The role of resonant plasma instabilities in the evolution of blazar induced pair beams
MMNRAS , 1โ15 (2021) Preprint 8 February 2021 Compiled using MNRAS L A TEX style ๏ฌle v3.0
The role of resonant plasma instabilities in the evolution of blazar inducedpair beams
Roy Perry , Yuri Lyubarsky Physics Department, Ben-Gurion University, P.O.B. 653, Beer-Sheva 84105, Israel
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
The fate of relativistic pair beams produced in the intergalactic medium by very-high energy emission from blazars remainscontroversial in the literature. The possible role of resonance beam plasma instability has been studied both analytically andnumerically but no consensus has been reached. In this paper, we thoroughly analyze the development of this type of instability.This analysis takes into account that a highly relativistic beam loses energy only due to interactions with the plasma wavespropagating within the opening angle of the beam (we call them parallel waves), whereas excitation of oblique waves resultsmerely in an angular spreading of the beam, which reduces the instability growth rate. For parallel waves, the growth rate isa few times larger than for oblique ones, so they grow faster than oblique waves and drain energy from the beam before itexpands. However, the speci๏ฌc property of extragalactic beams is that they are extraordinarily narrow; the opening angle isonly ฮ ๐ โผ โ โ โ . In this case, the width of the resonance for parallel waves, โ ฮ ๐ , is too small for them to grow inrealistic conditions. We perform both analytical estimates and numerical simulations in the quasilinear regime. These show thatfor extragalactic beams, the growth of the waves is incapable of taking a signi๏ฌcant portion of the beamโs energy. This type ofinstability could at best lead to an expansion of the beam by some factor but the beamโs energy remains nearly intact. Key words: instabilities, plasmas, intergalactic medium, jets
The high-energy relativistic jets from blazars emit gamma ray radi-ation, which travels along in the same direction as the jets. Duringtheir course through the intergalactic medium (IGM), the very highenergy (VHE) gamma photons interact with the extragalactic back-ground light (EBL) and the cosmic microwave background (CMB).This interaction has a mean free path ranging from several Mpc to1 Gpc, which leads to the production of ultra-relativistic electron-positron pairs with Lorentz factors of about ๐พ โผ โ . Theserelativistic beams propagate in the IGM and are expected to loseenergy by inverse-Compton scattering upon the CMB. However, theobserved gamma-ray bump associated with the product photons ofinverse-Compton cascades (ICC) is lower than predicted (e.g. Aha-ronian et al. 2006; Neronov & Vovk 2010; Broderick et al. 2012).One possible explanation is pair beam de๏ฌection by the inter-galacticmagnetic ๏ฌeld. Measurement of the GeV ๏ฌux on telescopes in con-junction with the estimation of blazar-originated gamma rays ๏ฌux, alower bound on the intergalactic magnetic ๏ฌeld is predicted. The in-consistencies between the predicted and measured ๏ฌux are attributedto the de๏ฌection of particles by the magnetic ๏ฌeld (Archambault et al.2017; Tiede et al. 2017; Ackermann et al. 2018). An alternate possi-bility (Broderick et al. 2012; Schlickeiser et al. 2012) is that beforethe beam pairs up-scatter photons, they lose energy through excita-tion of plasma waves via the resonance beam-plasma instability. Thebeam-plasma instability results in excitation of plasma oscillations(Langmuir waves) with a frequency close to the plasma frequency(e.g., Krall & Trivelpiece 1973). The IGM density is of the order of10 โ cm โ , so the plasma frequency is of the order of a few Hz, while the beam density ranges over many orders of magnitude, though es-timated to be no more than 10 โ cm โ (e.g., Broderick et al. 2012).The instability could develop in two di๏ฌerent regimes: reactive (hy-drodynamic) and kinetic (e.g., Boyd et al. 2003). The ๏ฌrst appliesto an essentially cold beam, whose thermal spread is small enoughsuch that all the particles are in resonance with the wave. The secondcorresponds to the case of a wide spread in the distribution functionof the beam. In that case, only a fraction of the particles contribute tothe growth of the wave. If the beam is initially narrow enough suchthat the instability develops in the hydrodynamic regime, then mostlyoblique waves are excited. Therefore the beam expands while losingalmost no energy (Fainberg et al. 1970). Eventually, the momentumspread of the beam becomes large enough for the kinetic regime tocome into play. During the kinetic regime, the beam could lose mostof its energy by exciting plasma waves which eventually heats thebackground plasma (Fainberg et al. 1970; Rudakov 1971; Breหizman& Ryutov 1971). The literature on the collective plasma processes inextragalactic pair beams is extensive but its conclusions still remaincontroversial. Miniati & Elyiv (2013) studied di๏ฌerent mechanismsthat a๏ฌect the development of resonance instability and concludedthat for typical beam parameters, either non-linear wave interactionsor inhomogeneities of the background plasma suppress this instabil-ity. However, no consensus has been reached about the role of thecollective plasma processes in the evolution of extragalactic beams(e.g., Broderick et al. 2012, 2014; Schlickeiser et al. 2012, 2013;Miniati & Elyiv 2013; Chang et al. 2014; Krakau & Schlickeiser2014; Supsar & Schlickeiser 2014; Chang et al. 2016; Sironi & Gi-annios 2014; Kempf et al. 2016; Ra๏ฌghi et al. 2017; Broderick et al. ยฉ a r X i v : . [ a s t r o - ph . H E ] F e b Perry et al.
The beam-plasma instability (e.g., Krall & Trivelpiece 1973) resultsin excitation of plasma oscillations (Langmuir waves) satisfying theresonance (Cerenkov) condition: ๐ = k ยท v , (1)where ๐ and k are the frequency and wave vector of the wave. Fordilute beams, ๐ ๐ (cid:28) ๐ ๐ , the frequency of the excited waves is close to the plasma frequency, ๐ ๐ = โ๏ธ ๐๐ ๐ ๐ ๐ ๐ , (2)where ๐ ๐ , ๐ are the electron mass and charge, respectively, and ๐ ๐ is the plasma density.The extragalactic beams are highly energetic; the typical particleLorentz factor is ๐พ โผ โ . Even though the beam energy spreadis large, ฮ ๐พ โผ ๐พ , the velocity spread is negligibly small, ฮ ๐ฃ โผ ฮ ๐พ / ๐พ .Therefore we can describe each beam particle velocity in sphericalcoordinates (on the azimuthal plane ๐ =
0) as v = ๐ (cid:16) โ ๐ / , ๐ (cid:17) , (3)where the ๏ฌrst and the second components correspond to the particlevelocity components along and perpendicular to the beam direc-tion, respectively. The wave-vector, k , can be described in a three-dimensional spherical coordinates space as: k = ( ๐, ๐ , ๐ ) . (4)The resonance condition (1) can then be written as:cos ๐ = ๐ ๐ / ๐๐ โ cos ๐ cos ๐ sin ๐ sin ๐ . (5)One can therefore describe the resonance wave-vector k with twoindependent parameters, ๐ and ๐ .The geometry of the particle-wave resonance in velocity space isshown in Figure 1. The arrows represent the directions of propagationof the Langmuir waves, where ๐ , ๐ (cid:48) are the angles of these vectorswith respect to the beam axis, chosen as the horizontal axis. The arcrepresents the distribution of the pair-beam particles, ranging over anangular spread of ฮ ๐ (cid:28)
1. The range of resonant waves is signi๏ฌedwith the bordered arrow between the two dashed lines. For any ๐ ,only waves in this range of k can exchange energy and momentumwith the beam particles. The ๏ฌgure shows that the range of resonantwaves, ฮ ๐ , decreases with decreasing wave vector angle. Indeed, to๏ฌrst order in ฮ ๐ , the range of resonance ฮ ๐ behaves as โผ tan ๐ ฮ ๐ .Therefore ฮ ๐ / ๐ โผ ฮ ๐ at ๐ โผ
1, but when the wave propagates withinthe opening angle of the beam, ๐ โผ ฮ ๐ , the resonance becomes verynarrow, ฮ ๐ / ๐ โผ ฮ ๐ .The angular spread of intergalactic beams is very small. After pair-production, the angular distribution of the particles is quite narrowbecause of their direct emission from a point-like source, far awayfrom the intergalactic plasma. Initially, the spread is about ฮ ๐ โผ ๐พ โ .The beam could expand as a result of the instability but not bymuch. Therefore the width of the resonance is very small even foroblique waves, whereas for waves nearly aligned with the beam, itis extraordinarily small. We show below that this has a profounde๏ฌect on the evolution of the beam because on the one hand, onlynearly aligned waves may be responsible for any energy loss ofthe beam, but on the other hand, they easily violate the resonanceconditions. Therefore, their growth may be easily suppressed byweak perturbations, such as inhomogeneity of the background plasmaor/and non-linear wave interactions. The particle distribution of the beam evolves because of the inducedemission and absorption of plasma waves. The energy and momen-tum conservation principles in a single emission/absorption event aredescribed via the following equations: ๐ธ = ๐ธ (cid:48) ยฑ โ ๐ ๐ ; p = p (cid:48) ยฑ โ k . (6) MNRAS , 1โ15 (2021) lazar induced pair beams Figure 1. (Color online) The beam-wave resonance condition in velocityspace. Shown are the particle velocity distribution (thick blue arc with anopening angle ฮ ๐ (cid:28)
1) and two plasma waves at two angles ๐ (cid:29) ฮ ๐ and ๐ (cid:48) โผ ฮ ๐ (red and green arrows). The range of the phase velocities which arein resonance with the beam is larger for the ๏ฌrst wave than for the second one. Here we denote the parameters of the beam particle before and afterthe wave excitation by unprimed and primed quantities, respectively.For concreteness, consider an emission event and assume for sim-plicity that the beam axis, the particle velocity and the wave vectorare all in the same plane. This imples: ๐ฟ๐พ = ๐พ (cid:48) โ ๐พ = โ โ ๐ ๐ ๐๐ , (7) ๐๐พ (cid:48) ๐ฃ (cid:48) sin ๐ (cid:48) โ ๐๐พ๐ฃ sin ๐ = โ โ ๐ sin ๐ ,๐๐พ (cid:48) ๐ฃ (cid:48) cos ๐ (cid:48) โ ๐๐พ๐ฃ cos ๐ = โ โ ๐ cos ๐ , (8)where ๐ , ๐ are the particle and wave vector angles with respect tothe beam axis, respectively. We consider a very narrow, ๐, ๐ (cid:48) (cid:28) ๐พ (cid:29)
