Probing Axions with Event Horizon Telescope Polarimetric Measurements

Yifan Chena,b, Jing Shua,c,d,e,f , Xiao Xuea,c, Qiang Yuanf,g,h, and Yue ZhaoiaCAS Key Laboratory of Theoretical Physics, Insitute of Theoretical Physics,

Chinese Academy of Sciences, Beijing 100190, P.R.ChinabLaboratoire de Physique Theorique et Hautes Energies (LPTHE),

UMR 7589, Sorbonne Universite et CNRS, 4 place Jussieu, 75252 Paris Cedex 05, FrancecSchool of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, P.R.China

dCAS Center for Excellence in Particle Physics, Beijing 100049, P.R.ChinaeSchool of Fundamental Physics and Mathematical Sciences, Hangzhou Institute for Advanced Study,

University of Chinese Academy of Sciences, Hangzhou 310024, ChinafCenter for High Energy Physics, Peking University, Beijing 100871, P.R.China

gKey Laboratory of Dark Matter and Space Astronomy, Purple Mountain Observatory,Chinese Academy of Sciences, Nanjing 210008, P.R.China

hSchool of Astronomy and Space Science, University of Science and Technology of China, Hefei 230026, P.R.ChinaiDepartment of Physics and Astronomy, University of Utah, Salt Lake City, UT 84112, USA

(Dated: February 18, 2020)

With high spatial resolution, polarimetric imaging of a supermassive black hole, like M87? orSgr A?, by the Event Horizon Telescope can be used to probe the existence of ultralight bosonicparticles, such as axions. Such particles can accumulate around a rotating black hole through thesuperradiance mechanism, forming an axion cloud. When linearly polarized photons are emittedfrom an accretion disk near the horizon, their position angles oscillate due to the birefringent effectwhen traveling through the axion background. In particular, the observations of supermassive blackholes M87? (Sgr A?) can probe the dimensionless axion-photon coupling c = 2πgaγfa for axionswith mass around O(10−20) eV (O(10−17) eV) and decay constant fa < O(1016) GeV, which iscomplimentary to other axion measurements.

Introduction The first ever image of the supermas-sive black hole (SMBH) M87? by the Event Horizon Tele-scope (EHT) [1] leads us to a new era of black holephysics. The high spatial resolution makes the direct vi-sual observation of an SMBH and its surroundings possi-ble. While it offers a new way to study the most extremeobjects in our universe predicted by Einstein’s theory ofgeneral relativity, we may wonder what else can we learn,especially for fundamental particle physics, from the richinformation extracted from the EHT under such an ex-treme environment.

The axion is a hypothetical particle beyond the stan-dard model (SM), which was originally motivated by thesolution of the strong CP problem [2] in QCD. Beyondthe QCD-axion, axion-like particles (ALPs) also generi-cally appear in fundamental theories [3], and serve as aviable dark matter candidate [4]. There are many searchstrategies proposed to look for axions, for example, viatheir conversion into photons [5–8], spectral oscillation ordistorsion of photons [9–11], nuclear magnetic resonance[12, 13], neutron star mergers [14] or various table-topexperiments [15–23].

When the Compton wavelength of an axion is at thesame order as the size of a rotating black hole, the ax-ion is expected to develop a large density near the hori-zon, forming an axion cloud through the superradiancemechanism [24–32] (for a review see [33]). Such superra-diance processes can be tested by black hole spin mea-surements [34–37], gravitational wave signals from bosen-ova [34, 35, 38–42] or electromagnetic emission from the

axion cloud [43, 44]. In this letter, we propose a novelway of detecting axion clouds around SMBHs by usingthe high spatial resolution, polarimetric measurementsof the EHT. Our proposed search strategy utilizes theunprecedent capability of the EHT and serves as a com-plimentary probe.

