Detecting Axion Stars with Radio Telescopes

When axion stars fly through an astrophysical magnetic background, the axion-to-photon conversion may generate a large electromagnetic radiation power. After including the interference effects of the spacially-extended axion-star source and the macroscopic medium effects, we estimate the radiation power when an axion star meets a neutron star. For a dense axion star with $10^{-13}\,M_\odot$, the radiated power is at the order of $10^{11}\,\mbox{W}\times(100\,\mu\mbox{eV}/m_a)^4\,(B/10^{10}\,\mbox{Gauss})^2$ with $m_a$ as the axion particle mass and $B$ the strength of the neutron star magnetic field. For axion stars occupy a large fraction of dark matter energy density, this encounter event with a transient $\mathcal{O}(0.1\,\mbox{s})$ radio signal may happen in our galaxy with the averaged source distance of one kiloparsec. The predicted spectral flux density is at the order of $\mu$Jy for a neutron star with $B\sim 10^{13}$ Gauss. The existing Arecibo, GBT, JVLA and FAST and the ongoing SKA radio telescopes have excellent discovery potential of dense axion stars.


Introduction
The QCD axion, as a byproduct of a solution to the strong CP problem [1][2][3][4][5][6][7], can also serve as a cold dark matter of the universe [8]. The axion is the pseudo Nambu-Goldstone boson corresponding to the spontaneous symmetry breaking of the Peccei-Quinn (PQ) symmetry. The PQ symmetry is slightly broken by the chiral anomaly, and hence the axion acquires a tiny mass through the physics at the QCD scale. The mass of the axion can be estimated as m a f a ≈ f π m π ≈ (10 8 eV) 2 [9], where f a(π) and m a(π) are the decay constant and mass of the axion(pion). The minimum of the axion potential corresponds to the CP conserving coupling constantθ = 0, and the strong CP problem is dynamically solved by introducing the new particle. Moreover, in the evolution of the universe, the universe may experience the phase transition of PQ symmetry breaking. At a very high temperature, the axion is essentially massless with all possible values ofθ and the axion field takes value which is different from θ = 0. When the universe cools down to a temperature around the QCD scale, the axion mass sits in and becomes comparable to the Hubble expansion rate. The axion field starts to oscillate around its minimum and behaves as a condensate of axions [10][11][12]. This vacuum misalignment mechanism results in the coherent oscillation of axions, which can explain the observed dark matter energy density. 1 If the PQ symmetry breaking occurs after the end of the inflation, the axion field value takes random value for each Hubble volume. This large spacial fluctuation of the axion field is further enhanced by the gravitational instability. As a result, the axion mini-cluster [16][17][18] is very likely to be formulated, and it becomes axion star after the gravitational thermalization [19,20]. Depending on the energy density, there are two types of axion star: diluted axion star and dense axion star [21]. The diluted axion star is formed by the balance between the kinetic pressure and gravitational attractive force, and the dense axion star is formed by the balance between the self interaction of the axion and the gravitational force. For a fixed axion-star mass, the radius of a diluted axion star can be several orders of magnitude larger than the dense one.
The searches for background axions, not inside an axion star, rely on the axion "invisible" couplings to the Standard Model (SM) particles, which are suppressed by the PQ scale, f a , above ∼ 10 8 GeV from experimental constraints [22]. Because of the suppressed coupling strength, it is challenging, but possible to have table-top experiments to detect axion dark matter particles [23][24][25][26][27]. To search for axion mini-clusters or stars, the gravitational lensing (micro-lensing) experiments could be sensitive for the mass range of QCD axion stars. Recently, the mirco-lensing-based searches for massive astrophysical compact halo object (MACHO) by Subaru Hyper Suprime-Cam in Ref. [28] has imposed a constraint on the fraction of axion-star energy in total dark matter energy f AS to be below a few percent for M a⊙ = 10 −12 M ⊙ to no constraint for M a⊙ ≈ 10 −14 M ⊙ . Future micro-lensing experiments will further constrain on the fraction of axion stars as a part of dark matter. Another novel way to look for axion stars is to look for the hydrogen axion star, which may well be formed by ordinary baryonic matter becoming gravitationally bound to an axion star [29].
