Fast Radio Bursts and Axion Miniclusters
FFast Radio Bursts and Axion Miniclusters
Igor I. Tkachev
Institute for Nuclear Research of the Russian Academy of Sciences, Moscow 117312, Russia
Non-linear effects in the evolution of the axion field in the early Universe may lead to the formationof gravitationally bound clumps of axions, known as “miniclusters.” Minicluster masses should bein the range M mc ∼ − M (cid:12) , and in plausible early-Universe scenarios a significant fraction ofthe mass density of the Universe may be in the form of axion miniclusters. Here I argue thatobserved properties (total energy release, duration, high brightness temperature, event rate) ofrecently discovered Fast Radio Bursts can be matched in a model which assumes explosive decay ofaxion miniclusters. I. INTRODUCTION
The recent detection of unusual radio pulses [1–5],known as Fast Radio Bursts (FRBs), has generatedstrong interest in identifying their origin and nature. Thebursts exhibit a frequency-dependent time delay, whichobeys a quadratic form so strictly, that the only expla-nation remains - signal dispersion in cold cosmic plasmaduring propagation. The magnitude of this delay, pro-portional to the electron column density along the line ofsight, and called dispersion measure, is so large that thecosmological distances are inferred for the sources.Thornton et al [3] deduce redshifts for four differentFRBs observed at Parkes radio telescope to be in therange from 0.45 to 0.96. Recently detected FRB atArecibo Observatory [4] (which has larger size antennaand therefore excludes atmospheric artefacts as possibil-ity for FRB) has derived redshift of z = 0 .
26. All ob-servers agree on a high rate of FRBs, ∼ events/dayfor the whole sky.FRBs are also characterized by extremely high fluxdensities ( ∼ Jy) over very short time scales (millisec-onds). Short time scales imply that the size of emittingregion is small, less then 300 km. Observed fluxes implythat the total energy radiated in the band of observa-tion was in the range 10 − ergs [3, 4], assumingisotropy and quoted redshifts. Derived redshifts, andtherefore the radiated energy, can be smaller if signifi-cant part of dispersion measure accumulates in the hostgalaxies. With this parameters, and assuming FRBs areat Gpc distances, their brightness temperature would be T B ∼ K, leading to the conclusion that radiationfrom FRB sources should be coherent [6–9].The source of the FRB signals is hotly debated in theliterature, with suggested progenitors ranging from ter-restrial interference to neutron star-neutron star mergers.Wide range of models was thoroughly discussed in Kulka-rni et al. [8], and refs. therein. While all models haveproblems, the verdict was that several arguments [10],which relate giant flares of young magnetars with FRBs,may offer plausible physical scenario [11] based on as-sumption that FRBs could be attributed to synchrotronmaser emission from relativistic, magnetized shocks.FRBs are so mysterious that a new physics modelswere also suggested and discussed in literature [12]-[15]. E.g., Ref. [7] even discussed, en route , the possibility thatFRBs are signals beamed at Earth by advanced civiliza-tions.In this paper I consider possible relation of FRBs andaxions [16]. In Ref. [17] it was suggested that axion fieldmay form gravitationally bound compact astrophysicalobjects, where under some conditions parametric insta-bility occurs, resulting in a powerful coherent burst ofmaser radiation. Such instability has been also studiedin Refs. [18, 19]. Here I reconsider this scenario and dis-cuss several mechanisms where axion miniclusters [20–23]are responsible for Fast Radio Bursts, see also [15] . I’dlike to note, that a general case of axion like particles(ALP, where particle mass, self coupling, and couplingto electromagnetic field are not tightly related to eachother), opens a lot of possibilities which are ruled outotherwise. I do not consider general case of ALP, stayinginstead with the standard QCD invisible axion model.Generalization to ALP is straightforward.
