RELATIVISTIC PAIR BEAMS FROM TeV BLAZARS: A SOURCE OF REPROCESSED GeV EMISSION RATHER THAN INTERGALACTIC HEATING

The evolution of the beam-plasma system during the oblique phase is presented in panels (a)–(d) of Figure 1. The beam is set up with a small thermal spread (k_BT_b/m_ec²  ≃  10⁻⁶), so that the oblique instability initially proceeds in the reactive regime, i.e., all the beam particles are in resonance with each harmonic of the packet of unstable modes, and the instability is the strongest. This is opposed to the kinetic regime, in which the beam velocity spread is considerable. Here, a number of unstable modes with a broad spectrum in phase velocity will be excited, with only a small number of beam particles being in resonance with each mode. This results in a slower growth, as compared to the reactive regime.

Zoom In Zoom Out Reset image size

In Figure 1, we confirm that the oblique instability is the fastest growing mode for ultra-relativistic dilute beams. The reactive phase of the instability governs the evolution of the system for $t\,{\lesssim}\, t_{\rm OBL}\,{\simeq}\,600\,\omega _e^{-1}$, where t_OBL is marked as a dash-dotted vertical blue line in panels (a) and (b). Here, $\omega _e=\sqrt{4\pi e^2 n_e/m_e}$ is the plasma frequency of the background electrons. The fastest mode grows on a scale ∼2πc/ω_e, where c/ω_e is the electron skin depth, and its wavevector is inclined at ∼45° with respect to the beam propagation. This is apparent in the 2D structure of the longitudinal electric field in Figure 1(c), as well as in the 2D plots of the transverse electric and magnetic fields (not shown).¹¹ The oblique mode is also captured in 3D simulations, as shown in the 3D structure of the longitudinal electric field of Figure 2(a).

Zoom In Zoom Out Reset image size

The growth rate of the oblique instability in the reactive regime is (e.g., Fainberg et al. 1970)

$$\begin{eqnarray} &&\omega _{\rm OBL}({\boldsymbol{k}})=\frac{\sqrt{3}}{2^{4/3}}\left(\frac{2 \alpha }{\gamma _b}\right)^{1/3}\left(\frac{k_\perp ^2}{k^2}+\frac{k_\parallel ^2}{\gamma _b^2k^2}\right)^{1/3}\omega _e, \end{eqnarray} \tag{ 3 }$$

where the different dependence on k_⊥ and k_∥ is related to the fact that for relativistic beams the transverse inertia is much smaller than the longitudinal inertia (by a factor of $\gamma _b^2$), so that the modes transverse to the beam are the easiest to be excited (for an intuitive physical description, see Nakar et al. 2011).¹² From the pattern in Figure 1(c), we infer $k_\perp \,{\sim}\,k_{\parallel }\,{\sim}\,k/\sqrt{2}$, so the growth rate of the fastest growing oblique mode will be

$$\begin{eqnarray} &&\omega _{\rm OBL}\sim \frac{\sqrt{3}}{2^{4/3}}\left(\frac{\alpha }{\gamma _b}\right)^{1/3}\omega _e\equiv \delta _{\rm OBL} \,\omega _e, \end{eqnarray} \tag{ 4 }$$

which nicely agrees with our results.¹³ In fact, in Figure 1(a) we show that the fraction of beam kinetic energy deposited into the background electrons (orange line), into the longitudinal (red) and transverse (blue) electric fields, and into the transverse magnetic field (green) all grow at the rate predicted by Equation (4) for $t\lesssim 600\,\omega _e^{-1}$ (dotted orange line in Figure 1(a)).

The oblique mode is quasi-electrostatic, i.e., roughly k∥E_k (e.g., Bret et al. 2010b). Since the angle between the wavevector and the beam is ≳ 45°, in agreement with analytical expectations (e.g., Bret et al. 2010b), it follows that the electric field component transverse to the beam is slightly larger than the longitudinal component, i.e., k_⊥ ≳ k_∥ implies that E_⊥ ≳ E_∥. This explains the small difference between the red and the blue lines in Figure 1(a). Also, since the mode is quasi-electrostatic, the magnetic component will be sub-dominant relative to the electric fields. In agreement with the analytical considerations of Lemoine & Pelletier (2010), we find that B_⊥  ∼  2 δ_OBL E_⊥. Given that δ_OBL ≪ 1 for ultra-relativistic dilute beams, it follows that B_⊥ ≪ E_⊥.