In this paper, we focus on the brightness of axions when they pass through some astrophysical environment with a large magnetic field. In the presence of the axion field, the electromagnetism becomes axion electromagnetism [30]. The axion coupling to two photons via a F µν F µν can convert axion to real photons in the magnetic background. If the conversion rate is sufficiently large, we could detect the signal as a monochromatic radio wave signal with a frequency of ν = m a /2π. In this respect, the axion star is the interesting object because it may enhance the signal due to its large energy density or field value at the core. To know the populations of axion stars in our galaxy or Universe, it is important to know the fraction of all axion particles inside axion stars, which requires a more realistic numerical calculation than Ref. [31]. In this paper, we treat the fraction of the axion star as a phenomenological parameter, and investigate the observational consequence. For the axion star mass, we concentrate on the range of 10 −14 -10 −12 M ⊙ , as estimated for the QCD axion [20,21,29].
We first estimate the radiated power from the homogeneous axion component interacting with the magnetic field in our galaxy/neutron star, which turns out to be too small to be detected by on-going and future radio telescope [32][33][34]. Afterwards, we estimate the radiation power of the axion star in the static magnetic field, 10 10 -10 15 Gauss, associated with the neutron star, which can be potentially large enough to be detected in the radio telescopes. In our calculation, we properly take account of the two important effects: the interference effect and medium effect. At the leading order in perturbation (the source size over its distance to the observer), the radiated field can be written in terms of the integration of the source term multiplied by the retarded Green's function. The integration over the source region leads to the interference effect, and the resultant radiation field is suppressed by powers of 1/(m a R AS ) depending on the geometry of integration range, where R AS is the radius of the axion star. Even for dense axion stars with R AS ∼ 1 m, the suppression factor is multiple power of 10 −3 . For the second medium effect, we consider the response of the electron-ion plasma, the state in the outer crust of neutron stars, to the axion-generated electric field. Using a simple Drude model, we calculate the electric susceptibility of the plasma medium and then the radiation power based on the macroscopical Maxwell equations. We have found that the power is not strong enough to explain the fast radio burst (FRB) [35], contrary to the claim in Refs. [36][37][38], where the above effects are not included. On the other hand and for a neutron star with a large magnetic field, the radiation power could be large enough to be detected by the on-going or future radio telescopes such as Arecibo [39], GBT [40], JVLA [41], FAST [42] and SKA [43]. This paper is organized as follows. In Section 2, we review the dilute and dense axion star properties. In Section 3, we estimate the radiation power from homogeneous axion dark matter in our galaxy. In Section 4, we calculate the radiation power of the axion star in the static magnetic field associated with the neutron star. In Section 5, we derive the encounter rate of the axion star with the neutron star, and investigate the detectability of the current and future observations. We then conclude in Section 6.

Axion-star Mass and Size
Depending on the axion star mass, M a⊙ , and other axion model parameters: m a and f a , one may have diluted or dense axion stars as components of dark matter, if the PQ-symmetry breaking scale is below the end of inflation scale H end Inflation 10 13 GeV [44]. For the diluted axion star, the axion self-interactions usually can be ignored. As studied in the literature, this requires the axion star mass to satisfy [45] with G N as the Newton constant. For the QCD axion, we anticipate m a f a ≈ m π f π ≈ (10 8 eV) 2 [9]. To account for all dark matter energy density, the axion mass is 10 −5 -10 −3 eV [8]. We therefore choose m a = 10 −4 eV and f a = 10 11 GeV as a benchmark model. Using a variation calculation based on the hydrogen 1S wave function, it is easy to show that the radius of the diluted axion star is Its energy density in the unit of m 2 a f 2 a is very small and has ρ a⊙ /m 2 a f 2 a ≈ 4.5 × 10 −21 for the same model parameters of the above equation.
