson from the spontaneous breaking of a U(1) global symmetry, the Peccei-Quinn(PQ) symmetry, to solve the strong CP problem in QCD [7,8]. It was later realized that there can be more Axion-like pseudo-Nambu-Goldstone bosons related to various global symmetry in UV theories [9,10]. If this spontaneous breaking of global symmetry happened before or during inflation, then the axion field value can be considered homogenous across the observed Universe, except for small iso-curvature and density perturbations. In this case, the axion density variation is not generically large. If * corresponding author: sichunssun@bit.edu.cn † corresponding author: zhangyunlong@nao.cas.cn the symmetry was broken after inflation, then the axion field can only be correlated across the local horizon size. Later when the temperature dropped to the order of the axion mass T ∼ m a , the axion began to oscillate at the bottom of the potential, which is acquired by nonperturbative effects such as instanton. At the same time axion topological defects decay. The oscillation is not coherent across the horizon. Notice that during the universe evolution history, the axion mass also depends on the temperature of the universe. These different horizon patches of the axion may have large density variations.
Some of the initial axion density variations grow under gravity, eventually collapse into denser objects and go through Bose-Einstein condensation [11][12][13][14][15][16][17][18][19][20][21] or even form black holes [22,23]. Another type of axion objects is the axion clouds around the spinning black hole. It is produced from the black hole superradiance or simply gravitational accretion, which can induce black hole instability [22]. The typical size and radius of the dense axion object depend on the mass of the axion, the decay constant Λ a , assumptions on the initial density fluctuations and the thermal dependence of the axion mass. The masses of such dense objects range from 10 −18 M ∼10 2 M [12 -17]. The formation and evolution of such objects include some highly non-linear process and can be studied by numerical calculation. Furthermore, if the axion self-interaction is included or the density is very large, such axion dense objects can go through Bose-Einstein condensation and become even denser. Depending on the model, the average size can be approximated in some case around 10 2 km for QCD axion, and could be possible candidates for the observed fast radio bursts (FRBs) [12][13][14][15][16][17]24]. Moreover, there are various kinds of multi-messenger signals of the axion-like dark matter [25][26][27][28][29], and the recent supermassive black hole image from the Event Horizon Telescope (EHT) can be used to probe the existence of ultralight bosonic particles that accumulate through the black hole superradiance effect [30,31].
Here in this paper, we study a scenario that high energy bursts or periodic bursts of gravitational waves can be generated around the axion objects, including axion mini-clusters and axion clouds around the black holes, with the gravitational Chern-Simons (CS) couplings between axion like particle and gravitons [32,33]. It is wellstudied that the electromagnetic radio bursts can be produced through similar processes, from the axion-photon couplings. We do a similar analysis with the gravitational CS coupling and the gravitational wave production from the drastic axion/photon energy oscillation and discuss their properties on the frequencies, strains, duration and the energy released. Such bursts have distinct signatures and interesting consequences and we can look for them with proposed different types of gravitational wave detectors [1,2]. This paper is organized as follows. In section II we discuss the cases with the axion clouds around the primordial black holes. In section III, we turn on both axionphoton and gravitational CS couplings and discuss different sources and gravitational strains produced. We end with discussions and comments in section IV. In the appendix A we review the FRB discussion for axion-photon couplings. In appendix B we calculate the gravitational waves induced by gravitational CS couplings on a flat background and discuss various features of such gravitational waves.

II. AXION CLOUDS AND PRIMORDIAL BLACK HOLES
In this section, we show that for the axion with mass µeV (QCD axion) surrounding the black holes, the mass regime of the black holes that the axion clouds can form is between 10 −7 M and 10 −3 M , supporting the parameters used in the previous sections. Rather than the stellar mass black holes, the masses of primordial black holes mostly fall into this mass region. See e.g. [34,35] for the formation of primordial black holes and recent constraints.