For electrostatic modes in the linear phase, it is expected that the amount of kinetic energy lost by the beam should be equally distributed between electric fields and plasma heating (e.g., Thode 1976). This explains why the energy _e of the background electrons in the exponential phase (orange line in Figure 1(a) at $t\lesssim t_{\rm OBL} \,{\simeq}\,600\,\omega _e^{-1}$) is comparable to the energy in electric fields (_{E, ∥} and _{E, ⊥}, respectively, red and blue lines in Figure 1(a)). We have verified that the fraction of beam energy transferred to the background protons is negligible as compared to the plasma electrons, so the evolution of the protons will be ignored hereafter.

Since E_⊥  ∼  E_∥ during the exponential phase of the oblique mode, both the plasma and the beam are heated quasi-isotropically, so that the momentum spreads in the longitudinal and transverse directions are nearly identical (compare the red and blue lines for t ≲ t_OBL in Figure 1(b); dashed lines refer to the plasma, solid lines to the beam). The transverse momentum spread of the beam can be related to the transverse electric field E_⊥ via the Lorentz force

$$\begin{eqnarray} &&\frac{\Delta p_{b,\perp }}{\Delta t}\sim \delta _{\rm OBL}\,\omega _e \Delta p_{b,\perp } \sim e E_{\perp }, \end{eqnarray} \tag{ 5 }$$

where we have assumed that the characteristic timescale is set by the oblique growth rate (i.e., Δt⁻¹  ∼  δ_OBL ω_e) and that the magnetic force is negligible compared to the electric force (in fact, B_⊥/E_⊥  ∼  2 δ_OBL ≪ 1).

Since the oblique mode is heating up the beam in the transverse direction (solid blue line in Figure 1(b)), the exponential growth at the reactive rate ω_OBL will necessarily terminate, when the assumption of a cold beam required by the reactive approximation becomes invalid. It is well known that the system will transition from the reactive phase to the kinetic phase when the beam velocity dispersion ${\boldsymbol {\Delta v_b}}$ reaches (e.g., Fainberg et al. 1970; Bret et al. 2010b)

$$\begin{eqnarray} &&|{\boldsymbol {k}}\cdot {\boldsymbol {\Delta v_b}}|\sim \omega _{\rm OBL}, \end{eqnarray} \tag{ 6 }$$

namely, when the beam, due to its velocity spread, can move across one wavelength of the most unstable mode during the growth time of the reactive instability. In this case, most of the beam particles will lose resonance with the unstable mode, and the instability will transition from the reactive to the kinetic regime. For ultra-relativistic beams with isotropic momentum dispersions (in fact, Figure 1(b) shows that Δp_{b, ⊥}  ∼  Δp_{b, ∥} during the oblique reactive phase), the transverse velocity spread Δv_{b, ⊥}/c  ∼  Δp_{b, ⊥}/γ_bm_ec is much larger than the longitudinal spread $\Delta v_{b,\parallel }/c\,{\sim}\,(\Delta v_{b,\perp }/c)^2+\Delta p_{b,\parallel }/\gamma _b^3m_e c$. Since k_⊥  ∼  k_∥  ∼  ω_e/c, Equation (6) above reduces to

$$\begin{eqnarray} &&\Delta p_{b,\rm OBL}\,{\sim}\,\delta _{\rm OBL} \gamma _b\,m_e c, \end{eqnarray} \tag{ 7 }$$

where Δp_{b, OBL} is the expected transverse dispersion in beam momentum at the end of the reactive oblique phase. The threshold in momentum dispersion Δp_{b, OBL} can also be recast as a limit in beam temperature (e.g., Bret et al. 2010a). We confirm that the reactive phase of the oblique mode terminates at $t\,{\sim}\,t_{\rm OBL}\,{\sim}\,600\,\omega _e^{-1}$, when the beam transverse momentum reaches the threshold Δp_{b, OBL} in Equation (7) (which is shown as a horizontal dash-dotted blue line in Figure 1(b)).