For a dense axion star, the self-interaction of axions is balanced by the gravitational attractive interaction. Following the calculation in Ref. [21] with the Thomas-Fermi approximation, we show the dense axion star size and mass in Fig. 1. We can fit the relation between the axion star radius and mass approximately as For the dense axion star, the energy density in the unit of m 2 a f 2 a can be order of unit. In our analysis later, we will treat M a⊙ , m a , and f a as three independent model parameters. We note that depending on the detailed non-perturbative effects to generate the axion particle mass, there could be some model-dependent relation among these three parameters.

Axion Electromagnetism
To observe the existence of the axion stars beyond its gravitational interaction [18,49], one could adopt other interactions of axion with SM particles. The relevant interactions are Here, we use ψ to represent SM fermions including electrons and quarks and F µν = 1 2 ǫ µνρσ F ρσ . With the existence of axion field, the Maxwell's equations are modified as In terms of #» B and #» E fields, we have the axion-generated current as Here, the axion field a( x, t) has variations in both time and space. In the Lorenz gauge with ∂ µ A µ = 0, the equation of motion of the gauge field is ∂ 2 A µ = j µ . Using the Green's function method and taking the retarded boundary condition, we have the gauge field as [50] A µ a ( x, t) = 1 4π If we ignore the time-dependence of the background #» B field and only keep the sinusoidal time dependence e i ma t of the axion field, the gauge field is the real part of Here, we have made the approximation of r ≡ | x| ≫ | y| with | x − y| ≃ r − n x · y and n x ≡ x/r =r. It is understood that the spacially-dependent current j µ a (y) has j a (y) = m a a(y) In the case with a homogeneous and static magnetic field and a zero electric field, we have If the axion field is also homogeneous (at least in the region of large magnetic field) with a(y) = a 0 , we have A 0 a = 0. Using the first equality of Eq. (8) and changing the integrating variable, it is easy to show that #» A a is independent of r. So, there is no axion-generated magnetic field, and hence no radiated power.
In reality, the axion field does not have correlation beyond its de Broglie wave length d a = 1/(m a v). As a result, we anticipate some inhomogeneity of the axion field and a nonzero A 0 a . For this general situation, there may not be correlation between the two directions of the generated #» E a and #» B a . For a volume, V , of constant magnetic field and after (randomly) averaging out V /d 3 a = V m 3 a v 3 small correlated axion filed boxes (within its de Broglie wave length), we have the radiated power as which is very small for a very far-away source like a neutron star.
For the axion dark matter in our galaxy, the large volume effect could enhance the flux, although the corresponding averaged magnetic field is much smaller, | #» B| galaxy ∼ O(10 µG). Following Ref. [51], we treat the magnetic field as a constant one within a simple cylinder-shape with a radius of 20 kpc and half-height of 2 kpc. For the dark matter distribution in our Milky Way Galaxy, we use the isotropic Einasto profile [52] with with ρ ⊙ = 0.4 GeV/cm 3 , r s = 20 kpc, r ⊙ = 8.5 kpc and β = 0.17. The spectral flux density that can be observed by a detector on the Earth is Here, the distance of a source to the observer is d( r) 2 = r 2 + r 2 ⊙ − 2 r r ⊙ cos θ, with θ as the angle between r and the direction from the galactic center to the Sun. In terms of the unit of Jansky with 1 Jy = 10 −26 W/(m 2 Hz), the total spectral flux density is at the order of 10 −19 Jy and very small (see Refs. [32][33][34] for similar calculations).

Without Medium
For the axion star, there are two effects that can potentially enhance the radiation power. One is the much higher energy density carried by the axion star. The second is that being a Bose-Einstein Condensation (BEC) state, the axion star is more compact and potentially has a larger contribution to radiation power from different regions. To simplify our discussion, we will simply assume the spacial wave-function of the axion star follows the 1S-state of the hydrogen atom. 3 So, the axion star field value is where we have chosen the normalization to have the total axion-star mass to be M a⊙ ≈ N a m a in the non-relativistic approximation. Performing a straight-forward integrations of Eqs. (9)(10) and in the limit of R AS ≫ 1/m a , we have the field value as Choosing

With Medium
If there is a medium surrounding the axion star and with a static magnetic field, the response of medium to the axion-generated electric field may generate a large radiation power. For a plasma of electrons and ions, the electron motions responding to the oscillating electric field behave as an oscillating electric dipole moment [53]. The large number density of electrons could potentially increase the radiation power. However, for a macroscopic medium, the induced electric field from the axion and static magnetic field is also suppressed by the plasma frequency, so the axion-star radiated power is not amplified in the medium.