II. DENSE AXION OBJECTS
The invisible axion is among the best motivated can-didates for cosmic dark matter [16]. The axion isthe pseudo-Nambu–Goldstone boson resulting from thespontaneous breaking of a U (1) global symmetry knownas the Peccei–Quinn, or PQ, symmetry introduced to ex-plain the apparent smallness of strong CP-violation inQCD [24].There are stringent astrophysical, cosmological, andlaboratory constraints on the properties of the axion [16].In particular, the combination of cosmological and astro-physical considerations restricts the axion mass m a to bein the window µ eV (cid:46) m a (cid:46) meV. Corresponding valueof the axion decay constant f a can be found using relation f a m a = f π m π , where π referes to pion. The contributionto the mean density of the Universe from axions in thiswindow is guaranteed to be cosmologically significant.Thus, if axions exist, they will be dynamically importantin the present evolution of the Universe.For what follows, it is important for PQ-symmetry tobe restored after inflationary stage of the Universe evo-lution. This happens if reheating temperature is largerthan the corresponding PQ-scale, but this may also hap- a r X i v : . [ a s t r o - ph . H E ] N ov pen [25] already during reheating, in a transient highlynon-equilibrium state, even if resulting temperature issmall. In this situation the axion field takes differentvalues in different casually disconnected regions at tem-peratures well above the QCD confinement, when theaxion is effectively massless. At the confinement temper-ature and below, QCD effects produce a potential for theaxion of the form V ( θ ) = m a f a [1 − cos( θ )], where theaxion field was parametrized as a dimensionless angu-lar variable θ ≡ a/f a . Axion oscillations commence andfield variations are transformed into density contrasts, ρ a , which later lead to tiny gravitationally bound “mini-clusters”. A. Axion Miniclusters
Since density variations in this circumstances are largefrom the very beginning, we do not refer to them asperturbations, and let us call corresponding regions as”clumps”. It is easy to understand that today thoseclumps, or miniclusters, will be very dense objects. Letus specify the density of a dark-matter clump prior tomatter-radiation equality as δρ a /ρ a ≡ Φ. In situationwhen Φ ∼ V ( θ ) = m a f a θ with random initial conditions), theseclumps separate from cosmological expansion and formgravitationally bound objects already at T = T eq , where T eq is the temperature of equal matter and radiation en-ergy densities. Density of such clump today will cor-respond to a matter density back then, i.e. will be 10 times larger than the local galactic halo dark matter den-sity.However, at the time when axion oscillations com-mence, in many regions θ ∼
1, and self-interaction isimportant. Numerical investigation of the dynamics ofthe axion field around the QCD epoch [21–23] had shownthat the non-linear effects result in regions with Φ muchlarger than unity, possibly as large as several hundred,leading to enormous minicluster densities. In such situ-ation a clump separates from cosmological expansion at T (cid:39) (1 + Φ) T eq which leads to a final minicluster densitytoday given by [22] ρ mc (cid:39) (1 + Φ)¯ ρ a ( T eq ) . (1)Even a relatively small increase in Φ is important becausethe final density depends upon Φ for Φ > ∼ T ≈ − M (cid:12) . Masses of miniclusters are rel-atively insensitive to the particular value of Φ associatedwith the minicluster. Corresponding minicluster radiusas a function of M and Φ: R mc ≈ × Φ (1 + Φ) / Ω a h (cid:18) M − M (cid:12) (cid:19) / km . (2) Since large-Φ miniclusters are very dense, form early, andare well separated from each other, they should escapetidal disruption and merging.According to Ref. [23], more than 13% of all axionicdark matter are in miniclusters with Φ > ∼
10, more thanabout 20% are in miniclusters with Φ > ∼ > N ∼ .At this point, an important relation M mc ∼ − M (cid:12) = 2 × ergs , (3)and the fact that gigahertz frequency radiation is withinallowed axion mass range, ν = m a / π ≈ . m a / µ eV)GHz, should tell us that if a fraction of axion miniclustermass is rapidly transformed into radiation, this will leadto something similar to observed FRB. Interestingly, inmy notes dating back to 1997 I have found the followingphrase: “Even if the tiny fraction 10 − M (cid:12) of the mini-cluster mass 10 − M (cid:12) will go into radiation on this fre-quency, it can be detected from anywhere in the Galaxyhalo (L. Rosenberg, private communication)”. Back thenthis (and several uncertainness which I will describe be-low) actually had prevented me from submitting alreadyprepared paper. Let us discuss farther evolution of ax-ion miniclusters and possible mechanisms of their masstransfer into radiation. B. Axion Bose-clusters
Miniclusters with Φ > ∼
30 undergo the Bose-condensation later on and consequently became evendenser and more compact [20]. Usually, in the relatedliterature, an existence of a Bose-star is just postulated,without questioning of how it can be formed, for a re-view of Bose-stars see, e.g. [26]. However, in the case ofinvisible axion all couplings are so small, that mere possi-bility of condensate formation has to be studied [27–30].Simple estimates has been done in Ref. [27], while Bose-condensation in the frameworks of Boltzmann equationwas studied numerically in Ref. [28]. In Boltzmann ap-proach Bose-condensate does not form actually, one cansee only an establishment of the Kolmogorov inverse cas-cade towards zero momenta (but this approach does al-low to estimate formation time). The problem was solvedin [29] by studying numerically the evolution of initiallyrandom classical fields, both for positive and negativeself-couplings (the later corresponds to the axion case).It is remarkable that in spite of the apparent small-ness of axion quartic self-coupling, | λ a | ≈ ( f π /f a ) ∼ − f − , the subsequent relaxation in an axion mini-cluster due to 2 a → a scattering can be significantas a consequence of the huge mean phase-space den-sity of axions. Then, instead of the classical expression, t − R ∼ σρ a v e m − a , where σ is the corresponding cross sec-tion and v e typical velocity in the gravitational well, onegets [27] for the relaxation time t − R ∼ λ a ρ a v − e m − a . (4)The relaxation time (4) is smaller then the present ageof the Universe for miniclusters with Φ > ∼
30 [20]. I willcall resulting objects Bose-clusters, not Bose-stars, sincetheir total mass is not in a stellar mass range.Characteristic sizes and limiting masses of resultingobjects can be estimated [17] analyzing simple equationof “hydrostatic equilibrium” in non-relativistic limit dP ( r ) dr = − ρ ( r ) M ( r ) M r . (5)The pressure P ( r ) and density ρ ( r ) has to be understoodhere as quantities averaged over the period of field oscil-lation.