We can now derive the expected fraction of beam kinetic energy transferred to the plasma electrons and to the electromagnetic fields at the end of the oblique reactive phase. By setting Δp_{b, ⊥} = Δp_{b, OBL} in Equation (5), we find that the fraction of beam energy converted into transverse electric fields at t  ∼  t_OBL is

$$\begin{eqnarray} &&\epsilon _{{E,\perp }}\equiv \frac{E_\perp ^2}{8 \pi \gamma _b n_b m_e c^2}\,{\sim}\,\frac{\sqrt{27}}{32}\delta _{\rm OBL}. \end{eqnarray} \tag{ 8 }$$

Since the oblique mode is quasi-electrostatic (i.e., E_⊥  ∼  E_∥), it follows that _{E, ⊥}  ∼  _{E, ∥}. Moreover, for electrostatic modes, the fraction _e of the beam kinetic energy converted into plasma heating is comparable to the energy in electric fields (e.g., Thode 1976), so _e  ∼  _{E, ⊥}  ∼  _{E, ∥}. Finally, since B_⊥  ∼  2 δ_OBL E_⊥, the magnetic energy fraction will be $\epsilon _{B,\perp }\,{\sim}\,\sqrt{27}\,\delta _{\rm OBL}^3/8$. We have extensively verified that the expected scalings of the efficiency parameters _e, _{E, ⊥}, _{E, ∥} and _{B, ⊥} with respect to δ_OBL are in agreement with the results of our simulations, across the whole range of beam Lorentz factors and density contrasts we have explored (see the various curves in Figure 1(a) at t  ∼  t_OBL, and also Section 4.2.1).

For t ≳ t_OBL, the evolution of the oblique mode will proceed in the kinetic (rather than reactive) regime. The kinetic oblique mode is indeed responsible for the peak in the electric field energy observed at ω_et  ∼  1000 in Figure 1(a) (red and blue lines), which produces a moderate increase in the fraction of beam energy transferred to the background electrons (orange line in Figure 1(a) at ω_et  ∼  1000). In this phase, the 2D structure of the longitudinal electric field in Figure 1(d) shows that the wavevector of the kinetic oblique mode is oriented at ∼20° relative to the beam propagation (as compared to the ∼45° angle observed during the reactive phase, see Figure 1(c)). A similar pattern is shown in the 3D plot of Figure 2(b). As expected, the increase in the transverse momentum dispersion has suppressed the modes having k_⊥ ≫ k_∥, which are most sensitive to transverse temperature effects (e.g., Bret et al. 2010b).

For a beam with initial transverse velocity dispersion Δv_{0, ⊥}, the growth rate of the kinetic oblique mode for k_⊥ ≲ ω_e/c is (e.g., Breiˇzman & Ryutov 1971)

$$\begin{eqnarray} &&\omega _k\,{\sim}\,\left(\frac{c}{\Delta v_{0,\perp }}\right)^2 \frac{\alpha }{\gamma _b}\,\omega _e\equiv \delta _{k} \,\omega _e. \end{eqnarray} \tag{ 9 }$$

At the end of the reactive oblique phase, Equation (7) prescribes that Δv_{0, ⊥}/c = Δp_{b, OBL}/γ_bm_ec  ∼  δ_OBL, so that the growth rate of the kinetic oblique mode will be ω_k∝δ_OBL ω_e, i.e., it will have the same scalings with α and γ_b as the reactive oblique mode (here, we have neglected factors of the order of unity).

We have explicitly verified that the peak in electric fields at ω_et  ∼  1000 is due to the kinetic oblique mode, by performing a dedicated simulation in which at ω_et  ∼  800 (i.e., shortly after the end of the reactive stage) we reset by hand the electromagnetic fields and the plasma temperature to their initial values (i.e., no seed fields and k_BT_e/m_ec²  ≃  10⁻⁸), yet we retain the beam momentum distribution that results self-consistently from the reactive oblique phase. In this setup, we find that the fastest growing mode has the same 2D pattern as in Figure 1(d) and its growth rate scales as ∝δ_OBL ω_e. This confirms that the peak in electric fields at ω_et  ∼  1000 (Figure 1(a)) is indeed associated to the kinetic oblique instability.