For macroscopic material, one need to calculate the applied electric field or the polarization field #» P , which is related to the electric field as #» P = χ #» E with χ as the electric susceptibility or the electric flux densities #» D = ǫ #» E with ǫ = 1 + χ (we have chosen vacuum permittivity ǫ 0 = 1 in the natural unit). Based on the simple Drude model, one can calculate the frequency-dependent permittivity as where ω is the frequency of the applied electric field; γ is a measure of the rate of collisions per unit time and for a typical conductor γ = O(10 14 s −1 ) = O(0.066 eV); the oscillator frequency ω 0 = 0 for free electrons and non-zero for bounded electrons; the plasma frequency ω p = (n e e 2 /m e ) 1/2 . Inside the medium and neglecting spatial derivative terms, the relevant Maxwell equations have 4 Solving the Maxwell equation, the effective electric field generated by the axion field around the source region is The corresponding axion-generated current in the medium has Similar to Eqs. (9)(10), we can use the above medium-responded current to calculate the radiation power. It is clear that if the medium has a large value of χ ≫ 1, the radiated power is suppressed by 1/χ 2 . Therefore, we conclude that the medium can not amplify the radiation power. 5 Around the surface region of the neutron star, the plasma frequency is ω p ≈ 1.1 × 10 5 eV for the nuclei (iron) density of 8 × 10 6 g/cm 3 [54]. For the oscillation frequency and using the radial potential of V = −Q ion α/r + l 2 /(2 m e r 2 ) with l as the angular momentum number, we can have For l = 1, we have |χ| ≈ ω 2 p /ω 2 0 ≈ 0.03. The medium effect for the axion-star-radiated power could be small, even when the axion star is actually overlapping with the neutron star. The radiated power in Eq. (17) without medium can be trusted as an order-of-magnitude estimation.
Before we end this section, we also comment on the possibility of using neutron electric dipole moment in Ref. [38]. We show that there is no volume-square or N 2 n (N n is the number of neutrons) effect from the integration or interference of different regions, as can be seen from Eq. (7). As a result, the neutron-medium-generated power is negligible. For completeness, we provide the orderof-magnitude estimation of the power from oscillating electric dipole moment of the neutron. In the background of classical axion field, the neutron electric dipole is [55,56] where a( r) is given by Eq. (14). Following the same calculation as before, we have the radiated power estimated as Here, n n is the number density of the neutron andn n is the averaged number density. In our estimation, we have assumed m −1 a ≪ R AS R NS . Because of the interference effects, the total power is suppressed by a factor of 1/(m a R AS ) 8 on top of N 2 n ≈ (n n R 3 AS ) 2 . Compared to the power generated from interacting with the magnetic field in Eq. (17), the power generated from oscillating electric dipole moment of neutrons is negligible.

Signal when Axion Stars Encounter Neutron Stars
After previous calculations, we have learned that the largest radiation power for the axion star can be obtained if it passes through a large magnetic field. Neutron stars are the obvious objects, but one could also consider other objects like white dwarfs with a smaller magnetic field but a larger population. Therefore, in this section we will estimate the encounter rates as well as the experimental detectability.