1, beam and expand up to ๏ฌrst order. Thepair of momentum equations now become:cos ( ๐ โ ๐ ) (cid:0) ๐พ (cid:48) ๐ (cid:48) โ ๐พ๐ (cid:1) = โ โ ๐ ๐ ๐๐ sin ๐ , cos ( ๐ โ ๐ ) (cid:0) ๐พ (cid:48) โ ๐พ (cid:1) = โ โ ๐ ๐ ๐๐ cos ๐ , (9)Upon dividing the ๏ฌrst equation in (9) by the second one and rear-ranging terms, we get the ratio between relative angular and energeticchange, (cid:12)(cid:12)(cid:12)(cid:12) ๐ฟ๐๐ ๐พ๐ฟ๐พ (cid:12)(cid:12)(cid:12)(cid:12) = tan ๐ ๐ . (10)Eq. (10) shows that the outcome of the emission/absorption processstrongly depends on the relative angles between the particle and thewave. Namely, the relative change in energy is comparable to therelative change in angle only when the wave is directed within theopening angle of the beam, ๐ โผ ฮ ๐ . In the case ๐ (cid:29) ฮ ๐ , theparticleโs angle varies signi๏ฌcantly more than its energy.This conclusion may be considered a preliminary one because itis based only on analysis of a single emission/absorption event. Theparticle distribution function evolves as a result of many interac-tions. The evolution may be described as di๏ฌusion in the momentumspace (e.g., Brejzman 1990). In section 5.1, we show rigorously, by analysing the momentum di๏ฌusion coe๏ฌcients, that only quasi-parallel waves ( ๐ โผ ฮ ๐ ) take away any energy e๏ฌciently whereasthe interaction with oblique waves ( ๐ (cid:29) ฮ ๐ ) leads only to beamexpansion.The evolution of the beam is governed by the total spectrum ofresonant plasma oscillations. The dominant part of the spectrum willdetermine whether the beam loses energy or expands in angle. Theinstability mechanism will enhance the spectrum of waves in eachdirection according to their respective growth rate. Those which growthe fastest are the ones who primarily control the evolution of thebeam. Since the growth is exponential, even a moderate excess ingrowth rate will make the corresponding range of the wave spectrumdominant. It follows from eq. (10), that the beam will lose its energy, ifsmall angle waves ( ๐ โผ ฮ ๐ ) grow faster. Indeed, according to kineticlinear theory, the growth rate is larger for such waves (Breหizman &Ryutov 1971; Rudakov 1971, see also section 3.2). However, theintergalactic beams are very narrow. Therefore the resonance widthis especially narrow, โผ ฮ ๐ , for these waves, as we have seen in theprevious subsection. In this case, even a weak inhomogeneity of theIGM or/and weak nonlinear processes result in loss of the resonancebetween these waves and the beam, which suppresses their growthrate. When a wave propagates through a steady but weakly inhomogeneousplasma, the wave vector varies according to the geometrical opticsprescription ๐ k ๐๐ก = โโ ๐ ๐ . (11)Taking into account that ๐ โผ ๐ ๐ / ๐ one sees that if the characteristicscale of the density variation is ๐ฟ , such that |โ ๐ ๐ | = ๐ ๐ / ๐ฟ , therelative change in the wave vector after time ๐ก would be ๐ฟ๐๐ โผ ๐๐ก๐ฟ . (12)Let a wave be excited initially in the resonance range, bounded bythe two dashed lines in Figure 1. As it travels through the plasma,its wave number moves along that line, towards one of the resonanceboundaries, until it crosses that boundary and exits the resonance re-gion. If the exit time is smaller than the instability time, the instabilitydoes not develop (Breizman & Ryutov 1970). Miniati & Elyiv (2013)have shown that inhomogeneity and other e๏ฌects (non-linearity, col-lisional damping) impose strong constraints on the development ofthe instability of intergalactic beams. Since then, these e๏ฌects werestudied by many authors quoted in our introduction. In this paper,we show that the constraints on the energy exchange between thebeam and the background plasma are much more severe because ofa special role played by quasi-parallel waves. We have seen that theresonance width decreases as the angle between the wave and beamdirections decreases. Hence, the inhomogeneity causes a loss of res-onance much faster for quasi-parallel waves than for oblique waves.We concentrate on the e๏ฌect of inhomogeneity. Non-linear e๏ฌectscould also suppress the instability by removing the waves from theresonance. However, we show in section 4 that in the context of in-tergalactic beams, inhomogeneity alone completely suppresses thegeneration of quasiparallel waves, responsible for energy loss of thebeam.It was claimed recently (Shalaby et al. 2018; Shalaby et al. 2020)that the inhomogeneity does not suppress the instability. The authors MNRAS000
1, beam and expand up to ๏ฌrst order. Thepair of momentum equations now become:cos ( ๐ โ ๐ ) (cid:0) ๐พ (cid:48) ๐ (cid:48) โ ๐พ๐ (cid:1) = โ โ ๐ ๐ ๐๐ sin ๐ , cos ( ๐ โ ๐ ) (cid:0) ๐พ (cid:48) โ ๐พ (cid:1) = โ โ ๐ ๐ ๐๐ cos ๐ , (9)Upon dividing the ๏ฌrst equation in (9) by the second one and rear-ranging terms, we get the ratio between relative angular and energeticchange, (cid:12)(cid:12)(cid:12)(cid:12) ๐ฟ๐๐ ๐พ๐ฟ๐พ (cid:12)(cid:12)(cid:12)(cid:12) = tan ๐ ๐ . (10)Eq. (10) shows that the outcome of the emission/absorption processstrongly depends on the relative angles between the particle and thewave. Namely, the relative change in energy is comparable to therelative change in angle only when the wave is directed within theopening angle of the beam, ๐ โผ ฮ ๐ . In the case ๐ (cid:29) ฮ ๐ , theparticleโs angle varies signi๏ฌcantly more than its energy.This conclusion may be considered a preliminary one because itis based only on analysis of a single emission/absorption event. Theparticle distribution function evolves as a result of many interac-tions. The evolution may be described as di๏ฌusion in the momentumspace (e.g., Brejzman 1990). In section 5.1, we show rigorously, by analysing the momentum di๏ฌusion coe๏ฌcients, that only quasi-parallel waves ( ๐ โผ ฮ ๐ ) take away any energy e๏ฌciently whereasthe interaction with oblique waves ( ๐ (cid:29) ฮ ๐ ) leads only to beamexpansion.The evolution of the beam is governed by the total spectrum ofresonant plasma oscillations. The dominant part of the spectrum willdetermine whether the beam loses energy or expands in angle. Theinstability mechanism will enhance the spectrum of waves in eachdirection according to their respective growth rate. Those which growthe fastest are the ones who primarily control the evolution of thebeam. Since the growth is exponential, even a moderate excess ingrowth rate will make the corresponding range of the wave spectrumdominant. It follows from eq. (10), that the beam will lose its energy, ifsmall angle waves ( ๐ โผ ฮ ๐ ) grow faster. Indeed, according to kineticlinear theory, the growth rate is larger for such waves (Breหizman &Ryutov 1971; Rudakov 1971, see also section 3.2). However, theintergalactic beams are very narrow. Therefore the resonance widthis especially narrow, โผ ฮ ๐ , for these waves, as we have seen in theprevious subsection. In this case, even a weak inhomogeneity of theIGM or/and weak nonlinear processes result in loss of the resonancebetween these waves and the beam, which suppresses their growthrate. When a wave propagates through a steady but weakly inhomogeneousplasma, the wave vector varies according to the geometrical opticsprescription ๐ k ๐๐ก = โโ ๐ ๐ . (11)Taking into account that ๐ โผ ๐ ๐ / ๐ one sees that if the characteristicscale of the density variation is ๐ฟ , such that |โ ๐ ๐ | = ๐ ๐ / ๐ฟ , therelative change in the wave vector after time ๐ก would be ๐ฟ๐๐ โผ ๐๐ก๐ฟ . (12)Let a wave be excited initially in the resonance range, bounded bythe two dashed lines in Figure 1. As it travels through the plasma,its wave number moves along that line, towards one of the resonanceboundaries, until it crosses that boundary and exits the resonance re-gion. If the exit time is smaller than the instability time, the instabilitydoes not develop (Breizman & Ryutov 1970). Miniati & Elyiv (2013)have shown that inhomogeneity and other e๏ฌects (non-linearity, col-lisional damping) impose strong constraints on the development ofthe instability of intergalactic beams. Since then, these e๏ฌects werestudied by many authors quoted in our introduction. In this paper,we show that the constraints on the energy exchange between thebeam and the background plasma are much more severe because ofa special role played by quasi-parallel waves. We have seen that theresonance width decreases as the angle between the wave and beamdirections decreases. Hence, the inhomogeneity causes a loss of res-onance much faster for quasi-parallel waves than for oblique waves.We concentrate on the e๏ฌect of inhomogeneity. Non-linear e๏ฌectscould also suppress the instability by removing the waves from theresonance. However, we show in section 4 that in the context of in-tergalactic beams, inhomogeneity alone completely suppresses thegeneration of quasiparallel waves, responsible for energy loss of thebeam.It was claimed recently (Shalaby et al. 2018; Shalaby et al. 2020)that the inhomogeneity does not suppress the instability. The authors MNRAS000 , 1โ15 (2021)
Perry et al. considered a monochromatic 1D beam propagating through the den-sity distribution with a local minimum and applied periodic boundaryconditions. They claim that within a "bowl", a standing plasma waveis formed between two turning points, which remain in resonancewith the beam all the time. This arti๏ฌcial example could not be con-sidered a disproof of the general e๏ฌect outlined above. First of all,the instability develops only in a small region in the vicinity of thelocal density minimum point. In the the 3D case, the e๏ฌects of wavelocking are irrelevant for most of the systemโs volume, even if thereare many density "bowls". Moreover, in 1D simulations with periodicboundary conditions, the beam enters the bowl already appropriatelymodulated during the previous passages, therefore the growth of per-turbations proceeds further, which does not happen in the real 3Dworld.Additionally, it is worth stressing that the work by Shalaby et al.2018; Shalaby et al. 2020 is based on an implicit assumption thatstanding waves are formed in the region of interest. In 1D, standingwaves are formed by back and forward directed waves bouncingbetween the turning points. In 3D, the wave changes direction atthe turning point and therefore could not remain in resonance withthe beam. The group velocity of such a non-resonant plasma waveis extremely small so that the global standing wave, in which thephases and the wave vectors of local oscillations are adjusted withinthe whole region, are not formed in the system.In any case, even if thetheory by Shalaby et al. (2018); Shalaby et al. (2020) is universallycorrect, their instability growth rate is still smaller than the collisionaldecay rate of the plasma waves in the IGM. In sect. 4 below, whereparameters of the beam are discussed, we present the correspondingestimate.