The growth of the kinetic oblique mode terminates due to self-heating of the beam, in analogy to the reactive oblique phase. In particular, the growth in the kinetic oblique phase cannot be sustained beyond the point where, due to the self-excited electric fields, the beam momentum dispersion in the transverse direction exceeds the initial value γ_bΔv_{0, ⊥}m_e. At this point, the expression for the growth rate in Equation (9) becomes clearly invalid. This happens when

$$\begin{eqnarray} &&e E_{\perp }\sim \omega _k \gamma _b\, \Delta v_{0,\perp } m_e \sim \delta _{\rm OBL}\, \omega _e\, \gamma _b \,\Delta v_{0,\perp } m_e, \end{eqnarray} \tag{ 10 }$$

which leads to the same scaling as in Equation (8), if we take Δv_{0, ⊥}/c = Δp_{b, OBL}/γ_bm_ec  ∼  δ_OBL, as appropriate for the beam velocity dispersion at the end of the reactive oblique phase. In summary, apart from factors of the order of unity, the electric fields at the end of the kinetic oblique phase will saturate at a level similar to the reactive oblique stage (compare the two peaks of electric energy in Figure 1(a) at ω_et  ∼  600 and ω_et  ∼  1000). It follows that the kinetic oblique instability will increase the fraction of beam kinetic energy transferred to the plasma electrons only by a factor of the order of unity, as compared to the reactive oblique mode (see the orange line in Figure 1(a), for ω_et ≳ 1000).

After the saturation of the oblique instability, further evolution of the beam-plasma system proceeds via quasi-longitudinal modes, as shown in the 2D plot of the longitudinal electric field in Figure 1(g), as well as in the 3D pattern of Figure 2(c). Being quasi-longitudinal, these modes are insensitive to thermal spreads in the direction perpendicular to the beam, so they can grow even after the end of the kinetic oblique phase.

The quasi-longitudinal oscillations shown in Figure 1(g) are a characteristic signature of the quasi-linear relaxation of the beam (see, e.g., Grognard 1975; Lesch & Schlickeiser 1987; Schlickeiser et al. 2002; Pavan et al. 2011). In the quasi-linear relaxation, the beam generates longitudinal Langmuir waves (see the peak in E_∥ at ω_et  ∼  10⁴ in Figure 1(e)), which scatter the beam particles and heat the background plasma (see the increase in the electron thermal energy shown by the orange line at 10⁴ ≲ ω_et ≲ 1.5 × 10⁴ in Figure 1(e)). Since E_∥ ≫ E_⊥ (compare the red and blue curves in Figure 1(e) at ω_et  ∼  10⁴), the background electrons will be heated preferentially in the direction of motion of the beam (see the increase in Δp_{e, ∥} at 10⁴ ≲ ω_et ≲ 1.5 × 10⁴ in Figure 1(f)). For the same reason, the quasi-linear relaxation is accompanied by a substantial increase in the beam momentum spread along the direction of propagation (red solid line in Figure 1(f), showing the growth of Δp_{b, ∥} at 5 × 10³ ≲ ω_et ≲ 1.5 × 10⁴). The spread in the parallel beam momentum is associated to the formation of phase space holes, that result from the trapping of beam particles by the longitudinal electric oscillations (e.g., O'Neil et al. 1971; Thode & Sudan 1975).

The quasi-linear relaxation occurs on a timescale much longer than the exponential oblique phase. Within the range of beam Lorentz factors and density contrasts probed by our simulations, we find that the characteristic relaxation time τ_R is at least two orders of magnitude longer than the exponential growth time of the oblique instability $\tau _{\rm OBL}=\omega _{\rm OBL}^{-1}$, in agreement with previous 1D simulations (Grognard 1975; Pavan et al. 2011). This emphasizes the importance of evolving our PIC simulations to sufficiently long times to capture the physics of the quasi-linear relaxation (see Section 4.2 for further details).