Encounter rate
Assuming that the total energy of axion stars is comparable to the total dark matter energy, the total number of axion stars in our galaxy is roughly 2 × 10 24 × [10 −13 M ⊙ /M a⊙ ]. Around our solar system, the axion-star density is around n AS ≈ f AS × 1.1 × 10 11 pc −3 [10 −13 M ⊙ /M a⊙ ] with f AS as the fraction of axion-star energy in total dark matter energy. 6 In the meanwhile, there are around 10 9 neutron stars in our galaxy with the magnetic field ranging from 10 8 to 10 15 Gauss [57]. Even slightly outside the neutron star, there is still a large magnetic field. Therefore, we do not require that the axion star strictly collides with the neutron star, but calculate the rate of the event where the minimal distance between the axion star and the neutron star is (R AS + ℓ R NS ) with ℓ ≥ 1. To estimate the encounter rate, we simply use the geometric cross section augmented by the "gravitation focus" effect or the increasing Safronov number [58]. The cross section has the form of with R NS ∼ 10 km and M NS ∼ M ⊙ as the neutron-star radius and mass, respectively. The relative speed is approximately the averaged axion-star speed in our galaxy and around 300 km/s. The escaping velocity is estimated to be v esc ≈ 1.6 × 10 5 ℓ −1/2 km/s. The encounter events per year are estimated to be For a lighter axion star, the encountering events can happen more frequently in our galaxy. From Eq. (29), the encounter events per year are not large. However, we note that the calculation here is just the order-of-magnitude estimation. The actual encounter rate could be increased significantly if a dark matter clump happens in the neutron-star-rich region. As can be seen from Eq. (29), the encounter rate can become large with a large value of ℓ, at the price of a suppressed magnetic field by ℓ −3 . 7 Furthermore, if the number of neutron stars is larger than 10 9 in our galaxy, then we may also obtain a larger encounter rate. Having these possibilities in mind, we will simply assume that encounter event can happen at the scale of one year and discuss the detectability. From Eq. (17) and when one axion star enters the vicinity of a neutron star, the radiated power can vary from 10 11 W to 10 21 W, depending on the magnetic field size from 10 10 Gauss to 10 15 Gauss. This kind of event is also likely to be a transient one with the time scale of O(few R NS /v rel ) ≈ O(0.1 s). Since this encounter event is like a point source from the astrophysical point of view, the closest event to our solar system may have the highest probability to be detected. For the ∼ 10 9 neutron stars in the our galaxy disk, the averaged density is roughly one per parsecs. For sure, the anticipated encounter rate at a distance of parsecs is very tiny.
Using Eq. (17) and assuming a typical source distance of 1 kpc, the estimated spectral flux density from axion-neutron-star encounter is which is applicable for the dense axion-star mass above the upper bound for the diluted axion mass in Eq. (1). The encounter rate is given in Eq. (29) with the typical source distance at kpc scale. The spectral density flux highly depends on the axion particle mass, the neutron-star magnetic field and the location of the encountering event.

Detectability
The signal from axion conversion to photon has the radio frequency peaked at ν = m a /2π = 24 GHz × (m a /10 −4 eV). Both the axion-star spacial profile and the inhomogeneity of the magnetic field can generate a non-zero but small bandwidth. For the interesting mass range, 10 −6 -10 −3 eV, of QCD axion, the corresponding frequency is 240 MHz to 240 GHz. It is intriguing to see if the on-going or future radio telescope can have chance to discover axion-neutron-star encounter event.
The existing Arecibo Observatory has a 305-meter single-aperture telescope with covered frequencies from 300 MHz to 10 GHz [39]. For instance, some fast radio burst was discovered by this radio telescope [59]. The sensitivity to the continuous spectral flux density is around 0.89 µJy (hr/t obs  [43]. To estimate the sensitivity of different telescopes on the axion-star generated power, we can convert the spectral flux density in Eq. (30) to the brightness temperature [60] Here, the factor of 1/2 takes into account that the telescope usually only receives one polarization; the parameter A eff is the effective area of the telescope. Depending on the telescope system temperature T sys (typically order of 10 K), which includes both sky noise and instrumental noise, 8 the minimal detectable brightness temperature is roughly Combining Eqs. (31) and (32), the rough sensitivity on the spectral flux density is For different telescopes, one can find a summary of different A eff /T sys in Ref. [43].