We define the typical scale of the black hole R BH = G N M BH /c 2 and the wavelength of the axions λ a = /(m a c). Then define the dimension-less number Based on the derivations in [3][4][5][6], for the near extremal rotating black hole, the formulation times of the axion clouds are estimated as τ a↑ 10 7 e 1.84α R BH , α 1, τ a↓ 24 α −9 R BH , α 1.
which are required to be smaller than the age of the universe ∼ 10 10 years, rescaled to be comparable to the ax-ion clouds formation time: In Figure 1, we plot the allowed mass range that the axion clouds can form for primordial black holes. In the shaded area, the upper bound is given by τ a↑ < τ U , and the lower bound is given by τ a↓ < τ U , with the expressions (2)(3)(4). Thus, for the clouds of axions with masses around µeV, the typical masses of host primordial black holes are around 10 −5 M , which in agreement with the early estimation in [36]. More intriguingly, it matches with the mass estimation of the novel object "Planet 9" with a mass 5 − 10M ⊕ in the outer Solar System [37,38], which is just around the order of 10 −5 M .

III. AMPLIFIED ELECTROMAGNETIC WAVES AND GRAVITATIONAL WAVES
In this section, we would like to discuss both the electromagnetic waves and gravitational waves as well as the interference effects coming from axion clumps. Especially, we consider the branching factor of the axion energy bursts into both waves and the gravitational waves created by the photon bursts. The total action of CS modified gravity with axion photon coupling is given by where L a , L aRR and L aFF are given below: whereR α γδ β ≡ 1 2 γδµν R α βµν andF µν ≡ 1 2 µνλρ F λρ . Again, we assume that the axion field a is only timedependent. Except for the axion resonance, there are two other possible sources for the gravitational waves, although they are expected to be much smaller than the resonance. Those two sources are the stress energy tensors of the Maxwell field and axion field. The equation of the motion becomes: where T (γ) At the meanwhile, the equations of motion for the electromagnetic field and axion field become It requires involved numerical studies to solve the coupled equations, especially around the axion objects.

A. Estimation of the branching ratios
In the following, we estimate the strength of both signals. One key property we would like to mention is the branching ratios from axion to both photons and gravitons, assuming both couplings turned on. Considering both photon and graviton are massless and have only two degrees of freedom, we have at the tree level at the zero temperature, Considering that the gravitational CS coupling α g is much less constrained than the photon-axion coupling α γ , we can expect that the energy of bursts can go to the gravitational sector dominantly. For example, if we take the typical values in α γ 10 −2 /f a ∼ 10 −14 GeV −1 . and ∼ 10 110 . See the appendix for more discussion on the constraints of these couplings. That means the gravitational wave bursts associated with the FRBs can be tremendously large. Observing this will put a new constraint for α g , apart from the current observation.
It is interesting to discuss the CS gravity coupling a little more here. In UV-completed models, the CS gravity and photon-axion couplings are considered at the same order from compactification [39]. However in recent study [40], as well as our concern here, the CS gravity is treated as the low energy effective field theory with a cut-off. The existence of the ghost mode constrains the theory in certain parameter space, see [40] for an interesting discussion. We refer to Fig. 2 to show some connections between two different kinds of CS couplings. This triangle diagram is divergent with 4 extra powers of momentum from aRR vertex and one power of momentum from each h µν T µν coupling (see the interaction forms in [41]), and evaluated as α γ ∼ α g (Λ c /M pl ) 4 , where Λ c is the cut-off for Chern-Simons theory. The cut-off scale Λ c is considered to be much lower than the Planck scale, which implies α g /α γ ∼ (M pl /Λ c ) 4 1. Moreover, since the photon-axion coupling is a lot much constrained, this diagram can also be used for constraining the effect of the low energy Chern-Simon gravity theory. Besides, aRR also induces an axion self-energy loop with two gravitons running in the loop. This diagram is also divergent and renormalizes the axion mass term. With a much lower cut-off scale Λ c M pl , the one loop diagram provides the sub-leading correction to the axion-photon coupling.