The growth rate of the instability ฮ can be calculated by makinguse of Maxwellโs equations and the linearized Vlasov equation (e.g.,Krall & Trivelpiece 1973). It is common in the literature to describethis instability mechanism in two di๏ฌerent regimes, i.e. the hydro(reactive) regime and the kinetic regime. The hydrodynamic regime applies to waves which resonate with thebeam as a whole. In other words, the beam is narrow enough suchthat the parallel velocity component of all the particles matches thephase velocity of the wave. For every wave vector whose parallelcomponent satis๏ฌes ๐ฃ ๐ ๐ (cid:107) = ๐ , where ๐ฃ ๐ is the typical velocity ofthe beam particles, the interaction is unstable but waves with di๏ฌerentperpendicular components grow at di๏ฌerent rates.The dispersion relation for unstable Langmuir waves in the elec-trostatic approximation looks like (Bludman et al. 1960):1 โ ๐ ๐ ๐ โ ๐ ๐ (cid:16) ๐ โ ๐ (cid:107) v ๐ (cid:17) (cid:18) ๐พ sin ๐ + cos ๐ ๐พ (cid:19) = , (13)where ๐ ๐ = โ๏ธ ๐๐ ๐ ๐ / ๐ ๐ is the beam plasma frequency. There is awide range of unstable solutions of this equation but in the case ofinterest, ๐ ๐ / ๐ ๐ โผ โ โ โ , ๐พ โผ , only the plasma waves inresonance with the beam grow fast enough (see Appendix A). Themaximal growth rate of the resonant plasma wave, is (Bludman et al. 1960; Fainberg et al. 1970) ฮ hydro = โ / ๐ ๐ ๐พ (cid:18) ๐ ๐ ๐ ๐ (cid:19) / ( cos ๐ ) / (cid:16) + ๐พ tan ๐ (cid:17) / . (14)This implies that in the hydrodynamic regime, the fastest growingwaves are oblique ones, i.e. waves at angle ๐ โผ
1. The growth ratefor such waves is roughly ฮ hydro ( ๐ โผ ) โผ ๐ ๐ (cid:18) ๐ ๐ ๐พ๐ ๐ (cid:19) / , (15)while for waves which are almost parallel to the beam axis, i.e. ๐ โผ ฮ ๐ โผ ๐พ โ , the growth rate reduces to ฮ hydro โผ ๐ ๐ ๐พ (cid:18) ๐ ๐ ๐ ๐ (cid:19) / . (16)Therefore the beam evolution is dominated by oblique waves. Inthis case, the beam angular spread will expand, without signi๏ฌcantlylosing energy (Fainberg et al. 1970).As the beam broadens, the transverse velocity dispersion, ฮ ๐ฃ โฅ โผ ๐ ฮ ๐ increases accordingly and eventually the beam could not beconsidered โcoldโ. The hydrodynamic regime is valid as long as allthe particles interact with the same wave, which implies: ฮ (cid:29) k ยท ฮ v , (17)where ฮ v is the width of the particle velocity distribution. For abeam with angular spread ฮ ๐ , the parallel and transverse velocityspreads are ฮ ๐ฃ โฅ = ๐ ฮ ๐ and ฮ ๐ฃ (cid:107) = ๐ ฮ ๐ /
2, respectively, so that thehydrodynamic regime condition reduces to ๐ ๐ (cid:18) ฮ ๐ + tan ๐ ฮ ๐ (cid:19) (cid:28) ฮ โ๐ฆ๐๐๐ , (18)where we have set the resonance condition, ๐ (cid:107) = ๐ ๐ / ๐ . Now onecan write ฮ ๐ (cid:28) (cid:18) ๐ ๐ ๐พ๐ ๐ (cid:19) / ; ๐ โผ ,๐พ ฮ ๐ (cid:28) (cid:18) ๐ ๐ ๐ ๐ (cid:19) / ; ๐ โผ ฮ ๐. (19)We show in section 4 that in agreement with Miniati & Elyiv (2013),the condition for the hydrodynamic regime is mostly violated forcharacteristic IGM and beam parameters so that the only applicableregime for the instability mechanism is the kinetic one. The growth rate of a Langmuir wave, in the kinetic regime is givenby (e.g., Brejzman 1990)
ฮ = ๐ ๐ ๐ ๐ ๐ โซ (cid:18) k ยท ๐ ๐๐ p (cid:19) ๐ฟ (cid:0) ๐ ๐ โ k ยท v (cid:1) ๐ ๐. (20)Here the beam distribution function is normalized such that โซ ๐ ๐ ๐ = ๐ ๐ . (21)The beam is highly relativistic, which means that the magnitude of v is ๐ up to second order in ๐พ โ . Hence the delta function argumentcontains only the angles of the beam momentum vector. Therefore,integration of the delta function is performed over the azimuthalangle, ๐ , and the growth rate function could be written as an integral MNRAS , 1โ15 (2021) lazar induced pair beams solely over the polar angle. If one introduces the angular distributionfunction: ๐ ( ๐ ) = โซ ๐ ๐๐ โซ โ ๐ ( p ) ๐๐๐, (22)the growth rate can be written as ฮ = ๐ ๐ (cid:18) ๐๐๐ ๐ (cid:19) (cid:18) ๐ ๐ ๐ ๐ (cid:19) (cid:16) ๐ ๐ ๐๐ (cid:17) โซ ๐ ๐ (cid:104) ๐ ๐๐๐๐ (cid:16) cos ๐ โ ๐๐๐ ๐ cos ๐ (cid:17) โ ๐ (cid:105)โ๏ธ ( cos ๐ โ cos ๐ ) ( cos ๐ โ cos ๐ ) ๐๐๐, (23)where the boundaries ๐ , are given by:cos ๐ , = ๐ ๐ ๐๐ cos ๐ ยฑ sin ๐ โ๏ธ โ (cid:16) ๐ ๐ ๐๐ (cid:17) . (24)There is a simple estimate (Breหizman & Ryutov 1971) for the maxi-mal growth rate for a general wave vector angle, ๐ : ฮ โผ ๐ ๐ (cid:18) ๐ ๐ ๐ ๐ (cid:19) cos ๐ ๐พ ฮ ๐ . (25)One sees that the growth rate decreases with increasing beam angularwidth, ฮ ๐ .The expression (25) is a good estimate but the exact dependenceof the growth rate on ๐ is rather important since the growth of theoscillations is exponential: a factor of two in the growth rate valuewill eventually lead to di๏ฌerence by orders of magnitude betweentwo wavesโ energies. The exact growth rate depends on the shape ofthe distribution function.To demonstrate the dependence of the growth rate on the wavevector, we consider a Gaussian distribution, ๐ ( ๐ ) = ๐ ๐ ๐พ๐๐ ฮ ๐ exp (cid:18) โ ๐ ฮ ๐ (cid:19) . (26)In this case, the growth rate may be found in the form (see AppendixB) ฮ = ๐ ๐ (cid:18) ๐ ๐ ๐ ๐ (cid:19) ๐๐พ ฮ ๐ cos ๐ exp (cid:32) โ ๐ + ๐ ฮ ๐ (cid:33) ร (cid:34)(cid:32) ๐ (cid:18) ฮ ๐ ๐ (cid:19) โ (cid:33) ๐ผ ( ๐ ) โ ๐ ๐ผ ( ๐ ) (cid:35) , , (27)where ๐ โก (cid:16) ๐ โ ๐ (cid:17) / ฮ ๐ and ๐ผ , ( ๐ฅ ) are the modi๏ฌed Besselfunctions of the ๏ฌrst kind. The growth rate depends on the directionof the wave, ๐ , and on the absolute value of the wave vector, ๐ .Figure 2 shows the maximum growth rate with respect to ๐ as afunction of ๐ . For large angles, ๐ (cid:29) ฮ ๐ , the maximum growth ratevaries as cos ๐ , according to the general estimate (25). Becauseof that dependence, the growth rate is constant in a wide range ofangles, ฮ ๐ (cid:28) ๐ (cid:28)
1. The shape of the maximum growth rate curvefor ๐ โผ ฮ ๐ is sensitive to the shape of the distribution function.Speci๏ฌcally for the Gaussian case, one gets a valley near ๐ โผ ฮ ๐ and the growth rate increases again to a plateau at ๐ (cid:28) ฮ ๐ . Theplateau at very small angles is larger than the intermediate anglesplateau ฮ ๐ (cid:28) ๐ (cid:28)
1, for this case by roughly 10%.For every wave vector directed at the angle ๐ , there exists a narrowrange of ๐ which is within resonance with the narrow particle beam.One can conveniently use instead of ๐ , the variable ๐ฅ = ๐ ๐ ๐๐ โ cos ๐ โ๏ธ โ ๐ ๐ ๐๐ cos ๐ . (28) Figure 2.
The maximum growth rate, ฮ max , as a function of wave direction, ๐ , for a Gaussian distribution with ฮ ๐ = โ . For any ๐ , the maximumwas found with respect to the resonance parameter ๐ฅ . The growth rate isnormalized with respect to ฮ = ๐ ๐ (cid:0) ๐ ๐ / ๐ ๐ (cid:1) (cid:16) ๐พ ฮ ๐ (cid:17) โ . For the remainder of this analysis, we shall use the wave vectorresonance parameter, ๐ฅ , as the wave parameter instead of ๐ . Thetransition from ๐ to ๐ฅ makes the various integration formulas moreconvenient. For large wave-vector angles, ๐ (cid:29) ๐ , | ๐ฅ | โค ๐, (29)which is independent of ๐ . For simple estimations, one can usuallyassume that | ๐ฅ | โผ ฮ ๐ , where ฮ ๐ is the angular spread of the beamvelocity distribution. Since ๐ฅ is of the same order as ๐ and ฮ ๐ , onecan safely assume that ๐ฅ is a small parameter, which considerablysimpli๏ฌes the derivation (see Appendix C).For ฮ ๐ (cid:28) ๐ and ๐ (cid:29) ฮ ( ๐ (cid:29) | ๐ฅ |) = ๐ ๐ โ ๐ (cid:18) ๐ ๐ ๐ ๐ (cid:19) (cid:16) โ ๐ฅ ฮ ๐ (cid:17) exp (cid:16) โ ๐ฅ ฮ ๐ (cid:17) ๐พ ฮ ๐ cos ๐ . (30)This growth rate is maximized for ๐ฅ = โ ฮ ๐ โ whose value is: ฮ max ( ๐ (cid:29) ฮ ๐ ) = ๐ ๐ โ๏ธ ๐ ๐ (cid:18) ๐ ๐ ๐ ๐ (cid:19) cos ๐ ๐พ ฮ ๐ . (31)For the other extreme case where | ๐ฅ | (cid:29) ๐ and ๐ฅ <
0, which corre-sponds to very small wave vector angles, the growth rate is ฮ (| ๐ฅ | (cid:29) ๐ ) = ๐๐ ๐ (cid:18) ๐ ๐ ๐ ๐ (cid:19) (cid:18) ๐ฅ ฮ ๐ โ (cid:19) ๐ โ ๐ฅ ฮ ๐ ๐พ ฮ ๐ , (32)which is maximized for ๐ฅ = โ ฮ ๐ and its value is: ฮ max ( ๐ โ ) = ๐๐ ๐ ๐ (cid:18) ๐ ๐ ๐ ๐ (cid:19) ๐พ ฮ ๐ . (33)One can show that the modes ๐ฅ > ๐ฅ .The behavior of the maximum curve at ๐ โผ ฮ ๐ signi๏ฌcantly de-pends on the shape of the distribution function (Breหizman & Ryutov1971; Rudakov 1971). In the general case, the growth rate could onlybe found numerically. Taking into account a rather non-trivial behav-ior of the growth rate at ๐ โผ ฮ ๐ , we could not use the same generalformula (23) and the same grid in order to ๏ฌnd the growth rate in MNRAS000
0, which corre-sponds to very small wave vector angles, the growth rate is ฮ (| ๐ฅ | (cid:29) ๐ ) = ๐๐ ๐ (cid:18) ๐ ๐ ๐ ๐ (cid:19) (cid:18) ๐ฅ ฮ ๐ โ (cid:19) ๐ โ ๐ฅ ฮ ๐ ๐พ ฮ ๐ , (32)which is maximized for ๐ฅ = โ ฮ ๐ and its value is: ฮ max ( ๐ โ ) = ๐๐ ๐ ๐ (cid:18) ๐ ๐ ๐ ๐ (cid:19) ๐พ ฮ ๐ . (33)One can show that the modes ๐ฅ > ๐ฅ .The behavior of the maximum curve at ๐ โผ ฮ ๐ signi๏ฌcantly de-pends on the shape of the distribution function (Breหizman & Ryutov1971; Rudakov 1971). In the general case, the growth rate could onlybe found numerically. Taking into account a rather non-trivial behav-ior of the growth rate at ๐ โผ ฮ ๐ , we could not use the same generalformula (23) and the same grid in order to ๏ฌnd the growth rate in MNRAS000 , 1โ15 (2021)
Perry et al. the entire range of possible resonance waves. Instead, we introducethree overlapping regions in the ( ๐ , ๐ฅ ) plane. Within each region weexpand the general formula to leading order in the small parameters.After a long derivation (see Appendix D) we get:(i) ๐ (cid:29) | ๐ฅ | ฮ = ๐ ๐ (cid:18) ๐๐๐ ๐ (cid:19) (cid:18) ๐ ๐ ๐ ๐ (cid:19) sgn ( ๐ฅ ) cos ๐ โซ โ ๐๐๐๐ง โ ๐ง โ ๐๐ง, (34)where ๐ง = ๐ / ๐ฅ . (35)Note that this function is anti-symmetric in ๐ฅ .(ii) ๐ โผ | ๐ฅ | For brevity one can de๏ฌne new parameters: ๐ โก ๐ / ๐ฅ ; ๐ โก sgn ( ๐ฅ ) โ๏ธ + ๐ โ . (36)The boundaries of the integral, eq. (24), are rewritten in the variable ๐ง as ๐ง , = ๐ โ ๐ ยฑ โ๏ธ ๐ ( ๐ โ ๐ ) , (37)The resonance range of ๐ฅ guarantees that the roots ๐ง , are real. Thegrowth rate in this case is given by ฮ = ๐ ๐ (cid:18) ๐๐๐ ๐ (cid:19) (cid:18) ๐ ๐ ๐ ๐ (cid:19) โซ ๐ง ๐ง ๐๐๐๐ง ( ๐ โ ๐ง ) โ ๐ โ๏ธ ( ๐ง โ ๐ง ) ( ๐ง โ ๐ง ) ๐๐ง. (38)(iii) ๐ (cid:28) | ๐ฅ | , ๐ฅ < ๐ง = ฮ = ๐๐ ๐ (cid:18) ๐๐๐ ๐ (cid:19) (cid:18) ๐ ๐ ๐ ๐ (cid:19) (cid:20)(cid:18) โ ๐๐๐๐ง (cid:19) โ ๐ (cid:21) ๐ง = . (39)The three cases show that the growth rate is always proportional tothe same basic value ฮ : ฮ = ๐ ๐ ๐ ๐ ๐พ ฮ ๐ ๐ ๐ , (40)which is consistent with the estimate given by eq. (25).As an example of a non-Gaussian angular distribution, we calculatethe instability growth rate for the function ๐ = ๐ exp (cid:34) โ (cid:18) ๐ ฮ ๐ (cid:19) (cid:35) , (41)which describes the self-similar expansion of the beam (Rudakov1971). Figure 3 shows the maximal growth rate with respect to thevariable ๐ฅ as a function of ๐ for the case ฮ ๐ = โ . One sees thatfor ๐ (cid:29) ฮ ๐ , the maximum growth rate is proportional to cos ๐ ,according to the general estimate (25); this part is not sensitive to theshape of the distribution function. The e๏ฌect of the beam distributionis seen clearly at ๐ โผ ฮ ๐ . In this range, the maximal growth rateincreases by a factor of three. This is an e๏ฌect of a steeper gradientin the distribution function at ๐ โผ ฮ ๐ . For distributions with a steepfront, the growth rate for the angles ๐ โผ ฮ ๐ turns out to be larger thanfor moderate angles ๐ โผ ๐ โผ ฮ ๐ implies that the spectrum ofexcited waves is dominated by the waves propagating within thebeam opening angle. These are the same waves which are capableof e๏ฌcient energy exchange with the beam. This subtle fact explains Figure 3.