The quasi-linear modes broaden the beam momentum spectrum in the longitudinal direction up to the point where Δp_{b, ∥}/γ_bm_ec  ∼  0.2 (see the red solid line in Figure 1(f), saturating at Δp_{b, ∥}/m_ec  ∼  0.2 γ_b  ∼  60). This in agreement with the so-called Penrose's criterion, stating that the beam will be stable to electrostatic modes only when the longitudinal dispersion in momentum approaches the initial beam Lorentz factor (e.g., Buschauer & Benford 1977). From Δp_{b, ∥}/γ_bm_ec  ∼  0.2, it follows that at the end of the relaxation phase, a fraction ∼10% of the beam energy has been transferred to the background electrons (see the orange line in Figure 1(e) at ω_et ≳ 2 × 10⁴). In Section 4.2.1 we demonstrate that, irrespective of the beam Lorentz factor or the beam-to-plasma density contrast, a generic by-product of the relaxation of cold ultra-relativistic dilute beams is the conversion of ∼10% of their energy into plasma heating.¹⁴

The quasi-linear relaxation significantly affects the shape of the beam and plasma longitudinal momentum spectrum, as shown in Figure 3. As a result of the quasi-longitudinal relaxation, the plasma distribution at late times (ω_et ≳ 2 × 10⁴) develops a pronounced high-energy tail (at 3 ≲ p_∥/m_ec ≲ 10²), that bridges the main thermal peak of the plasma electrons (at p_∥/m_ec  ∼  0.5) with the beam particles (that populate the isolated high-energy bump at p_∥/m_ec  ∼  300 in Figure 3). This high-energy component in the background electrons, which contains a significant amount of energy, is present only in the forward direction (i.e., along the beam propagation). In the backward direction, the spectrum of plasma electrons (red dotted line in Figure 3, at ω_et = 5 × 10⁴) is compatible with a Maxwellian.

Zoom In Zoom Out Reset image size

During the quasi-linear relaxation, the beam spectrum evolves from a quasi-monoenergetic distribution into a broad bump (from the black to the red curve at p_∥/m_ec  ∼  300 in Figure 3). In agreement with Figure 1(f) (red solid line), most of the evolution occurs at ω_et ≲ 1.5 × 10⁴, whereas the beam spectrum at longer times is remarkably steady (we have followed the system up to ω_et  ∼  1.5 × 10⁵, finding no further signs of evolution).

We point out that the beam momentum spectrum at late times does not approach the so-called "plateau" distribution ${dN}/{dp}_\parallel \propto p_\parallel ^0$ (indicated as a black dotted line in Figure 3), which is believed to be the ultimate outcome of the beam relaxation in 1D (e.g., Grognard 1975; Schlickeiser et al. 2002; Dieckmann et al. 2013). As opposed to earlier 1D claims, in our 2D and 3D simulations we find that the beam longitudinal relaxation leads to a momentum spectrum that is harder than the plateau distribution, yet the beam-plasma system appears stable.¹⁵ In turn, the fact that the beam spectrum is harder than ${dN}/{dp}_\parallel \propto p_\parallel ^0$ explains why the amount of beam energy transferred to the plasma is only ∼10% (it should be ∼50% for a plateau distribution extending up to γ_bm_ec, see Thode & Sudan 1975).

We argue that the transverse dispersion in beam momentum, which could not be properly captured in previous 1D studies, prevents the longitudinal beam spectrum from relaxing to the plateau distribution (see Appendix B for further details). Our claim is supported by the following experiment. At the end of the kinetic oblique phase (ω_et  ∼  2000), we artificially set the beam transverse dispersion to be Δp_{b, ⊥}/m_ec ≪ 1 (for comparison, the self-consistent evolution in Figure 1(b) yields Δp_{b, ⊥}/m_ec  ∼  10 at the end of the kinetic oblique phase, see the solid blue line). Also, in the subsequent evolution, we inhibit any growth in the transverse beam momentum. In this setup, in which any transverse dispersion effects are artificially neglected, the beam relaxation should proceed as in 1D. So, it is not surprising that, in agreement with previous 1D studies, at late times the beam relaxes to the plateau distribution (red dashed line in Figure 3). In other words, we find that the relaxation to the plateau distribution is not a general result of the multi-dimensional evolution of ultra-relativistic beams, but it only occurs when the beam transverse dispersion stays sufficiently small, so the system is quasi-1D. We now provide a qualitative explanation of this important aspect of the physics of beam-plasma interactions, deferring a more detailed analytical treatment to a future publication.