1. Non-interacting field.
Positive contribution to thepressure comes from the field gradients, and can be ap-proximated as P grad ∼ a /R , where R is the averagedsize of the configuration, and subscript “0” in what fol-lows will mean amplitude of field oscillations. This gives R ≈ m a v e ≈
300 10 − M (cid:12) M bc (cid:18) µ eV m a (cid:19) km , (6)which, depending upon axion parameters and mass of thecluster can be comparable or less then FRB’s emittingregion. The maximum possible mass of a stable Bose-cluster corresponds to v e ∼ M max ( λ = 0) ≈ M /m a . For non-interacting axions this would be in therange of ∼ − M (cid:12) .
2. Positive self-coupling.
The self-coupling may betiny, but its contribution to the pressure, P λ ∼ λa , cannot be neglected in a certain parameter range. Usingthis expression in Eq. (5) one finds [17] M max ( λ >
0) = √ λM /m . With the positive self-coupling (which cor-responds to a repulsive interaction) the maximum massof a stable Bose-cluster can be significantly bigger thanfor non-interacting particles.
2. Negative self-coupling.
The self-coupling of ax-ions is negative and their interaction is attractive. Con-sequently, there will be no-stable configuration when | P λ | > P grad . We find that the instability develops when | λ | a R > M (cid:38) M Pl / (cid:112) | λ | . For axions | λ | = m a /f a and this condition reduces to M max ( λ <
0) = f a M Pl /m a ∼ − M (cid:12) (10 µ eV /m a ) . Instability con-dition can be also re-written as θ > v e .During Bose-relaxation the mass of the Bose-condensed core in the clump grows, while its radiusshrinks. When the mass exceeds M max ( λ < R min ∼ M Pl /f a m a ≈
200 km , (7)regardless of m a . Note that the maximum mass for astable axion Bose-cluster at m a = 10 µ eV is of the orderof the typical mass of the axion minicluster. III. TRANSFER OF ENERGY INTORADIATION
Electromagnetic properties of axions are described byMaxwell’s equations ∇ [ E + α π θ B ] = 0 , (8) ∇× [ B − α π θ E ] − ∂ [ E + α π θ B ] = 0 , (9) ∇ B = 0 , ∇ × E + ∂ B = 0 , (10)where α ≈ / a → γγ were studied in Refs. [17–20].
1. Explosive maser effect.
In homogeneous axionicmedium the Fourier amplitudes, g k , of left and right-polarized photons will obey the equation¨ g k + ( k ± kα ˙ θ ) g k = 0 , (11)This equation can be reduced to the standard form forthe Mathieu equation ¨ g k + [ A − q cos(2 τ )] g k = 0 with A ≡ k /m and q ≡ kαθ /m . The number of photonsin certain momentum bands will grow exponentially intime, n k = exp( µ k t ). Maximum amplification is achievedfor k = m/ µ = αθ m/ a → γγ ), and radiation is amplified in theband δk = µ . When incident radiation pass through thecluster, it will be amplified along the photon path. It isimportant that the position of the resonance, k = m/ µ does.It is convenient to introduce the amplification coeffi-cient for the whole cluster, D ≡ µR . If at some momentof time the condition D (cid:29) D ∼ αθ / v e for the equilibriumBose-cluster, R ∼ /mv e . In the axion case the clusteris in ”hydrostatic” equilibrium when θ < v e , therefore D < α/
2. While clusters with
D < ≈ θ ∼ π .Indeed, physical pion and axion are mass eigenstates ofsmall oscillations around the minima of the common po-tential (see e.g. [33]) V ( θ, β ) = f π m π [cos θ cos β − ξ sin θ sin β + 1] , (12)where ξ ≡ ( m d − m u ) / ( m d + m u ) ≈ . β ≡ π /f π .With θ ∼
1, the pion field will develop too. Which par-ticle species and with what spectra will be created, andwhat would be the rate of corresponding explosions areinteresting problems to study.