The recently reported direct image of the shadow ofM87? illustrates that the EHT is capable of resolving theemission ring close to the event horizon [1] where the ax-ion cloud can concentrate. A linearly polarized photonemitted from the innermost region of the accretion diskwhich lies in a dense axion background experiences thebirefringent effect, and its position angle oscillates, witha period being equal to the axion oscillation period [45–52]. The amplitude is proportional to the local axionfield value. Interestingly, the predicted periodic oscil-lation of the position angle due to the axion cloud canbe the same order as the background Faraday rotationat mm wavelengths for both M87? and Sgr A? [53, 54],making the spatially-resolved polarimetric measurementsof the SMBH by the EHT not only optimal to observe thedisk magnetic field structure, but also to probe axions.

Photon Polarization Variation from BirefringenceIn the presence of an axion-photon interaction and axionpotential V (a), the relevant Lagrangian is

L = −1

4FµνF

µν−1

2gaγaFµν F

µν+1

2∇µa∇µa−V (a), (1)

arX

iv:1

905.

0221

3v2

[he

p-ph

] 1

5 Fe

b 20

20

2

in which gaγ is the axion-electromagnetic-field coupling.This modifies the equation of motion for a photon prop-agating in an axion background field which leads to peri-odic oscillation of a linearly polarized photon’s positionangle [45, 46]. More explicitly, we assume that the vari-ation of the axion field in space and time is much slowerthan the photon’s frequency ωγ , i.e., µ� ωγ , where µ isthe axion mass. In later discussions, it will become clearthat the spacetime is approximately flat in the region weare interested in.

In the Lorentz gauge, the modified Maxwell’s equationfrom Eq. (1) for photons propagating along the z-axis is

2A± = ±2igaγ [∂zaA± − a∂zA±], (2)

where ± denotes two opposite helicity states with A0 =A3 = 0, A± = (A1 ∓ iA2)/

√2, with solutions

A±(t, z) = A±(t′, z′) exp [−iωγ(t− t′) + iωγ(z − z′)± igaγ(a(t, z)− a(t′, z′))] . (3)

The leading order effect of the axion background fieldcomes from the last term of Eq. (3), which results ina rotation of the position angle for a linearly polarizedphoton

∆Θ = gaγ∆a(tobs,xobs; temit,xemit)

= gaγ

∫ obs

emit

ds nµ ∂µa

= gaγ [a(tobs,xobs)− a(temit,xemit)]. (4)

Here nµ is the null vector along the path. Note that thisonly depends on the initial and final axion field values.

In the following discussion, we consider photons emit-ted from the accretion disk near the horizon of an SMBH,which are linearly polarized due to synchrotron in thehighly ordered magnetic field in the disk [55], and ob-served at the Earth by, e.g. the EHT. Thus we can safelyneglect a(tobs,xobs) since the axion field can be very largesurrounding the SMBH due to superradiance (see below).Then Eq. (4) becomes

∆Θ ' −gaγa(temit,xemit)

= −gaγa0(xemit) cos [ωtemit + β(xemit)]. (5)

Here ω is the oscillation frequency of the axion field whichdepends on the axion mass µ. The amplitude a0 andphase factor β, are set by the energy density and thephase of the axion cloud at the emission point whosespatial dependence will be discussed later.

Superradiance and Bosenova The axion equationof motion from Eq. (1) in a Kerr background is

2a = µ2a, (6)

where we take V (a) = 12µ

2a2, and neglect the self-interaction for now. After imposing infalling boundaryconditions at the black hole horizon, superradiance oc-curs when the axion frequency ω is below the criticalvalue

|ω| < ωc =aJm

2Mr+, (7)

where m is the azimuthal number, aJ is the dimension-less spin of the black hole, and r+ is the black hole outerhorizon radius. In Planck units (GN = c = ~ = 1), the

outer and inner radii are r± = rg

(1±

√1− a2J

), with

M being the black hole mass and rg = M . In this re-gion, one gets a positive value of Im(ω) which leads to anexponential growth of the wavefunction a ∼ exp(t/τSR)with a superradiance timescale τSR ' 1/Im(ω).