The same as in Fig. 2 for the beam distribution function (41) with ฮ ๐ = โ . why in the kinetic regime, relativistic beams e๏ฌciently lose theirenergy (Fainberg et al. 1970; Breหizman & Ryutov 1971; Rudakov1971). However, we have seen above that in very narrow beams, thewidth of resonance for these waves is especially low. Therefore in thiscrucial range of angles, the instability is easily suppressed. Belowwe show that no such condition exists which allows the excitation ofthese waves by relativistic intergalactic beams. Now let us apply these general results to extragalactic beams inparticular. The IGM plasma density is (Broderick et al. 2012) ๐ ๐ = . ร โ ( + ๐ง ) cm โ , (42)where ๐ง is the redshift of the TeV blazar. The beam density ๐ ๐ isdetermined by balance between cooling and creation processes. Theupper limit for the beamโs density is found when the cooling isdetermined by inverse Compton (IC) scattering. The IC scatteringrate is given by (Broderick et al. 2012): ฮ IC = . ร โ ( + ๐ง ) ๐พ s โ , (43)and the corresponding upper limit for the beam density derived fromthe cooling rate is: ๐ max ๐ โผ . ร โ ( + ๐ง ) . (cid:18) ๐ฟ ๐ธ erg s โ (cid:19) (cid:18) ๐ธ TeV (cid:19) cm โ . (44)Here ๐ฟ ๐ธ is the gamma-ray luminosity of the blazar at the photonenergy ๐ธ . The Lorenz factors of the produced pairs are found in therange ๐พ โผ โ with an average of ๐พ โผ (Miniati & Elyiv2013). The pairs are produced at an angular spread ฮ ๐ โผ ๐พ โ .Substituting the above plasma and beam densities into the condi-tion for the hydrodynamic instability regime, Eq. (19), and takinginto account that ฮ ๐ โผ ๐พ โ , one ๏ฌnds that the hydrodynamic regimeis valid only for large Lorentz factor values: ๐พ (cid:29) . ร ( + ๐ง ) โ / (cid:18) ๐ฟ ๐ธ erg s โ (cid:19) โ / ; ๐ โผ ,๐พ (cid:29) ร ( + ๐ง ) โ / (cid:18) ๐ฟ ๐ธ erg s โ (cid:19) โ / ; ๐ โผ ฮ ๐. (45) MNRAS , 1โ15 (2021) lazar induced pair beams One sees that the condition for the hydrodynamic regime is notsatis๏ฌed for oblique waves and could only be marginally satis๏ฌedfor quasi-parallel waves. This means that the treatment of the beaminstability in the kinetic regime is justi๏ฌed for typical beams in theIGM. Now let us estimate the role of plasma inhomogeneity on thedevelopment of the instability. It was discussed in section 2.3 thatthe e๏ฌects of plasma inhomogeneities on the instability result in lossof resonance between the particles and the Langmuir wave causedby changes in the wave vector at a rate given by (12). The resonantwave grows exponentially with a rate ฮ from the initial thermal noiselevel. In order for the wave to grow signi๏ฌcantly, the exponent has tobe rather large. Therefore the time of growth must exceed ๐ก = ฮ / ฮ ,where ฮ is the Coulomb logarithm.It was shown in section 2.1 that the resonance width of wavenumbers strongly depends on the angle between the wave and thebeam axis, ๐ , and may be estimated as ๐ฟ๐๐ โผ (cid:26) ฮ ๐ ; ๐ โผ ( ฮ ๐ ) ; ๐ โผ ฮ ๐. (46)Now the condition for the development of the instability can bewritten as: ฮ ๐๐ฟ ฮ โค ๐ฟ๐๐ โผ (cid:26) ฮ ๐ ; ๐ โผ ( ฮ ๐ ) ; ๐ โผ ฮ ๐. (47)Making use of the estimate (31) for the kinetic growth rate andintroducing the following short-hand notations: ๐ ๐ = ๐ ๐, โ ร โ cm โ , ๐ ๐ = ๐ ๐, โ ร โ cm โ , we can write the condition in the form:0 . ( ๐พ ฮ ๐ ) (cid:18) ฮ (cid:19) (cid:169)(cid:173)(cid:171) ๐ / ๐, โ ๐ ๐, โ (cid:170)(cid:174)(cid:172) (cid:18) ๐ ๐๐๐ฟ (cid:19) < ๐ โผ . ๐พ (cid:18) ฮ (cid:19) (cid:169)(cid:173)(cid:171) ๐ / ๐, โ ๐ ๐, โ (cid:170)(cid:174)(cid:172) (cid:18) ๐ ๐๐๐ฟ (cid:19) < ๐ โผ ฮ ๐. (49)We normalize ๐ฟ by the maximal inhomogeneity scale, of the orderof the cosmological void side. One can see that even in this case,only oblique waves could grow. Interaction of the beam with thesewaves leads only to expansion of the beam without noticeable lossof energy. The instability ceases when the beam expands about anorder of magnitude, ๐พ ฮ ๐ โผ
10. Nearly parallel modes, ๐ โผ ฮ ๐ ,which could take the energy o๏ฌ the beam, would not grow at therelevant Lorentz factors of the beam, ๐พ โผ โ . These arethe only waves which could take a signi๏ฌcant fraction of the beamenergy, but one can see that their growth is completely suppressed bythe inhomogeneity condition. Therefore even the weakest possibleinhomogeneity makes any energy loss of the beam ine๏ฌcient.Now let us come back to the theory proposed by Shalaby et al.(2018); Shalaby et al. (2020), which is discussed in sect. 4. Even iftheir theory were universlly correct, their instability growth rate (seeeq. 4.16 in Shalaby et al. 2020) would only be ฮ โผ (cid:32) ๐ ๐ ๐ ๐ ๐พ (cid:33) / ๐ ๐ = . ร โ ๐ . ๐, โ ๐ . ๐, โ ๐พ . s โ , (50)which is less rapid than the collisional decay time of the plasmawaves (e.g., Miniati & Elyiv 2013) ๐ coll โ โ ๐ ๐, โ ๐ โ / s โ , (51)where ๐ = ๐ ร K. We stress out that their result was obtainedin the hydrodynamic approximation of the instability, even though it should be described by kinetic theory at these parameters. Thenthe kinetic instability is weaker than the hydrodynamic one at thesame beam density because in the kinetic regime, only a fractionof electrons resonantly excite the wave. This means that the wavespropagating along the beam are not excited in any case. As to obliquewaves, it was explained above and will be con๏ฌrmed by the rigoroustheory in the next section that the excitation of these waves resultsmerely in the expansion of the beam, but doesnโt lead to energy loss.In the next section, we demonstrate straightforwardly that even ifthe oblique instability develops, the beam just expands, while energyloss remains negligibly small.
When excited wavesโ energy is small in comparsion with the plasmaโsthermal energy, the evolution of the beam distribution function is gov-erned by the di๏ฌusion equation in momentum space (e.g. Breหizman& Ryutov 1971; Brejzman 1990) ๐ ๐๐๐ก = ๐๐ ๐ ๐ผ (cid:18) ๐ท ๐ผ๐ฝ ๐ ๐๐ ๐ ๐ฝ (cid:19) , (52)where ๐ท ๐ผ๐ฝ is the resonant momentum-di๏ฌusion tensor de๏ฌned by ๐ท ๐ผ๐ฝ = ๐๐ โซ ๐ ( k , ๐ก ) ๐ ๐ผ ๐ ๐ฝ ๐ ๐ฟ ( k ยท v โ ๐ ) ๐ ๐. (53)For the spherical, azimuthal-symmetric case, the di๏ฌusion equationis written as (under the previous assumption that the beam is narrow, ๐ (cid:28) ๐ ๐๐๐ก = ๐ ๐ ๐๐๐ (cid:16) ๐๐ท ๐ ๐ ๐ ๐๐๐ (cid:17) + ๐๐ ๐๐๐ (cid:16) ๐๐ท ๐ ๐ ๐ ๐๐๐ (cid:17) + ๐ ๐๐๐ (cid:16) ๐๐ท ๐ ๐ ๐ ๐๐๐ (cid:17) + ๐ ๐๐๐ (cid:16) ๐ ๐ท ๐ ๐ ๐ ๐๐๐ (cid:17) . (54)The di๏ฌusion coe๏ฌcients become ๐ท ๐ = ๐๐ โซ ๐ ( k , ๐ก ) (cid:18) ๐ ๐ ๐ (cid:19) ๐ ๐ฟ ( k ยท v โ ๐ ) ๐ ๐, (55)where ๐ = , , ๐ท ๐ ๐ , ๐ท ๐ ๐ , ๐ท ๐ ๐ , respectively.It follows immediately from eq. (55), that if the spectral energyof oscillations, ๐ ( k ) , is dominated by waves propagating within theangle ๐ , the di๏ฌusion coe๏ฌcients are related as ๐ท ๐ ๐ : ๐ท ๐ ๐ : ๐ท ๐ ๐ โผ ๐ : ๐ : 1 . (56)The intergalactic beams are extremely narrow, ฮ ๐ โผ โ , whereastheir energy spread is large ฮ ๐ โผ ๐ . In this case, the di๏ฌerent termsin the R.H.S. of eq. (54) are related as1 ๐ ๐๐๐ (cid:18) ๐๐ท ๐ ๐ ๐ ๐๐๐ (cid:19) : ๐๐ ๐๐๐ (cid:18) ๐๐ท ๐ ๐ ๐ ๐๐ ๐ (cid:19) : ๐๐ ๐ (cid:18) ๐๐ท ๐ ๐ ๐ ๐๐๐ (cid:19) : ๐๐ ๐ (cid:18) ๐ ๐ท ๐ ๐ ๐ ๐๐ ๐ (cid:19) โผ (cid:18) ๐ ฮ ๐ (cid:19) : (cid:18) ๐ ฮ ๐ (cid:19) : (cid:18) ๐ ฮ ๐ (cid:19) : 1 (57)One sees that if the energy of the excited waves is dominated byoblique waves, ๐ (cid:29) ฮ ๐ , the ๏ฌrst term in the R.H.S. of eq. (54) signif-icantly exceeds the others, which means that the beam expands with-out losing energy. Only if the waves are excited predominantly withinthe beam opening angle, ๐ โผ ฮ ๐ , all the terms on the R.H.S. arecomparable, and then the beamโs energy is e๏ฌciently redistributed.A special feature of the kinetic beam instability is that at ๐ < ฮ ๐ , MNRAS000
When excited wavesโ energy is small in comparsion with the plasmaโsthermal energy, the evolution of the beam distribution function is gov-erned by the di๏ฌusion equation in momentum space (e.g. Breหizman& Ryutov 1971; Brejzman 1990) ๐ ๐๐๐ก = ๐๐ ๐ ๐ผ (cid:18) ๐ท ๐ผ๐ฝ ๐ ๐๐ ๐ ๐ฝ (cid:19) , (52)where ๐ท ๐ผ๐ฝ is the resonant momentum-di๏ฌusion tensor de๏ฌned by ๐ท ๐ผ๐ฝ = ๐๐ โซ ๐ ( k , ๐ก ) ๐ ๐ผ ๐ ๐ฝ ๐ ๐ฟ ( k ยท v โ ๐ ) ๐ ๐. (53)For the spherical, azimuthal-symmetric case, the di๏ฌusion equationis written as (under the previous assumption that the beam is narrow, ๐ (cid:28) ๐ ๐๐๐ก = ๐ ๐ ๐๐๐ (cid:16) ๐๐ท ๐ ๐ ๐ ๐๐๐ (cid:17) + ๐๐ ๐๐๐ (cid:16) ๐๐ท ๐ ๐ ๐ ๐๐๐ (cid:17) + ๐ ๐๐๐ (cid:16) ๐๐ท ๐ ๐ ๐ ๐๐๐ (cid:17) + ๐ ๐๐๐ (cid:16) ๐ ๐ท ๐ ๐ ๐ ๐๐๐ (cid:17) . (54)The di๏ฌusion coe๏ฌcients become ๐ท ๐ = ๐๐ โซ ๐ ( k , ๐ก ) (cid:18) ๐ ๐ ๐ (cid:19) ๐ ๐ฟ ( k ยท v โ ๐ ) ๐ ๐, (55)where ๐ = , , ๐ท ๐ ๐ , ๐ท ๐ ๐ , ๐ท ๐ ๐ , respectively.It follows immediately from eq. (55), that if the spectral energyof oscillations, ๐ ( k ) , is dominated by waves propagating within theangle ๐ , the di๏ฌusion coe๏ฌcients are related as ๐ท ๐ ๐ : ๐ท ๐ ๐ : ๐ท ๐ ๐ โผ ๐ : ๐ : 1 . (56)The intergalactic beams are extremely narrow, ฮ ๐ โผ โ , whereastheir energy spread is large ฮ ๐ โผ ๐ . In this case, the di๏ฌerent termsin the R.H.S. of eq. (54) are related as1 ๐ ๐๐๐ (cid:18) ๐๐ท ๐ ๐ ๐ ๐๐๐ (cid:19) : ๐๐ ๐๐๐ (cid:18) ๐๐ท ๐ ๐ ๐ ๐๐ ๐ (cid:19) : ๐๐ ๐ (cid:18) ๐๐ท ๐ ๐ ๐ ๐๐๐ (cid:19) : ๐๐ ๐ (cid:18) ๐ ๐ท ๐ ๐ ๐ ๐๐ ๐ (cid:19) โผ (cid:18) ๐ ฮ ๐ (cid:19) : (cid:18) ๐ ฮ ๐ (cid:19) : (cid:18) ๐ ฮ ๐ (cid:19) : 1 (57)One sees that if the energy of the excited waves is dominated byoblique waves, ๐ (cid:29) ฮ ๐ , the ๏ฌrst term in the R.H.S. of eq. (54) signif-icantly exceeds the others, which means that the beam expands with-out losing energy. Only if the waves are excited predominantly withinthe beam opening angle, ๐ โผ ฮ ๐ , all the terms on the R.H.S. arecomparable, and then the beamโs energy is e๏ฌciently redistributed.A special feature of the kinetic beam instability is that at ๐ < ฮ ๐ , MNRAS000 , 1โ15 (2021)