In Fig. 2, we show the dense-axion-star generated radiation power when it encounters neutron stars with a magnetic field of 10 10 Gauss and 10 15 Gauss, for different axion particle masses. We fix the product of m a f a = (10 8 eV) 2 to approximately match the QCD axion relation. For a large value of magnetic field above ≈ 10 13 Gauss, the sensitivities of radio telescopes are good enough to potentially observe the transient O(0.1 s) radio source, very similar to the fast burst radio, except located within our galaxy.

Discussion and Conclusions
So far, we have concentrated on the radiation power from a dense axion star. Our calculation has been kept very general such that our Eq. (17) can be applicable to other axion stars like a diluted axion star. For sure, the diluted-axion-star radius R AS , given by the Eq. (2), is much larger than the one for a dense axion star in Eq. (3). Since the flux is proportional to R −5 AS as shown in Eq. (26), the signal of a diluted axion star is dramatically suppressed compared with that of a dense axion star and is unlikely to be observed. This large suppression factor is due to the interference effect when one integrates the radiation power generated from different regions of the axion star.
We also comment on the case of the non-QCD axion. In this case, the relation of m a f a ≈ (10 8 eV) 2 does not need to be satisfied. Requiring the symmetry breaking scale below the end-of-inflation scale and satisfying the astrophysical bound, the decay constant should be still in the range of 10 8 GeV < f a < 10 13 GeV. Keeping the axion star mass around 10 −13 M ⊙ and from Eq. (1), one could have f a = 10 8 GeV to saturate the astrophysical bound. To have a dense axion star, the axion particle mass is bounded from below m a 10 −7 eV. The maximum power for a dense non-QCD axion star could be 9 orders of magnitude higher than a dense QCD axion star, as can be seen from Eq. (17). Therefore, the radio telescopes could have a good coverage for some non-QCD dense axion stars if they search for spectral-line signals with a wide range of frequency, order of 10 −2 -10 2 GHz. The transient radio signal when an axion star meets a neutron star or other astrophysical object is very similar to the FRB. Both of them have the signal lasting below one second. So far, the discovered FRB's have their origin at an extra-galactic distance [63]. The axion-neutron-star encounter event has a smaller radiation power than a FRB, and is likely in the nearby region of our galaxy if discovered. Therefore, a future discovery of a galactic FRB should deserve a study of its spectrum to see if it has a clear spectra line peaked at the preferred QCD or other non-QCD axion mass.
As shown in Eq. (29), the axion-neutron-star encounter rate is not that high because of the small geometric size and population of neutron stars. An axion star could just enter the vicinity of a neutron star and could still generate a large enough radiation power. Far away from the neutron star, the magnetic field decreases as ∝ r −3 if we treat the neutron star as a point-like magnetic dipole source. The encounter rate increases as ∝ r while the spectral flux density decreases as ∝ r −6 . For sure, we have not considered the situation with some local clump of axion stars in our galaxy disk where most of neutrons sit, which will possibly increase our estimation of the encounter rate.
Other than neutron stars, one could also consider other astrophysical objects with a large magnetic field. White dwarfs are the obviously next candidate. In our galaxy, we anticipate O(10 11 ) white dwarfs, which has their radius of O(10 4 km) and magnetic field of 10 6 -10 9 Gauss [64,65]. Using Eq. (29), the encounter rate could be at the order of 100 ℓ f AS /year. Because of its small magnetic field, the encounter event for a dense QCD axion star can be observed by radio telescopes only if it happens in a very nearby region, O(1 pc) from our solar system. Finally, the dense axion star may also enter our solar system and has an encounter event with our Sun. Both the generated radiation power and the encounter rate are too small to be observed.
In summary, we have estimated the radiation power when a dense axion star encounters a neutron star or other astrophysical objects with a large magnetic field. The spectral flux density is at the order of µJy for a neutron star with B ∼ 10 13 Gauss and the encounter event happening within one kpc from our solar system. The existing Arecibo, GBT, JVLA and FAST and the ongoing SKA radio telescopes have excellent discovery potential for the dense axion stars.