In Ref. [30], it was shown that the superradiance takesplace efficiently when the Compton wavelength of theaxion (λC) is comparable to the size of the rotating blackhole within an order one factor:

rgλC

= µM ≡ α ∈ (0.1, 1), (8)

where α ≡ µM is written in Planck units. In this region,the superradiance rate Im(ω)/µ ranges between 10−10 to10−7 according to the simulation in Ref. [30]. For dif-ferent SMBHs like M87∗ and Sgr A∗, the correspondingaxion mass windows are different. The axion field pro-duced through superradiance forms a bound state withthe SMBH as a “gravitational atom”. In the α� l limit,Eq. (6) for the bound state reduces to the hydrogenicSchrodinger equation with a discrete spectrum

Re(ω) '(

1− α2

2n2

)µ, (9)

where n = n + l + 1 is the principal quantum number.For m < l, τ−1SR is negligible compared to the m = l stateand so in the following discussion, we take m = l.

So far we have neglected the axion self-interaction,which, as well as the axion mass, arises from instan-ton corrections induced by associated quantum anoma-lies. The axion potential generically takes the followingform,

V (a) = µ2f2a

(1− cos

a

fa

), (10)

where the leading order expansion around the minimumgives the axion mass term 1

2µ2a2.

Since superradiance continually populates the axioncloud, one needs to carefully examine whether self-interaction remains negligible. When fa is sufficientlylarge (> 1016 GeV) [35], gravity will always dominate,and the angular momentum of a black hole will decreaseuntil Eq. (7) no longer holds. Therefore the existenceof black holes with high angular momentum has been

3

used to constrain the parameter space of axions in thiscase [34–37]. On the other hand, when fa is small (< 1016

GeV), the amplitude of the axion field in the axion cloudgrows to fa first before the black hole spin aJ is de-creased, and the self-interactions among these bosonsfrom Eq. (10) become important compared with gravity,leading to the non-linear regime [34, 35, 38–40].

In [38–40], simulations are performed to study thisnon-linear behavior of the axion cloud. After enteringthe non-linear regime, the axion cloud either ends asa bosenova explosion or continues to saturate the non-linear region with a steady outflow. In the former case,self-interactions make the axion cloud collapse and theaxion field value decreases by an O(1) factor, then theaxion cloud starts to build up again until it reaches thenon-linear region at a later time. This process could pre-vent superradiance from persisting, if after a final bosen-ova explosion a black hole is left unable to reenter thesuperradiant regime due to environmental effects. In an-other case, the loss from the gradual scattering towardsthe far region balances the extraction of the energy fromthe black hole without triggering an explosion. Althoughthe axion cloud may end up in two very different regimes,the simulations in [38, 40] indicate that the axion fieldamplitude in the densest region amax is always aroundfa.

From Eq. (5), the maximal change of the position anglecan be written as:

∆Θmax ' −bgaγfa cos [µtemit + β(|xemit| = rmax)],(11)

by using a0 ≈ fa and ω ≈ µ from Eq. (9). Here b ≡amax/fa, which is an O(1) number as discussed above.

For general axion, the magnitude of the axion-photoncoupling gaγ varies depending on the underlying theoryand we can redefine gaγ to cγ in order to extract thecommon factors

gaγ ≡c

2πfa≡ cγαem

4πfa, (12)

where αem is the fine-structure constant. In the simplestcase, one can imagine N copies of fermions with elec-tric charge Q coupling to the axion with coupling gaf ,then the axion-photon coupling gaγ is induced throughthe fermion loop and cγ ∼ NQ2. In extended theoriessuch as clockwork axions [56], however, the axion-photoncoupling gaγ could be exponentially large. Consider anN -site clockwork model with scalar charge q, and let asingle set of color neutral vector-like fermions couple atsite M (M < N). The axion-photon coupling at lowenergy is then [57]

cγ ∼ 2Q2qN−M . (13)

Therefore, we can see that a large cγ can compensatethe loop suppression factor αem so that gaγ can be evenlarger than 1/fa. Notice that our ma/fa range is well

outside that normally considered for QCD axions, so ouraxions must be axion like particles (ALPs).