Perry et al. the growth rate is larger than at a larger ๐ โผ ฮ ๐ (Breหizman & Ryutov1971; Rudakov 1971, see also section 3.2). Due to the exponentialgrowth from a very low level of initial thermal ๏ฌuctuations, nearlyparallel waves could reach a much larger energy than oblique ones.This subtle fact leads to the widely known statement (Fainberg et al.1970; Breหizman & Ryutov 1971; Rudakov 1971) that in the kineticregime, relativistic beams e๏ฌciently lose their energy. However, thewidth of resonance for these waves is extremely small. Therefore theinstability of these waves may be easily suppressed by, e.g., weakinhomogeneity (Breizman & Ryutov 1970, see also section 2.2).The estimates show (see section 4) that for typical parameters ofintergalactic beams, this e๏ฌect completely suppresses the excitationof nearly parallel waves, so the beam could only expand withoutcooling. In section 5.3 we present numerical simulations supportingthe estimates above. The evolution equation for the oscillation spectrum density, ๐ ( k ) ,is written in the quasilinear approximation as ๐๐๐๐ก + v ๐ ยท โ ๐ โ โ ๐ ยท ๐๐๐ k = ( ฮ โ ๐ coll ) ๐, (58)where ๐ coll is the collisional decay rate of the plasma waves (51). Thel.h.s. of this equation is the full time derivative of the plasma wavespectral energy density along the rays which obey the geometricaloptics law: ๐ r ๐๐ก = ๐๐๐ k ; ๐ k ๐๐ก = โโ ๐. (59)The ๏ฌrst term in (58) represents the time evolution, the second isspatial advection, and the third term describes a spectral shift in ๐ -space due to inhomogeneity. The r.h.s. of this equation describes thegrowth or damping of wavesโ energy due to resonance interactionwith the beam and decay due to Coulomb collisions (see Eq. (51)).We do not take into account non-linear processes such as inducedscattering and modulation instability of the plasma waves. Theseprocesses redistribute the energy of the waves over the spectrum, thusremoving them from the resonance region. Therefore they provideadditional mechanisms for the stabilization of the beam instability.We show that the inhomogeneity solely restricts the e๏ฌciency ofenergy loss via the beam instability. Thus, there is no special need toconsider additional processes.A formal solution to eq. (58) is ๐ ( ๐ก, r , k ) = ๐ exp (cid:26) โซ ๐ก (cid:2) ฮ ( ๐ก (cid:48) , r (cid:48) , k (cid:48) ) โ ๐ coll ( r (cid:48) ) (cid:3) ๐๐ก (cid:48) (cid:27) , (60)where the integral is along the characteristics r (cid:48) = r + v ๐ ( ๐ก (cid:48) โ ๐ก ) , k (cid:48) = k โ โ ๐ ( ๐ก (cid:48) โ ๐ก ) . The growth rate ฮ is given by eq. (23). Thetime-dependence of the growth rate, ฮ , is determined by the timevariation of the distribution function. The dependence on k is verystrong because the resonance range ฮ ๐ is very narrow. Both ฮ and ๐ coll depend on the spatial coordinate only via the plasma density,which enters the expressions for ฮ and ๐ coll as a multiplier and variesat a very large scale ๐ฟ . Taking into account that the light traveltime across the spatial inhomogeneity scale signi๏ฌcantly exceeds theinstability time, ๐ก (cid:28) ๐ฟ / ๐ , one can neglect this dependence in theintegrand. Assuming that the initial ๏ฌuctuation rate, ๐ , is spatiallyhomogeneous, one ๏ฌnds that the dependence on r , and therefore on v ๐ , drops out of the solution. This means that one can drop thesecond term in the l.h.s. of eq. (58): the evolution of the wave densityis determined by the time evolution of k according to the secondequation in (59) and not by the spatial transfer. For numerical purposes, we will use eq. (58) but not eq. (60).Now we can rewrite this equation in a simpli๏ฌed form: ๐๐๐๐ก + ๐ ๐ ๐ฟ ๐๐๐๐ = ( ฮ โ ๐ coll ) ๐. (61)Taking into account that the waves are concentrated in a very narrowrange of ๐ , one can conveniently rewrite, for numerical purposes, theequation in terms of the resonant parameter (29), which is roughlybounded by | ๐ฅ | โค ฮ ๐ . Then for the case ๐ (cid:29) ฮ ๐ , we get ๐๐๐๐ก โ ๐๐ฟ cos ๐ sin ๐ ๐๐๐๐ฅ = ( ฮ โ ๐ coll ) ๐ ; (62)whereas for ๐ โผ ฮ ๐ , the equation may be written as ๐๐๐๐ก โ ๐ ๐๐ฟ ๐๐๐๐ฆ = ( ฮ โ ๐ coll ) ๐, (63)where ๐ฆ = ๐ฅ โ๏ธ ๐ฅ + ๐ โ ๐ฅ ๐ + . (64)One can show that the resonance width ฮ ๐ฆ is roughly ๐ / ๐ so that ฮ ๐ฆ โผ ๐ฅ โผ ๐ . In this section we present the results of our numerical simulation ofthe beam-plasma interaction in the quasi-linear regime.The extragalactic beams are extremely narrow, ฮ ๐ โผ โ . Ourprevious estimates show that only oblique waves, ๐ โผ
1, are excited,in the presence of inhomogeneity. This implies that the ๏ฌrst term onthe R.H.S. of the momentum di๏ฌusion equation (54) exceeds otherterms by 5-10 orders of magnitude. Therefore we initially only con-sider the evolution of the angular distribution of particles accordingto the equation ๐๐๐๐ก = ( ๐พ๐๐ ) ๐๐๐ (cid:18) ๐ ๐ท ๐ ๐ ๐๐๐๐ (cid:19) . (65)where the angular distribution function, ๐ , has been de๏ฌned in (22).Having found the angular evolution of the beam, together with thespectral evolution of the wave, we evaluate the other di๏ฌusion coef-๏ฌcients in the momentum di๏ฌusion equation. If they are comparableto ๐ท ๐ ๐ , then according to our estimates the other di๏ฌusion processesare negligible, which is self-consistent with our assumption.For each time step, we ๏ฌrst solve the di๏ฌusion equation (65) giventhe di๏ฌusion coe๏ฌcients from the last time step. After the distribu-tion function is updated, we calculate the growth rates for each wavevector, making use of the formulas presented in section 3.2. After thegrowth rates are calculated in the current time step, we next calculatethe new values of the oscillations energy density, ๐ , according to theevolution equation (eq. 62 or, for small ๐ , 63). Using the updatedvalues of ๐ , we calculate the di๏ฌusion coe๏ฌcients (the procedureis described in Appendix E). The new values of the di๏ฌusion coef-๏ฌcients are used in the next time step for a new cycle as describedabove.Each separate simulation is characterized by a set of (cid:0) ๐พ, ๐ ๐ , ๐ ๐ , ๐ฟ (cid:1) .The simulation time is measured in units of the characteristic insta-bility time ๐ก = ฮ โ = (cid:0) ๐ ๐ / ๐ ๐ (cid:1) ๐ โ ๐ ๐พ โ . (66)The initial angular distribution function is chosen in the form ๐ ( ๐ ) = exp (cid:16) โ . ( ๐๐พ ) (cid:17) , (67) MNRAS , 1โ15 (2021) lazar induced pair beams Figure 4.
The angular spread of the beam distribution function (solid) as afunction of normalized time and max ( ๐ ) (dashed) as a function of normalizedtime. ๐พ = , ๐ ๐, โ = ๐ฟ = ๐ ๐๐ . so that the initial angular spread is always ฮ ๐ โ ๐พ โ , and the functionhas a sharp gradient around ๐ โผ ๐พ โ . The oscillations energy densityis initialized uniformly to a low level such that the angular di๏ฌusionrate due to the initial background is negligibly slow (see below). Inall of the simulations we performed, we have ๏ฌxed the value of theplasma density to be ๐ ๐ = . ร โ cm โ , which corresponds tono redshift. We consider the maximal possible plasma inhomogene-ity length scale, ๐ฟ = ๐ ๐๐ , providing the minimal in๏ฌuence ofinhomogeneity.Figure 4 shows the angular spread, ฮ ๐ , and the maximum valueof the angular distribution function, max [ ๐ ] , as a function of thenormalized time. The beam is spreading in angle while the distribu-tion ๏ฌattens, which means that particles gain transverse momentumand they are drifting away from the initial direction of the beam.The spreading begins after about a dozen instability times, whenthe resonant wave grew enough from the initial low level. After thebeam spreads by about an order of magnitude, the growth rate of theinstability falls below the collision decay rate of the plasma waves,and the process is saturated.In ๏ฌgure 5 we present simulations of the beam with the sameparameters but with di๏ฌerent initial plasma wave energy levels, ห ๐ ,normalized byห ๐ = ๐๐ ๐ ๐ ๐ก (cid:18) ๐ ๐๐ ๐ ๐ ๐ (cid:19) โ ๐ , (68)where ๐ is dimensionless. One sees that the results are weaklydependent on ๐ , the initial level of oscillations, provided it is smallenough.Figure 6 shows the evolution of the beam width for di๏ฌerent valuesof beam density and Lorentz factors. One sees that in all cases, thebehavior of the beam is essentially the same: the expansion saturatesafter the beam expands a few times over. This agrees with the resultof our qualitative estimates presented in section 4.In our numerical simulation, we calculate the growth and decayof the resonant Langmuir waves energy. Figures 7 and 8 show themaximum oscillation energy density with respect to ๐ as a functionof ๐ at di๏ฌerent times. We measure the oscillations energy densitywith respect to their initial value, ๐ which is considered as the Figure 5.