Here we only focus on the first triangle diagram, and assume that the axion-photon coupling is generated from axion-graviton coupling. For the aRR term, people have mostly studied its impact on the binary systems in the early inspiral phase, above the LISA sensitivities. The released power P (g) of the gravitational wave bursts can be related to the strain h (g) at the detector through h 2 (g) ∼ κ4P (g) [3][4][5][6]. From the FRB information, the power P (γ) is roughly 10 42 ergs in 10 −3 seconds [13][14][15][16][17]. Assuming that P (g) /P (γ) The branching ratio in (12) implies Then it is possible that h (g) can be greatly enhanced at the same frequency of photons ν = k * /2π = m a /4π ∼ GHz. Though we need to assume a much larger axion clump, instead of the size of mini-cluster estimated only from the power P (γ) ∼ 10 −12 M /ms of the FRBs. Now let us estimate the secondary effects of the gravitational waves productions from the photon burst process, with the source given by the stress energy tensor T (γ) µν of the Maxwell field in (9). The discussion of the stress-energy tensor of axion field T (a) µν in (9) is similar and generalization is straightforward. The quadrupole of the stress energy tensor is defined as The quadrupole integration is over the size of the axion object. And for the detector at the distance of L from the source and assuming Notice that a sphere has zero quadruple moments and only out of the equilibrium decay process of the axion clump can induce the gravitational waves. We approximate is the total observed released energy of the photons in FRBs, and E (γ) is the non-spherical contribution. Then h ij ∼ πκ4 L E (γ) d 2 a ν 2 , and we will compare this to the direct gravitational wave production from the CS gravitational term later. Considering ∼ 10 −8 , the strain h (a) and h (γ) that our detector can receive are bounded by the dimensionless number Another way to estimate the quadrupole moment is considering the non-spherical contributionÏ Notice that the typical frequency k are m a and 2m a respectively for h (γ) and h (a) , generated from different sources T We plot the schematic diagrams of these possible signals in Fig. 3. In particular, we expect the signals around the spinning black hole is even stronger [22] and may be within the range of the high frequency gravitational waves detection proposals [1,2]. When the size of the black hole is at the order of 1/(m a v a ), we have the superradiance effects and the axion clouds form. For m a ∼ GHz gravitational waves, the corresponding black hole mass is around 10 where c is the portion of the energy stored in the black hole clouds, and can be taken as 10 −4 . It is foreseeable that CS term corrects the black hole superradiance effects by some factor, e.g. see [33], or induce Kerr black hole instability [42], although the overall order should not change much.  (13) have been taken as αg/αγ 10 10 and L 1kpc. The light blue line is the expected strain sensitivity of advanced LIGO [43], and the dashed blue line is based on the proposed sensitivities in [44] One can also see more discussions on h (a) in a black hole background in [1,2], corresponding to a a → g annihilation. Also, notice here the estimations for the stain h are very rough, since the collapsing and bursting processes are actually very complicated. The more precise estimation will require some involved numerical studies. From the estimations above and related pulsars or supernovae burst processes informations, these secondary gravitational effects without the CS gravity terms can release around 1% of total released energy.
Above the strains are estimated for GHz fast gravitational wave bursts. We expect that h (a) ∼ h (γ) can be different from h (g) for MHz gravitational waves, and notice that the normal drastic FRB energy oscillations induce gravitational bursts h (a) ∼ h (γ) as well without the CS gravitational coupling. For aFF the particle decay process a → γγ or aa → γγ gives rise to a gravitational wave signal directly, although the signal is very small without the presence of a black hole. The authors of [1,2] discussed this kind of gravitational wave production enhanced around the black hole. Especially for the black hole superradiance the process a + → a − + g happens, where a superradiant cloud axion emits a graviton and jumps onto a lower level, then may get captured by the black hole horizon. A benchmark frequency of this process can be 10 −2 Hz for a 10 7 M black hole and a 10 −17 eV axion. And this could possibly be detected by LISA. Considering the distance between the detector and the source L, the strain is given by ∼ 10 −22 10 −2 Hz

IV. DISCUSSION AND SUMMARY
We have studied the gravitational wave bursts from the CS coupling with axion and graviton as well as the associated FGB from FRB sources without the CS term, though we only performed the calculation on the flat background to demonstrate the features of the signals. For the case with axion clouds around the spinning black hole, the situation is more complicated. Especially the black hole superradiance can keep on copiously producing axions, and induce the instability around the axion clouds, where the gravitational bursts/radio bursts can happen periodically.