Axion Field Profile In this section, we study theaxion cloud spatial profile. We focus on the black holevicinity, especially the region with the ring feature pre-sented by the EHT, i.e. rring ' 5.5 rg [1]. We note thatthe current results published by the EHT group do nothave polarization information, but the polarization datais expected to be available in the future [58].

R [r] / R [rmax]

Arg [R [r]]

5 10 15 200

0.5

1

-1

-0.5

0

r / rg

R[r]/R[rmax]

Arg

[R[r]]

rring rmaxr+

FIG. 1: The absolute value and the complex phase of R(r)for the l = 1,m = 1 state from Eq. (15). We take α = 0.4,aJ = 0.99.

The general solution for Eq. (6) in the Kerr back-ground can be written as

a(xµ) = e−iωteimφSlm(θ)Rlm(r), (14)

where xµ = [t, r, θ, φ] in Boyer-Lindquist coordinates.The θ dependence is characterized by spheroidal har-

monics Slm = Sml

(cos θ, aJM

2√ω2 − µ2

), which sim-

plify to spherical harmonics Y ml in the non-rotating ornon-relativistic limit.

Imposing ingoing boundary condition at the outer hori-zon r+ and setting the axion field to zero at infinity, theradial part can be written as

R(r) = (r− r+)−iσ(r− r−)iσ+χ−1eqr∞∑n=0

an

(r − r+r − r−

)n,

(15)

with σ = 2r+(ω − ωc)/(r+ − r−), q = −√µ2 − ω2, and

χ = (µ2 − 2ω2)/q. The expansion coefficients, an, aresolved using Leaver’s nomenclature in [30].

One may worry about the non-linear effects discussedin the last section, which come from the generalization ofthe Klein-Gordon equation to the Sine-Gordon equationarising from Eq. (10). This subtlety is studied in the Ap-pendix of [38] where the deviation from Eq. (14) is calcu-lated with the Green’s function method. It is shown that

4

a perturbative transition to other modes not satisfyingthe superradiance condition is only significantly inducedwhen there is a bosenova. After the bosenova, these ad-ditional modes are expected to fall back into the blackhole such that one regains a state similar to the initialperturbative one within a relatively short astrophysicaltimescale (although this has not yet been unambiguouslydemonstrated in the simulations [38–40]). Since the su-perradiance timescale τSR is much longer than τBN , thesolution of the Klein-Gordon equation is valid for most ofthe time during the large period between each bosenova.

Another issue is whether such a bound state is stableagainst environmental effects. We can estimate the ratiobetween self and gravitational interactions at a given ra-dius; in the non-relativistic limit, it is maximised aroundO(15)rg. This implies that the point which first trig-gers bosenova is quite far away from the ring positionrring ' 5.5 rg observed by EHT, and thus the inner re-gion we have studied should be relatively weakly coupledand stable. This back-of-the-envelope estimation is inagreement with simulations (c.f. Fig. 9 of [38]). There-fore, for the state we are likely to be observing long afterthe final bosenova, the axion field value at the ring posi-tion should remain of order fa.

In Fig. 1, we show the axion field profile for an l =1,m = 1 state which enjoys the largest superradiancerate. Note that the mixture of different modes with otherquantum numbers would not change our following dis-cussing qualitatively since the amplitude for l = 1 andm = 1 mode is much higher than other modes and thespatial distributions of the superradiance clouds of dif-ferent modes are also different. We take α = 0.4 andaJ = 0.99 as benchmark. At rring = 5.5rg [1], the axionfield value is not significantly different from the maximalvalue, i.e. R(rring) ' 0.9 R(rmax). Notice that the com-plex phase in R(r) is almost a constant for r > 2r+ andSlm does not contribute to a complex phase. Thus thespace-dependent complex phase in Eq. (5) is dominatedby mφ in Eq. (14),

β(xemit) ' mφ. (16)

The ∆Θ(r, θ, φ) dependence on the time and spatial po-sition of the radio sources can be obtained from Eq. (5),(14) and (15),

∆Θ(t, r, θ, φ) ≈ −bgaγfaR11(r)

R11(rmax)sin θ cos [ωt−mφ]. (17)

Taking b = 1 and θ = π/2 which specifies the plane ofthe accretion disk, the position angle variation is between±8c◦. For the time just before the bosenova explosion, itcan even reach ±25c◦.