The normalized angular spread, ๐พ ฮ ๐ , as a function of normalizedtime for di๏ฌerent initial levels of oscillations, ๐ , see eq. (68). The beamparameters are the same as in the previous ๏ฌgure. Figure 6.
The normalized angular spread, ๐พ ฮ ๐ as a function of normalizedtime for ๐พ = , ๐ ๐, โ = ๐พ = , ๐ ๐, โ = ๐พ = , ๐ ๐, โ = . ๐พ = , ๐ ๐, โ = . ๐ฟ = ๐ ๐๐ . dimensionless noise level. The ๏ฌrst ๏ฌgure shows results for ๐ ๐ = โ cm โ and the second for ๐ ๐ = โ cm โ . One sees that eachcurve has a maximum at some angle ๐ โผ
1. This peak is a result ofthe competition between two terms: the growth, which is proportionalto cos ๐ , thus stronger for smaller angles, and the inhomogeneityterm, which suppresses the instability more e๏ฌciently for smallerangles. Both cases show that small angles waves do not grow, asexpected. Note that whereas the densities of two beams di๏ฌer bya factor of 10, the level of the oscillations energy di๏ฌers by fourorders of magnitude. The reason is that the instability growth ratefor the smaller density beam is ten times smaller than that of thedenser beam. Therefore for the more dilute beam, inhomogeneity MNRAS000
1. This peak is a result ofthe competition between two terms: the growth, which is proportionalto cos ๐ , thus stronger for smaller angles, and the inhomogeneityterm, which suppresses the instability more e๏ฌciently for smallerangles. Both cases show that small angles waves do not grow, asexpected. Note that whereas the densities of two beams di๏ฌer bya factor of 10, the level of the oscillations energy di๏ฌers by fourorders of magnitude. The reason is that the instability growth ratefor the smaller density beam is ten times smaller than that of thedenser beam. Therefore for the more dilute beam, inhomogeneity MNRAS000 , 1โ15 (2021) Perry et al.
Figure 7.
The maximum oscillation energy density ๐ osc with respect to theresonance parameter ๐ฅ as a function of ๐ for di๏ฌerent times (normalized bythe instability time (66)). ๐ ๐, โ = ๐พ = , ๐ฟ = ๐ ๐๐ . Figure 8.
Same as ๏ฌgure 7, but with ๐ ๐, โ = โ . also succeeds in suppressing the growth of oblique waves, ๐ โผ ๐ osc initially grows but eventually beginsto decrease. This occurs when inhomogeneity overcomes the lineargrowth, yet the competition between the two processes is dynamicand eventually the instability is completely suppressed. The maxi-mum energy density with respect to the whole resonant spectrumstrongly depends on the beam density, where a change in ๐ ๐ by afactor of 10 changes the possible growth of the energy density byorders of magnitude.In the above simulations, the beam expands while the instabilitygrowth rate remains larger than the collisional decay time, after whichthe collisionless evolution ceases. We also examine an arti๏ฌcial casewhere collisional decay is absent, i.e. ๐ coll =
0. For this case, waveswill decay only if they exit the resonance range due to inhomogeneity.The waves propagating nearly perpendicularly to the direction of the
Figure 9.
The maximum oscillation energy density ๐ ๐๐ ๐ , normalized by ๐ with respect to the entire resonant spectrum as a function of normalized time(66) for ๐พ = , ๐ ๐, โ = ๐พ = , ๐ ๐, โ = ๐พ = , ๐ ๐, โ = . ๐พ = , ๐ ๐, โ = . ๐ฟ = ๐ ๐๐ . density gradient still grow so that the beam expands further out. Theinstability growth rate, and therefore the rate of evolution, decreaseswith the beam width. Eventually the beam expansion time becomessmaller than the Compton scattering time, after which the collision-less relaxation does not a๏ฌect the beam any more: the beam will cooldown and deposit its energy to the background light (Broderick et al.2012; Sironi & Giannios 2014). Figure 10 shows the expansion of thebeam for this arti๏ฌcial case, for which ๐พ = and ๐ ๐, โ = .
1. Thesimulations were stopped when the expansion time became equal tothe characteristic Compton time. One sees that the beams expandsa few times more than in the case shown in ๏ฌgure 6. In any case,the beam evolution is determined by obliquely propagating waves,therefore the beam practically does not loose energy but just expandsuntil the Compton scattering comes into the play.
In the previous subsection, we studied the angular expansion of thebeam. In addition to angular di๏ฌusion, the beam particles couldexchange energy with the excited oscillations, which leads to a radialdi๏ฌusion in momentum space. The energy exchange is governed bythe small angles part of the spectrum of oscillations. In section 5.1 wehave estimated the relative rate of the angular and energy di๏ฌusionand have shown that it depends both on the ratio between the di๏ฌusioncoe๏ฌcients and on the beam width, so that if the di๏ฌusion coe๏ฌcientsare of the same order of magnitude, then energy di๏ฌusion is negligiblewith respect to angular di๏ฌusion, see eq. (57). According to ourestimates, the di๏ฌusion coe๏ฌcients are of the same order unless theplasma wave spectrum is dominated by small angle waves. Takinginto account that growth of these waves is totally suppressed even bythe slightest possible inhomogeneity, we retained only the angularterms in the momentum di๏ฌusion equation. Now we could justifyour conjecture by directly calculating all the di๏ฌusion coe๏ฌcientsmaking use of the spectrum as obtained in simulations.Figure 11 shows the angular dependence of the di๏ฌusion coe๏ฌ-cients for di๏ฌerent times whereas ๏ฌgure 12 shows the time evolution
MNRAS , 1โ15 (2021) lazar induced pair beams Figure 10.
The evolution of the beam width when the collisional decay of theplasma waves is absent. Shown is the normalized angular spread, ๐พ ฮ ๐ , as afunction of normalized time for ๐พ = , ๐ ๐, โ = . ๐ฟ =
100 Mpc. Thetypical inverse-Compton scattering time is shown as a vertical black line.
Figure 11.
Angular dependence of the di๏ฌusion coe๏ฌcients for di๏ฌerenttimes: ๐ก = . ๐ก (squares); ๐ก = . ๐ก (circles); ๐ก = . ๐ก (dashes); ๐ก = . ๐ก (triangles). ๐ ๐, โ = ๐พ = , ๐ฟ = ๐ ๐๐ . of the di๏ฌusion coe๏ฌcients for di๏ฌerent values of ๐ . One sees thatthe di๏ฌusion coe๏ฌcients remain of the same order all the time for allthe relevant particle angles. This shows that the terms on the R.H.S.of equation (54) are related as 1 : ฮ ๐ : ฮ ๐ : ( ฮ ๐ ) , correspondingly.Taking into account that ฮ ๐ โผ โ โ โ , this justi๏ฌes the neglectof the energy evolution of the beam. In this paper, we have examined the beam-plasma instability of Lang-muir waves in the intergalactic medium. The source of the instabilityis the resonant energy exchange between a narrow, dilute, energeticbeam of ultra-relativistic pairs traveling through the plasma and theLangmuir oscillations of the background plasma. The beams of elec-trons and positrons which travel through the IGM are created by pair
Figure 12.
The time evolution of the di๏ฌusion coe๏ฌcients at ๐พ ๐ = . ๐พ ๐ = ๐พ ๐ =
10 (dotted).Beam parameters are the sameas in the previous ๏ฌgure. production processes of highly energetic gamma ray photons with thebackground radiation in the plasma. The evolution of the pair beamsand the deposition of their energy to the IGM and back to the gammaray ๏ฌux is important to understand in order to explain the lack of anyGeV halos around blazars (Aharonian et al. 2006; Neronov & Vovk2010; Broderick et al. 2012).The crucial point is that the energetic evolution of the beam,whether it loses energy or gains transverse momentum, depends onthe direction of the wave vector: if the interaction is dominated bywaves propagating within the opening angle of the beam, ๐ โผ ฮ ๐ ,then the beam loses a large fraction of its energy. However, whenthe interaction is dominated by oblique waves, the momentum lossis small and the beam simply expands.For realistic parameters of the IGM, the beam instability developsin the kinetic regime. The kinetic growth rate dependence on thewave vector angle shows that the fastest growing waves are oneswhich travel at small angles ( ๐ โผ ฮ ๐ ); their growth rate is a fewtimes larger than that of oblique waves ( ๐ โผ (โผ ( ฮ ๐ ) ) , which means that in a very narrow beam, thegrowth of these waves is easily suppressed. We have shown thateven for the slightest density gradient with a length scale as large as100 Mpc, the quasi-parallel waves lose resonance at any reasonableparameters of intergalactic beams. Therefore the growth of the samewaves which contribute to the decelerating mechanism is forbiddenby the characteristics of the inhomogenous medium.We have solved the quasi-linear equations for the momentum dif-fusion of pair beams with a time evolution of the plasma oscillations,including the instability mechanism and the inhomogeneity term.We have found that the beam broadens in angle by about an orderof magnitude but practically does not lose energy. This leads to theconclusion that the lack of GeV bumps in the blazar spectra cannotbe attributed to the beam decay due to plasma instabilities. MNRAS000
10 (dotted).Beam parameters are the sameas in the previous ๏ฌgure. production processes of highly energetic gamma ray photons with thebackground radiation in the plasma. The evolution of the pair beamsand the deposition of their energy to the IGM and back to the gammaray ๏ฌux is important to understand in order to explain the lack of anyGeV halos around blazars (Aharonian et al. 2006; Neronov & Vovk2010; Broderick et al. 2012).The crucial point is that the energetic evolution of the beam,whether it loses energy or gains transverse momentum, depends onthe direction of the wave vector: if the interaction is dominated bywaves propagating within the opening angle of the beam, ๐ โผ ฮ ๐ ,then the beam loses a large fraction of its energy. However, whenthe interaction is dominated by oblique waves, the momentum lossis small and the beam simply expands.For realistic parameters of the IGM, the beam instability developsin the kinetic regime. The kinetic growth rate dependence on thewave vector angle shows that the fastest growing waves are oneswhich travel at small angles ( ๐ โผ ฮ ๐ ); their growth rate is a fewtimes larger than that of oblique waves ( ๐ โผ (โผ ( ฮ ๐ ) ) , which means that in a very narrow beam, thegrowth of these waves is easily suppressed. We have shown thateven for the slightest density gradient with a length scale as large as100 Mpc, the quasi-parallel waves lose resonance at any reasonableparameters of intergalactic beams. Therefore the growth of the samewaves which contribute to the decelerating mechanism is forbiddenby the characteristics of the inhomogenous medium.We have solved the quasi-linear equations for the momentum dif-fusion of pair beams with a time evolution of the plasma oscillations,including the instability mechanism and the inhomogeneity term.We have found that the beam broadens in angle by about an orderof magnitude but practically does not lose energy. This leads to theconclusion that the lack of GeV bumps in the blazar spectra cannotbe attributed to the beam decay due to plasma instabilities. MNRAS000 , 1โ15 (2021) Perry et al.
ACKNOWLEDGEMENTS
This research was supported by the grant 2067/19 from the IsraeliScience Foundation.