We would also like to comment on the fact that the CS gravity has a ghost in the UV, which can be fixed by higher order terms, e.g. [45][46][47]. Moreover, our discussions, even around the black holes, are in the IR region, and it is perturbative in the gravitational release case. The parametric resonance effects of both electromagnetic and gravitational fields are important ingredients. Especially the gravitational wave bursts can be dominant in the energy release. In that case, the radio burst signals can be a reference point for the gravitational releases. Notice that the CS gravity coupling can be much larger than the axion-photon coupling, due to the limited studies and experiments on the bound.
For the cosmological inflation with axion term, the GW signal/photon signal is usually circularly polarized, since one mode is growing and the other mode is decaying. This is due to the fact that the background field dθ dt is with the fixed sign. However, in our case with the oscillating background θ ∝ sin(mt), both of modes are growing exponentially. So the signals do not have a certain polarization. The oscillating background only modifies the phase factor of the signal modes, e.g. see [48] for a related discussion.
In summary, the gravitational wave burst signals proposed here happened in the late universe and can be strongly enhanced by the gravitational CS couplings inside the dense axion regions. Its characteristics include high frequencies and burst features within a short period. For this process around the spinning black holes, we expect some technical issues to overcome [49]. The spinning black hole solution is corrected in CS gravity. And the black hole accretion process and the axion cloud dynamics are modified from this enhanced decays of a → gg. Detecting the gravitational waves of high frequencies around kHz to GHz is also an interesting topic for future detectors. In addition, there could be entangled signals from particles decays, which might be detected with the HBT interferometers, as suggested in [50,51]. For the discussion on gravitational waves from light primordial black holes one can refer to [52], and recent progresses on the FRBs and resonant instability as well as superradiance around primordial black holes can also be found in [53][54][55][56][57][58][59][60].

Appendix A: Axion dense objects and fast radio bursts
Let us consider the model of electromagnetic and axion field with an axion photon interaction term, where F 2 ≡ F µν F µν and the electromagnetic tensor The Lagrangian density of the axion field a is given by The usual choice of the potential is V (a) = m 2 a f 2 a 1 − cos a fa , where m a is the axion mass and f a is the axion decay constant. We are interested in the region a f a , such that the potential in (A2) is approximately V (a) 1 2 m a a 2 . The interaction term between the axion and Maxwell field is whereF µν ≡ 1 2 µνλρ F λρ . The current experiments are now probing the parameter region 10 −16 GeV −1 α γ 10 −8 GeV −1 and m a 10 4 eV and part of the region has already been ruled out [3,25]. For the QCD axion, we have to impose the condition f a m a = f π m π , where f π and m π are the decay constant and mass of the pion, respectively. However, for general axion-like particles, the relation between f a and m a is model dependent and we consider two parameters as independent of each others. For a model of largely homogenous axion vacuum dark matter and considering that the oscillating axion accounts for no more than all dark matter, we have the relic abundant constraint f a θ i −2 × 10 12 GeV with θ i ∼ O(1) for initial misalignment.