In Fig. 2, assuming that the rotation axis of the diskpoints to the observer, we show ∆Θ in Eq. (17), at afixed time, as a function of position. Therefore, bothtime dependence of the position angle at a fixed spatial

point and the position angle as a function of position ata fixed time can be used to test the existence of an axioncloud by the high-resolution polarimetric imaging fromthe EHT.

Δ Θ (°)

[t=0, θ=π

2, r, ϕ]

-10 c

-5 c

0

5 c

10 c

FIG. 2: ∆Θ(t = 0, θ = π/2, r, φ) viewed along the rotatingaxis of the black hole. The amplitude of oscillation is around8c◦ at rring for l = 1, m = 1, α = 0.4, and aJ = 0.99. Theregion of r < r+ is masked.

Detectability As shown in previous sections, takingc ∼ O(1) as an example, the maximal oscillation ampli-tude of the position angle is O(10)◦. This is expected tobe well within the capability of the EHT. The previousobservations of Sgr A?, with a subset of the EHT config-uration and an exposure of tens of minutes, measure theposition angle at a precision of δΘ ∼ 3◦ [55]. It is rea-sonable to expect that a better precision can be achievedwith even shorter exposure time for the upgraded EHTobservations. For Sgr A?, the expected oscillation periodis 100 ∼ 1000 s (In Table I, we give a summary of pa-rameters for two SMBHs, M87? and Sgr A?). It mightbe challenging to have a good enough sampling of theobservations within one period which typically requiresan exposure of e.g. tens of seconds. The situation forM87? is more promising, due to a substantially longeroscillation period (a few days). The upcoming analysisof the polarization data, particularly for M87?, should beable to provide valuable information about the possibleaxion superradiance around the SMBH.

In order to give a realistic estimation, one needs to takeinto account the spatial resolution of the EHT. The inte-grals along the r and θ directions always give a positivecontribution to the signal, especially when the accretiondisk is face-on. Only the average over φ within the res-olution may wash out the axion induced position anglevariation. The spatial resolution of the M87? image isabout 20 µas (full width at half maximum; FWHM) [1],which corresponds to a region of ∼ 2rg at a distance of

5

SMBH M aJ µ range µ for α = 0.4 τa τSR

M87? 6.5 × 109M� 0.99 2.1 × (10−21 ∼ 10−20) eV 8.2 × 10−21 eV 5.0 × 105 s > 1.5 × 1012s

Sgr A? 4.3 × 106M� · · · 3.1 × (10−18 ∼ 10−17) eV 1.2 × 10−17 eV 3.3 × 102 s > 1.0 × 109s

TABLE I: Typical parameters of the axion superradiance of the two SMBHs, M87? and Sgr A?.

∼ 17 Mpc. Assuming a nearly face-on emission disk,which is similar to the case of M87? with an inclinationangle of ∼ 17◦ [59], this spatial resolution translates toδφ ' 4rg/rring = 0.7 rad. Without losing generality,taking φ = 0 and considering the average effect withinδφ, one obtains the wash-out factor caused by the spatialresolution as

1

δφ

∫ δφ/2

−δφ/2cos(µt+mφ)dφ =

sin (mδφ/2)

mδφ/2cosµt. (18)

Without the high spatial resolution provided by the EHT,one has to perform the integration on the whole accretiondisk, i.e. δφ = 2π, and the change of the position angle isaveraged out. For the EHT, taking m = 1 and δφ ' 0.7,the wash-out factor is 0.98, and the effect is negligible.