DATA AVAILABILITY
The data underlying this article will be shared on reasonable requestto the corresponding author.
REFERENCES
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APPENDIX A: HYDRODYNAMIC INSTABILITY
The dispersion equation (3.1) for a mono-energetic beam is easilysolved if ๐ ๐ (cid:28) ๐ ๐ (e.g., ยง61 in Lifshitz & Pitaevskii 1981). Onecan conveniently write the dispersion equation in the form1 โ ๐ ๐ ๐ = ๐ ( ๐ โ k ยท v ๐ ) ; (A1)where ๐ = ๐ ๐ ๐พ sin ๐ + cos ๐ ๐พ . (A2)At ๐ ๐ (cid:28) ๐ ๐ , the rhs is non-negligible only if the denominator isclose to zero, therefore one can present the solution in the form ๐ = k ยท v ๐ + ๐ฟ, (A3)where ๐ฟ (cid:28) ๐ . Substituting this into (A1), one ๏ฌnds ๐ฟ = ยฑ (cid:118)(cid:117)(cid:116) ๐ โ ๐ ๐ ๐ ๐ฃ ๐ . (A4)This solution represents the well known beam mode. The wave isunstable at ๐๐ฃ < ๐ ๐ ; the growth rate is typically ฮ โผ ๐ / , (A5)but it increases when ๐๐ฃ ๐ approaches ๐ ๐ . The maximal growth rateis achieved at ๐๐ฃ โ ๐ โ ๐ ๐ . In this case, we deal with the plasmawave in resonance with the beam. The solution is now presented as ๐ = ๐ ๐ + ๐ , where ๐ (cid:28) ๐ ๐ . The standard procedure yields thegrowth rate of the plasma wave ฮ โผ ( ๐๐ ๐ ) / . (A6)The ratio of the growth rate of the beam mode and of the plasmawave is ฮ ฮ โผ ๐ / ๐ / ๐ = (cid:32) ๐ ๐ ๐ ๐ ๐พ sin ๐ + cos ๐ ๐พ ๐ (cid:33) / . (A7)Due to the power of 1 /
6, the growth rates of two modes could beroughly of the same order for not very large Lorentz factors andmoderate density ratios. Therefore in simulations, one can observethe beam modes growing together with the resonant plasma wave(e.g., Bret et al. 2010). However, in the extragalactic beams, ๐ ๐ / ๐ ๐ โผ โ โ โ , ๐พ ๐ โผ ; therefore the above ratio (A7) is as smallas 10 โ โ โ .The instability with such a small growth rate as that of the beammodes could not develop in the IGM because of the collisional damp-ing. Regardless f this reason, one can readily see that the beam modesdo not in fact exist in the IGM. Namely, the condition for these modesto exist is that the beam may be considered as monoenergetic, so thatall the beam particles are in resonance with the wave, ฮ > ๐ ฮ ๐ฃ. (A8)In our case, ฮ ๐ฃ โผ ๐ ฮ ๐ , ๐ โผ ๐ ๐ / ๐ . Even for oblique waves, for whichthe growth rate is larger than for parallel ones, we get ฮ ๐ ฮ ๐ โผ โ๏ธ ๐ ๐ ๐ ๐ ๐พ ๐ ฮ ๐ โผ โ โ โ , (A9) MNRAS , 1โ15 (2021) lazar induced pair beams so that the condition (A8) is violated by large margin.The above consideration shows that only resonant plasma wavescould be excited in the extragalactic beam-plasma systems thereforewe consider only these waves in the paper. APPENDIX B: KINETIC GROWTH RATE FOR AGAUSSIAN DISTRIBUTION
Substituting into the general formula for the growth rate (23) thedistribution function (26) and taking into account that ๐ (cid:28)
1, onegets:
ฮ = ๐ ๐ (cid:18) ๐พ ฮ ๐ (cid:19) (cid:18) ๐ ๐ ๐ ๐ (cid:19) (cid:16) ๐ ๐ ๐๐ (cid:17) ร โซ ๐ ๐ ๐๐ ๐ + (cid:18) ๐ ๐ + ๐ โ โ ฮ ๐ (cid:19)โ๏ธ(cid:16) ๐ โ ๐ (cid:17) (cid:16) ๐ โ ๐ (cid:17) ๐ โ ๐ ฮ ๐ . (B1)Using the integral formulas of the modi๏ฌed Bessel functions of the๏ฌrst kind ๐ผ ๐ ( ๐ฅ ) , ๐ผ ๐ ( ๐ฅ ) = ๐ โซ ๐ ๐ ๐ฅ cos ๐ cos ( ๐๐ ) ๐๐, (B2)it is possible to evaluate the integrals โซ ๐ฅ ๐ฅ ๐ ๐ฅ ฮ ๐ โ๏ธ ( ๐ฅ โ ๐ฅ ) ( ๐ฅ โ ๐ฅ ) ๐๐ฅ = ๐๐ ๐ฅ + ๐ฅ ฮ ๐ ๐ผ (cid:18) ๐ฅ โ ๐ฅ ฮ ๐ (cid:19) , (B3a) โซ ๐ฅ ๐ฅ ๐ฅ๐ ๐ฅ ฮ ๐ โ๏ธ ( ๐ฅ โ ๐ฅ ) ( ๐ฅ โ ๐ฅ ) ๐๐ฅ = ๐๐ ๐ฅ + ๐ฅ ฮ ๐ ร (cid:20) ๐ฅ โ ๐ฅ ๐ผ (cid:18) ๐ฅ โ ๐ฅ ฮ ๐ (cid:19) + ๐ฅ + ๐ฅ ๐ผ (cid:18) ๐ฅ โ ๐ฅ ฮ ๐ (cid:19)(cid:21) . (B3b)Finally, we now get ฮ = ๐ ๐ (cid:18) ๐ ๐ ๐ ๐ (cid:19) ๐๐พ ฮ ๐ cos ๐ exp (cid:32) โ ๐ + ๐ ฮ ๐ (cid:33) ร (cid:34)(cid:32) ๐ (cid:18) ฮ ๐ ๐ (cid:19) โ (cid:33) ๐ผ ( ๐ ) โ ๐ ๐ผ ( ๐ ) (cid:35) , , (B4)where ๐ โก (cid:16) ๐ โ ๐ (cid:17) / ฮ ๐ . APPENDIX C: RESONANCE PARAMETER
The resonance parameter, ๐ฅ , is de๏ฌned for each wave vector ( ๐, ๐ ) as ๐ฅ = ๐ ๐ ๐๐ โ cos ๐ โ๏ธ โ ๐ ๐ ๐๐ cos ๐ . (C1)First, we prove that | ๐ฅ | โค ๐ for resonant waves with particles at apolar angle ๐ . We begin with the resonance condition โ โค ๐ ๐ / ๐๐ โ cos ๐ cos ๐ sin ๐ sin ๐ โค . (C2)Given that the polar angles are in the range [ , ๐ ] , one can get โ ๐ sin ๐ โ cos ๐ ๐ โค ๐ ๐ ๐๐ โ cos ๐ โค ๐ sin ๐ โ cos ๐ ๐ . (C3) After some arithmetics, we ๏ฌnd the complete resonance range of ๐ฅ as โ ๐ (cid:16) + ๐ ๐ (cid:17)โ๏ธ + ๐ tan ๐ + ๐ ๐ โค ๐ฅ โค ๏ฃฑ๏ฃด๏ฃด๏ฃด๏ฃด๏ฃฒ๏ฃด๏ฃด๏ฃด๏ฃด๏ฃณ ๐ (cid:16) โ ๐ ๐ (cid:17)โ๏ธ โ ๐ tan ๐ + ๐ ๐ ๐ โฅ ๐๐ /โ ๐ โค ๐ . (C4)In order to express ๐ via ๐ฅ and ๐ , we write eq. (C1) as a quadraticequation: (cid:16) ๐ ๐ ๐๐ (cid:17) + (cid:16) ๐ฅ โ (cid:17) cos ๐ ๐ ๐ ๐๐ + cos ๐ โ ๐ฅ = . (C5)The solution is ๐ ๐ ๐๐ = (cid:18) โ ๐ฅ (cid:19) cos ๐ + ๐ฅ โ๏ธ ๐ฅ cos ๐ + ๐ , (C6)where we kept only the solution which is consistent with (C1). Wedivide the form of this solution to three di๏ฌerent cases, or ranges of ๐ฅ / ๐ : ๐ ๐ ๐๐ = cos ๐ ( + ๐ฅ tan ๐ ) ; ๐ (cid:29) | ๐ฅ | ,๐ ๐ ๐๐ = โ ๐ + ๐ฅ (cid:169)(cid:173)(cid:171) sgn ( ๐ฅ ) โ๏ธ + ๐ ๐ฅ โ (cid:170)(cid:174)(cid:172) ; ๐ โผ | ๐ฅ | ,๐ ๐ ๐๐ = โ ๐ฅ ; ๐ (cid:28) | ๐ฅ | , ๐ฅ < , (C7)where in the second and third cases we assumed that ๐ (cid:28) APPENDIX D: KINETIC GROWTH RATE IN THEGENERAL CASE
Proceeding to a general distribution function, we ๏ฌrst need to simplifythe di๏ฌerent terms in the growth rate formula. We begin by expressingthe denominator in the growth rate formula (23) via ๐ฅ and ๐ : ( cos ๐ โ cos ๐ ) ( cos ๐ โ cos ๐ ) = sin ๐ ๏ฃฎ๏ฃฏ๏ฃฏ๏ฃฏ๏ฃฏ๏ฃฏ๏ฃฐ โ ๐ฅ tan ๐ + ๐ฅ tan ๐ โ ๐ฅ (cid:16) โ ๐ฅ ๐ (cid:17) โ (cid:110) โ ๐ฅ tan ๐ + ๐ฅ ๐ โ ๐ ๐ (cid:111) ๏ฃน๏ฃบ๏ฃบ๏ฃบ๏ฃบ๏ฃบ๏ฃป . (D1)We next consider the formula in three di๏ฌerent regimes of ๐ฅ / ๐ .(i) | ๐ฅ | (cid:28) ๐ ๐ ๐ ๐๐ = cos ๐ + ๐ฅ sin ๐ , (D2) ( cos ๐ โ cos ๐ ) ( cos ๐ โ cos ๐ ) = sin ๐ โ cos ๐ โ (cid:16) ๐ ๐ ๐๐ (cid:17) + ๐ ๐ ๐๐ cos ๐ cos ๐. (D3)After some arithmetics, one gets: ( cos ๐ โ cos ๐ ) ( cos ๐ โ cos ๐ ) = sin ๐ (cid:16) ๐ โ ๐ฅ (cid:17) . (D4)The nominator can be rewritten with a change of variables: (cid:104) ๐ ๐๐๐๐ (cid:16) cos ๐ โ ๐๐๐ ๐ cos ๐ (cid:17) โ ๐ (cid:105) ๐๐๐ = (cid:104) ๐๐๐ ( ๐ / ๐ฅ ) (cid:16) ๐ฅ tan ๐ โ ๐ฅ (cid:16) tan ๐ + ๐ ๐ฅ (cid:17)(cid:17) โ ๐๐ฅ (cid:105) ๐ (cid:16) ๐ ๐ฅ (cid:17) . (D5) MNRAS000
Proceeding to a general distribution function, we ๏ฌrst need to simplifythe di๏ฌerent terms in the growth rate formula. We begin by expressingthe denominator in the growth rate formula (23) via ๐ฅ and ๐ : ( cos ๐ โ cos ๐ ) ( cos ๐ โ cos ๐ ) = sin ๐ ๏ฃฎ๏ฃฏ๏ฃฏ๏ฃฏ๏ฃฏ๏ฃฏ๏ฃฐ โ ๐ฅ tan ๐ + ๐ฅ tan ๐ โ ๐ฅ (cid:16) โ ๐ฅ ๐ (cid:17) โ (cid:110) โ ๐ฅ tan ๐ + ๐ฅ ๐ โ ๐ ๐ (cid:111) ๏ฃน๏ฃบ๏ฃบ๏ฃบ๏ฃบ๏ฃบ๏ฃป . (D1)We next consider the formula in three di๏ฌerent regimes of ๐ฅ / ๐ .(i) | ๐ฅ | (cid:28) ๐ ๐ ๐ ๐๐ = cos ๐ + ๐ฅ sin ๐ , (D2) ( cos ๐ โ cos ๐ ) ( cos ๐ โ cos ๐ ) = sin ๐ โ cos ๐ โ (cid:16) ๐ ๐ ๐๐ (cid:17) + ๐ ๐ ๐๐ cos ๐ cos ๐. (D3)After some arithmetics, one gets: ( cos ๐ โ cos ๐ ) ( cos ๐ โ cos ๐ ) = sin ๐ (cid:16) ๐ โ ๐ฅ (cid:17) . (D4)The nominator can be rewritten with a change of variables: (cid:104) ๐ ๐๐๐๐ (cid:16) cos ๐ โ ๐๐๐ ๐ cos ๐ (cid:17) โ ๐ (cid:105) ๐๐๐ = (cid:104) ๐๐๐ ( ๐ / ๐ฅ ) (cid:16) ๐ฅ tan ๐ โ ๐ฅ (cid:16) tan ๐ + ๐ ๐ฅ (cid:17)(cid:17) โ ๐๐ฅ (cid:105) ๐ (cid:16) ๐ ๐ฅ (cid:17) . (D5) MNRAS000 , 1โ15 (2021) Perry et al.