Consider the flat background ds 2 = η µν dx µ dx ν , where η µν = diag[−1, 1, 1, 1]. From the action in (A1), the equations of motion for the axion field and Maxwell field are We consider the case that the effects from axion coupling term is very small, such that the right-hand side of (A5) can be ignored at the background level. Then we assume a coherent oscillation of the axion field as below, where ϑ 0 ≡ f a θ i . Without loss of generality, we will adopt the phase φ 0 = 0. Note that for now, we do not consider the spatial dependence of the axion field value for simplicity, although the axion field today is assumed to be incoherent across space. It means that any large scale resonance is nearly impossible. The incoherence is due to the realization of the axion oscillation in the early universe, and the axions field at the different spatial points may pick up different phases factors. However, the local axion object can play an important role in this electromagnetic amplification effect and has been discussed extensively, see e.g. [12][13][14][15][16][17]. Thus, we consider the coherent solution (A7) above as the background solution.
We assume that the gauge potential A µ (t, z) is homogeneous in x-and y-directions and define the helicity ± modes A ± (t, z) ≡ [A x (t, z) ± iA y (t, z)]/ √ 2. In the Coulomb gauge, the equations of motion for the Maxwell field (A6) are reduced to the formula With the ansatz of the axion field in (A7),ȧ(t) is a cosine type function. Consider the following mode, In the momentum space, the equation of motion (A8) is reformulated as Redefine τ ≡ m a t/2, we can write down (A10) as 2 ). The resonance can be achieved when a decays to the diphoton, at the resonance frequency k * = m a /2. And a typical amplification happens when µ k t > 1.
Consider the size of the axion stars or mini-cluster with average size d a as the light travels through, then the total where v a is the typical velocity inside of the clumps and the typical relation d a 1/(m a v a ) where the equilibrium cluster has been used [12][13][14][15][16][17]. Thus, the amplification factor for the Maxwell field is e Γγ tγ , with We can see that the explosive decay of the axion clumps or axion clouds can happen, with certain axion parameters.
At the meanwhile, we have the dynamic equation of motion for the axion field (A5). The complete solution of both axion and electromagnetic fields dynamics requires numerical studies. For example, see [22] for the curved background solution. Here we only present the qualitative description of the process that axion oscillation induces a burst of energy in the electromagnetic sector at the resonance frequency and the axion fields lost energy. With the ansatz of the axion field with the typical wave number k * a(t, z) =ā(t) + δa(t, z), the spacial factor in (A5) can be decomposed. It is enough to see a couple of features in this effect to explain the FRBs. In Figure 4, we plot the schematic diagrams of the amplification of electromagnetic waves. The energy density of the photons ρ γ = T When the Maxwell fields pass through these axion dense objects or the axion clouds, due to the photonaxion interaction, the coherent oscillation of axions amplifies the Maxwell field. The result is an explosion of the photon number in the resonance frequency k * = m a /2, as in the mechanism a → γγ described in the introduction. This effect has been amazingly linked to the observed multiple FRBs in the galaxy, in both the axion dense object case [12][13][14][15][16][17] and black hole superradiance [22,29,31]. Especially, it is for the peak frequency, with a benchmark point frequency around GHz(∼ µeV).

Appendix B: Axion amplified gravitational waves
In this section, we discuss the similar amplification effects of gravitational waves in the dynamical CS modified gravity [32,33]. The total action is given by where κ 4 = 8πG The Lagrange density of the axion field L a is the same as in (A2), and the CS term is given by whereR α γδ β ≡ 1 2 γδµν R α βµν . We consider the flat background with the gravitational wave perturbation h ij , And we take the transverse and traceless (TT) gauge, h i i = δ ij h ij = 0 and ∂ i h ij = 0. Assuming that a is only time-dependent as in (A7), the linearized Einstein field equations are given by [33,[61][62][63][64], where = ∇ µ ∇ µ and˜ pki = tpki . Let us now concentrate on gravitational wave perturbations, for which one can make the ansatz The circular polarization tensors are defined as e R kl = where ε R = +1 and ε L = −1. Then in the momentum space with I = R, L, (B4) becomes At the meanwhile, the dynamical equation of motion for the axion field is |h Top: the amplitude growth of the strength of the gravitational wave, which is normalized by the initial value of h0. Bottom: the growth of energy densities of axion ρa and GW ρg, which are normalized by the initial value of GW ρ0 ≡ ρg(t = 0). Notice the different growth phase for the R, L modes of the gravitational wave is due to the parity breaking of the gravitational CS term. Now we can discuss the bursts of the gravitational waves around the axion dense object or the axion clouds around the black hole. We use the conventions in [63,64], and the typical peak frequency of such GW is around k * = m a /2 with a bandwidth κ 4 α g ϑ 0 m 3 a 4 . It corresponds to the process a → gg, where an axion decays into two gravitons. This is a different and more drastic process comparing to two axion annihilation to the graviton process proposed in [1,2], where no gravitational CS coupling is assumed.