-22 -20 -18 -16 -14-3

-2

-1

0

1

2

3

Log10[ma(eV)]

Log 10c

fa=1012 GeV

δ Θ = 3°

δ Θ = 1°

δ Θ = 0.3°

CAST SN1987A M87* SgrA*

FIG. 3: The expected parameter space probed by polarimetricobservations of M87? and Sgr A? assuming b = 1 for positionangle precisions 3◦, 1◦ and 0.3◦. We also compare our sensi-tivity with the bounds from CAST [7] and Supernova 1987A[6], assuming fa = 1012 GeV.

The emission from the accretion disk is expected tobe unstable, which makes the identification of the ax-ion induced signal somehow challenging. Nevertheless,the astrophysical variability of the disk is usually non-periodic. As illustrated in [55], the position angle of thelinearly polarized emission shows intra-hour variabilitiesin the vicinity of a few Schwarzschild radii. However,the variabilities are quite diverse for different observationtimes. Therefore, the unique behavior of periodic oscil-

lation of the position angle due to the axion field is po-tentially detectable with high-precision measurements bythe EHT, through e.g., a periodicity search after Fouriertransforming the data in the time domain. On the otherhand, a null-detection (similar to the Day 82 observa-tion of SgrA? [55] which shows relatively steady positionangles [60]) can give effective constraints on the axionparameters.

In Fig. 3, we show the axion parameter space which ispotentially probed by M87? and Sgr A?, assuming b = 1for different position angle precisions. Notice that thismethod is complementary to the constraints from blackhole spin measurements [35], which potentially excludethe region of large fa (> 1016 GeV). A non-observationof periodic oscillation angles will put an upper boundon the value of c and rule out the corresponding masswindow for fa < 1016 GeV.

Conclusion and Discussion Dense axion cloud canbe induced by rapidly rotating black holes through su-perradiance. The position angles of linearly polarizedphotons emitted near the horizon oscillate periodicallydue to the existence of the axion cloud. A polarimet-ric measurement with good spatial resolution by e.g., theEHT, is particularly crucial for such a test. The periodicchange of the position angle can be tested both tempo-rally and spatially, which would give strong hints of theexistence of axion superradiance.

Our proposed search strategy is complimentary toblack hole spin measurements where the axion self-interaction cannot be too strong. In addition, when theaxion cloud enters the non-linear region, either a drasticbosenova or a steady outflow gives amax/fa ∼ O(1) formost of the time. Thus our observable does not rely onthe detailed dynamics of the axion cloud.

The main constraint on model-dependent factors is thedimensionless axion photon coupling c = 2πgaγfa, whichis rescaled by the scale of the associated symmetry break-ing. This makes our experimental constraint unique com-paring to other experiments like CAST and SN1987Awhich only constrain the axion photon coupling gaγ .

Finally, we note that the position angle oscillationinduced by the axion background does not depend onphoton frequency. This is a unique property distinctfrom the Faraday rotation induced by the galacticmagnetic field, where the position angle is proportionalto the square of photon wavelength. Polarimetric

6

measurements at different frequencies in the future canthus be used to distinguish astrophysical backgroundand improve the sensitivity of tests of the axion super-radiance scenario.

Acknowledgements We are grateful to Nick Hous-ton, Siming Liu, Ru-Sen Lu, and Hirotaka Yoshino foruseful discussions. We also thank the anonymous ref-erees for helpful comments and suggestions. Y.C. issupported by the Labex “Institut Lagrange de Paris”(IDEX-0004-02). J.S. is supported by the National Nat-ural Science Foundation of China (NSFC) under GrantsNo.11947302, No.11690022, No.11851302, No.11675243and No.11761141011, and by the Strategic Priority Re-search Program of the Chinese Academy of Sciencesunder Grants No.XDB21010200 and No.XDB23000000.Q.Y. is supported by the NSFC under GrantsNo.11722328, No.11851305, and the 100 Talents programof Chinese Academy of Sciences. YZ is supported by U.S.Department of Energy under Award No. DESC0009959.Y.C. and Y.Z would like to thank the ITP-CAS for theirkind hospitality.

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