Combining all these together, the growth rate is written as an integralover a single variable ๐ง = ๐ / ๐ฅ : ฮ = ๐ ๐ (cid:18) ๐๐๐ ๐ (cid:19) (cid:18) ๐ ๐ ๐ ๐ (cid:19) cos ๐ ( + ๐ฅ tan ๐ ) ร โซ โ ๐๐๐๐ง (cid:104) sgn ( ๐ฅ ) โ (cid:16) | ๐ฅ | tan ๐ (cid:17) (cid:16) tan ๐ + ๐ง (cid:17)(cid:105) โ ๐ (cid:16) | ๐ฅ | tan ๐ (cid:17) โ ๐ง โ ๐๐ง โ ๐ ๐ (cid:18) ๐๐๐ ๐ (cid:19) (cid:18) ๐ ๐ ๐ ๐ (cid:19) sgn ( ๐ฅ ) cos ๐ โซ โ ๐๐๐๐ง โ ๐ง โ ๐๐ง. (D6)(ii) | ๐ฅ | โผ ๐ First we evaluate the denominator terms:cos ๐ , โ cos ๐ = ๐ ๐ ๐๐ cos ๐ ยฑ sin ๐ โ๏ธ โ (cid:16) ๐ ๐ ๐๐ (cid:17) โ cos ๐ = ๐ โ ๐ + ๐ฅ ๏ฃฎ๏ฃฏ๏ฃฏ๏ฃฏ๏ฃฏ๏ฃฐ sgn ( ๐ฅ ) โ๏ธ + ๐ ๐ฅ โ ๏ฃน๏ฃบ๏ฃบ๏ฃบ๏ฃบ๏ฃป ยฑ ๐ | ๐ฅ | (cid:118)(cid:117)(cid:116) ๐ ๐ฅ + โ sgn ( ๐ฅ ) โ๏ธ + ๐ ๐ฅ == ๐ฅ ๏ฃฑ๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃฒ๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃณ ๐ ๐ฅ โ ๐ ๐ฅ + ๏ฃฎ๏ฃฏ๏ฃฏ๏ฃฏ๏ฃฏ๏ฃฐ sgn ( ๐ฅ ) โ๏ธ + ๐ ๐ฅ โ ๏ฃน๏ฃบ๏ฃบ๏ฃบ๏ฃบ๏ฃป ยฑ ๐ | ๐ฅ | (cid:118)(cid:117)(cid:116) ๐ ๐ฅ + โ sgn ( ๐ฅ ) โ๏ธ + ๐ ๐ฅ ๏ฃผ๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃฝ๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃพ . . (D7)Then the denominator term reduces to ( cos ๐ โ cos ๐ ) ( cos ๐ โ cos ๐ ) = ๐ฅ ๏ฃฎ๏ฃฏ๏ฃฏ๏ฃฏ๏ฃฏ๏ฃฏ๏ฃฐ ๐ (cid:104) ๐ + โ sgn ( ๐ฅ ) โ๏ธ + ๐ (cid:105) โ (cid:16) ๐ง โ ๐ + (cid:104) sgn ( ๐ฅ ) โ๏ธ + ๐ โ (cid:105)(cid:17) ๏ฃน๏ฃบ๏ฃบ๏ฃบ๏ฃบ๏ฃบ๏ฃป = ๐ฅ (cid:20) ( ๐ โ ๐ ) ( ๐ง โ ) โ ๐ง (cid:21) , , (D8)where we have de๏ฌned ๐ง = ๐ / ๐ฅ , ๐ = ๐ / ๐ฅ and ๐ = sgn ( ๐ฅ ) โ๏ธ + ๐ โ ๐ง : ๐ง , = ๐ โ ๐ ยฑ โ๏ธ ๐ ( ๐ โ ๐ ) . (D9)The nominator is evaluated using the de๏ฌnitions of ๐ง, ๐ and ๐ above.Given the expression ๐ ๐ ๐๐ = โ ๐ฅ ( ๐ โ ๐ ) , (D10)the nominator can be written as (cid:20) ๐ ๐๐๐๐ (cid:18) cos ๐ โ ๐๐๐ ๐ cos ๐ (cid:19) โ ๐ (cid:21) ๐๐๐ = (cid:20) ๐๐๐๐ง ( ๐ โ ๐ง ) โ ๐ (cid:21) ๐ฅ ๐๐ง. (D11)Combining all these together, the expression for the growth rate is ฮ = ๐ ๐ (cid:18) ๐๐๐ ๐ (cid:19) (cid:18) ๐ ๐ ๐ ๐ (cid:19) โซ ๐ง ๐ง ๐๐๐๐ง ( ๐ โ ๐ง ) โ ๐ โ๏ธ ( ๐ง โ ๐ง ) ( ๐ง โ ๐ง ) ๐๐ง. (D12) (iii) | ๐ฅ | (cid:29) ๐ , ๐ฅ < ๐ ๐ ๐๐ โ โ ๐ฅ โ ๐ , โ๏ธ ( cos ๐ โ cos ๐ ) ( cos ๐ โ cos ๐ ) = ๐ฅ โ๏ธ ( ๐ง โ ) ( + ๐ โ ๐ง ) , (cid:20) ๐ ๐๐๐๐ (cid:18) cos ๐ โ ๐๐๐ ๐ cos ๐ (cid:19) โ ๐ (cid:21) ๐๐๐ = โ (cid:20) ๐๐๐๐ง (cid:16) ๐ง + (cid:17) + ๐ (cid:21) ๐ฅ ๐๐ง. (D13)All together, we get ฮ = ๐๐ ๐ (cid:18) ๐๐๐ ๐ (cid:19) (cid:18) ๐ ๐ ๐ ๐ (cid:19) (cid:20)(cid:18) โ ๐๐๐๐ง (cid:19) โ ๐ (cid:21) ๐ง = . (D14)The evaluation of the growth rates in section 3.2 and in the numericalsimulation is done by using these three cases, given that the formulasmatch each other at intermediate ranges of ๐ฅ / ๐ . APPENDIX E: DIFFUSION COEFFICIENTS
The evaluation of di๏ฌusion coe๏ฌcients (55) as a function of thebeam angle ๐ requires an integration over the entire range of wavesresonant with a pair-beam particle with polar angle ๐ . After the ๐ integration, using the delta function, the di๏ฌusion coe๏ฌcient takesthe form ๐ท ๐ = ๐๐ โซ ๐ ( k , ๐ก ) ๐ ๐ (cid:16) ๐ ๐ ๐๐ โ cos ๐ โ ๐ cos ๐ (cid:17) ๐ ๐ ๐๐๐ cos ๐ ๐๐ โ๏ธ(cid:2) cos ๐ โ ( cos ๐ ) (cid:3) (cid:2) ( cos ๐ ) โ cos ๐ (cid:3) . (E1)This expression is too cumbersome to evaluate, even numerically.One has to simplify the expressions in the integral by dividing theintegration space to two non-overlapping sub-spaces, based on themagnitude of ๐ฅ / ๐ and ๐ / ๐ . The complete di๏ฌusion coe๏ฌcientintegral is then presented as a sum of two expressions, ๐ท ๐ = ๐ท ,๐ + ๐ท ,๐ , (E2)where subscript โ1โ corresponds to large values of ๐ / ๐ฅ or ๐ / ๐ andโ2โ to the ( ๐ฅ, ๐ ) space complementary. The ๏ฌrst subspace, โ1โ, con-tains all points ( ๐ฅ, ๐ ) for which ๐ is much larger than ๐ or than | ๐ฅ | .The second subspace contains all points ( ๐ฅ, ๐ ) for which ๐ is notmuch larger than | ๐ฅ | and ๐ . Subspace โ1โ is separated further into twosmaller non-overlapping sub-spaces โ1aโ and โ1bโ as follows: Sub-space โ1aโ , ๐ ๐ , contains all points ( ๐ฅ, ๐ ) such that 50 | ๐ฅ | < ๐ and ๐ โค ๐ . Subspace โ1bโ , ๐ ๐ , contains all points ( ๐ฅ, ๐ ) such that ๐ > ๐ . We have found that the arbitrary choice of the large factorof 50 is suitable for the approximations made in the simpli๏ฌcationsof the algebraic expressions. Subspace โ2โ contains all the rest. Thesub-division of subspace โ1โ is mainly for convenience, while the in-tegration formula is the same. An example for the integration space ( ๐ฅ, ๐ ) for the di๏ฌusion coe๏ฌcients is described in ๏ฌgure E1. Thesolid curves correspond to the boundaries of the resonance parame-ter, ๐ฅ , as determined by formula (C4). The complete integration spaceis then the area between these two curves (ranging over all values of ๐ ). The dashed lines correspond to the separation between the twosub-spaces, where the numbers โ1โ and โ2โ denote the subspace.We now proceed to algebraic development of the di๏ฌusion in-tegration formulas for each subspace separately. In the subspace 1, ๐, | ๐ฅ | (cid:28) ๐ . Expanding in small parameters to the ๏ฌrst non-vanishing MNRAS , 1โ15 (2021) lazar induced pair beams Figure E1.
Integration space for the di๏ฌusion coe๏ฌcient at ๐ = ฮ ๐ = โ .The solid lines are the curves for the upper and lower bounds of ๐ฅ as afunction of ๐ , respectively. The dashed lines represent the boundaries of thetwo integration subspaces, denoted by 1 and 2 in the ๏ฌgure, respectively. Thesubdivision of subspace โ1โ is denoted by โ1aโ and โ1bโ and their boundariesare also denoted by dashed lines. The vertical dashed line represents the line ๐ = ๐ . order, one gets, after some work, ๐ท ,๐ = ๐๐ ๐ ๐ ๐ โซ ๐ ( ๐ ) (cid:18) tan ๐ + tan ๐ (cid:19) ๐ / ๐ง ๐ โ๏ธ โ ๐ง ๐ ( k , ๐ก ) ๐๐ง๐ tan ๐ , (E3)where ๐ง = ๐ฅ / ๐ .In the subspace 2, we de๏ฌne new variables ๐ฆ = ๐ฅ โ๏ธ ๐ฅ + ๐ โ ๐ฅ ๐ + ๐ค = ๐ ๐ . (E4)Then the di๏ฌusion coe๏ฌcient can be written, to zeroth order in ๐ , as ๐ท ,๐ = ๐๐ ๐ ๐ ๐ (cid:18) ๐ (cid:19) ๐ + โซ ๐ ( ๐ ) ๐ ( k , ๐ก ) ( ๐ฆ โ ) ๐ โ๏ธ ๐ค โ ๐ฆ ๐๐ค ๐๐ฆ. (E5) This paper has been typeset from a TEX/L A TEX ๏ฌle prepared by the author. MNRAS000