The duration of the energy released in such burst is related to the axion objects and clouds properties. In the FRB cases, time duration is very short in milliseconds and the energy released is around 10 38 ∼ 10 40 ergs with high flux densities [13][14][15][16][17]. If we assume the same clumps formed by axions also has the gravitational CS couplings, a portion of the axion energy, if not all, can be released to the gravitational waves. Notice here these two couplings are not related to each other, without assuming certain UV theory. If we assume such axion-like particles only have the gravitational couplings rather than photon couplings, we can expect the gravitational wave bursts with the similar total energy released as the FRBs, although the duration might be different, depending on the CS coupling.

The amplification factor
The amplification factor is another important parameter of such phenomenon, especially for the expected signal to background ratios. The gravitational waves traveled through the axion objects pick up an exponential factor, while the other gravitational waves remain the background. We can estimate that for the peak frequency, the exponentiated factor is e Γgtg , where For an estimation of the order, we have Γ g m a ∼ κ 4 α g 1eV −3 m a 10 −9 eV 2 ϑ 0 10 9 GeV .
We can now see that the observation of such bursts can put severe constraints on the gravitational CS coupling and axion properties. We take the current upper bound of CS modified gravity coupling from [33], that 10 11 m 5 × 10 17 eV −1 . In our notations, it turns out to be Thus, if taking κ 4 α g ∼ 10 −4 eV −3 , m a ∼ µeV and ϑ 0 ∼ 10 9 GeV in (B9), we can see that the current upper bound of Γg ma is around 10 2 , which means a very large amplification factor e Γgtg ∼ e 100/va , considering that v a ∼ 10 −3 . In Figure 5, we plot the schematic diagrams of the amplification of the gravitational wave. The energy density of the photons ρ g = T (g) tt are read out from the stress energy tensor of the gravitational wave T (g) µν = c 4 4κ4 (∂ µ h ij )(∂ ν h ij ).

Various Gravitational waves signals from axions
For the early universe, one thing we would like to comment here is the gravitational wave production from aFF or aRR terms during inflation. They are at the frequency of 10 −16 Hz to 10 −10 Hz, which has been discussed in [33] and [65][66][67][68] for the interests of the primordial gravitational waves. For both cases, one can solve the EM or GW modes in an FRW geometry and take the inflationary background. For the electromagnetic case with aFF , the growing modes of the EM waves act like the sources for the background cosmological scalar fluctuations, which give rise to the near scale-invariant power spectrum and can be probed by the CMB and large scale structure. Especially, the tensor to scalar ratio becomes r ξ = 8.1 × 10 7 H 2 . Moreover, this tensor fluctuation can be detected in much smaller scales when these modes exit the horizon close to the end of inflation. And a stochastic gravitational background can be possibly detected by advanced LIGO/VIRGO [69,70]. For the aRR term one has the gravitational wave production directly, one needs to solve the field equations in an inflationary background. In the regime of Θ ≡ 2αgH 2 Mp √ 2 < 10 −5 with the slow roll parameter , one can reach the tensor to scalar ratio rCS rnon-CS ∼ 1 + 0.022Θ 2 . For the larger Θ the analysis has some technical difficulties, for example, the ghosts arise and the result is unknown currently [33,71].