2. Decay in magnetic field.
In strong magnetic fieldsaxions can oscillate into photons. These oscillations areemployed in axion laboratory searches and may happenin astrophysical environment too [34].Resonant conversion happens when the photon plasmamass equals to the axion mass. In changing magneticfields and electron plasma density this condition may bemet at some distance from the astrophysical object. Iwill not question all range of posible astrophysical envi-ronments, but simply use resent work [35] where resonantconversion of dark matter axions to photons in magneto-spheres of neutron stars has been considered.Calculations were presented for a particular example ofthe neutron star with magnitude of magnetic field on itssurface 10 G, and spin period 10 s. It was found, thatconversion probability reaches P = 0 . m a = 5 µeV at a distance r = 3 . r ns , where magneticfield equals 2 . × G. Conversion reaches maximumpossible value of P = 0 . m a = 7 µeV , and staysat this value for larger axion masses. Bandwidth of thesignal was found to be in the range of 5 MHz, however,calculations in Ref. [35] where done for the case of un-clustered halo axions. In the case of explosive conversionof the (fraction) of minicluster, with all accompanyingplasma effects the bandwidth will be higher. E.g., inRef. [7] it was noted that electric fields near FRBs whichare at cosmological distances would be so strong thatthey could accelerate free electrons from rest to relativis-tic energies in a single wave period. Questions relatedto resulting spectrum, burst duration and energy releaserequire further detailed study.Similarly to the case of a free minicluster, stimulatedaxion-photon conversion is possible in external magneticfield as well. To find the rate of stimulated emission onehas simply to solve classical field equations, see Ref. [36],which in the present case will be Eq. (8)-Eq. (10). Axionminicuster is not a collection of a particles, but, as a con-sequence of a huge phase-space density, can be describedas a random classical field, therefore this approach isvalid. (Eq. (8) in external magnetic field suggests partic-ular solution E = − αθ B ext / π , but this can be mislead-ing since the source for electric field is actually zero inhomogeneous axion field. The whole system of equationshas to be solved.) When stimulated process develop infull strength, such that back reaction is important, re-sulting spectra of created particles are not narrow, but S / N n (MHz) FIG. 1. Signal to noise ratios for detected FRB’s after fre-quency shift by cosmological redshifts, (1 + z ). Solid line cor-responds to data from Ref. [4], while dashed lines to Ref.[3]. display series of peaks [36], which evolve later on [37] intopower law spectra of Kolmogorov turbulence.Let us estimate event rate in this case. Number densityof miniclusters is n ≈ pc − . Gravitational captureradius for a neutron star in a halo with typical velocities v h ∼ − gives for cross-section σ ≈ × km . Thisgives for event rate nσv h ∼ × − day − for a colli-sions with single neutron star. Taking 10 as an estimatefor a number of neutron stars with strong magnetic field,and multiplying by the number of galaxies in a visibleUniverse we obtain 4 × day − for the event rate inthis scenario. IV. DISCUSSION
We have shown that observed properties of FRB’s (to-tal energy release, duration, high brightness temperature,event rate) can be matched in a model which assumes ex-plosive decay of axion miniclusters. Primary frequencyof radiation will correspond to m a / m a for the decay in externalmagnetic field. Comparison of predicted and observedspectra may help to falsify the model. In particular, the(narrow) frequency band of resulting signal, in the refer-ence frame of the source, should be always at the sameposition in suggested model, while in pure astrophysi-cal scenarios, e.g. in the model of Ref. [11], there is noparticular reason for this to hold.In Fig. (1) the signal to noise ratios for several FRBsobserved in Refs. [3, 4] are presented. With respect to theoriginal data, I have made a frequency shift by (1 + z ),using cosmological redshifts quoted for each FRB. TheAresibo burst, shown by the solid line, has unusual (as itwas noted in Ref. [4]), steeply rising spectrum, S ∝ ν α ,with the best fit value α = 11. This peculiarity of thesignal was interpreted as a consequence of being detectedin a sidelobe of the receiver. For the highest S/N event inRef. [3] another peculliary has been stressed: spectrumhas well defined bands, of 100 MHz width. For the otherthree events, the verdict was that they do not have suf-ficiently high S/N to say something definite about theirspectra. All of the above can be true. On the other hand,as Fig. 1 suggests, the S/N for all of this bursts repeatsimilar pattern, which may be actually due to a narrowbandwidth of a maser emission. Visible, rather narrowpeaks are not at the same frequency. However, spectra ofwell developed stimulated emission may display several peaks [36]. Also, redshifts to FRBs are not really knownbecause of a possible significant dispersion at the sources. ACKNOWLEDGMENTS
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