As we further discuss in Appendix B, the beam relaxation produces a plateau distribution only if Δp_{b, ⊥}/m_ec ≪ 1, due to the following argument. Complete stabilization of the beam-plasma system is achieved when the longitudinal velocity spread of the ultra-relativistic beam reaches Δv_{b, ∥}/c  ∼  1 (more precisely, when the velocity spread is comparable to the beam speed). The spread in longitudinal velocity includes contributions from both the longitudinal and the transverse momentum dispersions:

$$\begin{eqnarray} &&\frac{\Delta v_{b,\parallel }}{c}\,{\sim}\,\frac{\Delta p_{b,\parallel }}{\gamma _b^3 m_e c} +\left(\frac{\Delta p_{b,\perp }}{\gamma _b m_e c}\right)^2, \end{eqnarray} \tag{ 11 }$$

where the second term on the right hand side is absent in the case of 1D relaxation. It follows that the transverse momentum spread can appreciably modify the relaxation process only if $\Delta p_{b,\perp }/m_e c\gtrsim \sqrt{\Delta p_{b,\parallel }/\gamma _b m_e c}\,{\sim}\,1$, where we have used that Δp_{b, ∥}/γ_bm_ec  ∼  0.2 at the end of the relaxation phase. Then, the fact that in the case studied in Figure 1(b) the transverse beam dispersion after the oblique phase is Δp_{b, ⊥}/m_ec  ∼  10 explains why the beam relaxation cannot lead to a plateau distribution. In Appendix B, we provide further evidence that the relaxation to a plateau distribution requires Δp_{b, ⊥}/m_ec ≪ 1.

In the previous subsections, we have primarily focused on the electrostatic character of the growing modes, which determines the coupling efficiency between the beam energy and the plasma thermal energy. Here, we comment on the generation of magnetic fields associated with the evolution of the beam-plasma system.

As shown in Figure 1(a), the growth of the quasi-electrostatic oblique mode (both in the reactive and in the kinetic regime) is accompanied by a minor magnetic component. In Section 4.1.1, we have estimated that the fraction of beam kinetic energy transferred to the magnetic fields at the end of the oblique phase is $\epsilon _{B,\perp }\,{\sim}\,\delta _{\rm OBL}^3$, apart from factors of the order of unity. Since δ_OBL ≪ 1 for ultra-relativistic dilute beams, the magnetic fields generated by the oblique instability are generally unimportant.

At the end of the relaxation phase, the beam and the plasma are highly anisotropic, with the longitudinal momentum spread much larger than the transverse one (see Figure 1(f) at ω_et  ∼  1.5 × 10⁴). As a result, the system is prone to the Weibel instability (e.g., Weibel 1959; Yoon & Davidson 1987; Sakai et al. 2000; Silva et al. 2002, 2003; Jaroschek et al. 2005), which generates the transverse magnetic field pattern shown in Figure 1(h).¹⁶ As a result of the Weibel instability, the magnetic field energy increases (see the green line in Figure 1(e), at 10⁴ ≲ ω_et ≲ 2 × 10⁴), and the beam and plasma anisotropy is reduced by increasing the transverse momentum spread (see the solid and dashed blue lines at 10⁴ ≲ ω_et ≲ 2 × 10⁴ in Figure 1(f)).

The Weibel instability is predominantly magnetic, so it does not mediate any significant exchange of energy from the beam to the plasma electrons, and the heating efficiency _e does not increase beyond the value _e  ∼  10% attained at the end of the relaxation phase. However, the Weibel instability might be a promising source for the generation of magnetic fields, as discussed by Schlickeiser et al. (2012b). However, the evolution of the magnetic filaments shown in Figure 1(h) can only be captured with large-scale 3D simulations (e.g., Bret et al. 2008; Kong et al. 2009). A detailed 3D investigation of the strength and scale of the magnetic fields resulting from ultra-relativistic dilute pair beams from TeV blazars will be presented elsewhere.

In Figure 4, we show how the evolution of the beam-plasma system depends on the beam Lorentz factor (that we vary from γ_b = 3 up to γ_b = 1000) and on the beam-to-plasma density contrast (from α = 3 × 10⁻² down to α = 3 × 10⁻³). Most of the previous studies have focused on moderately relativistic electron beams (γ_b = 3–6) with α = 10⁻¹ (e.g., Gremillet et al. 2007; Bret et al. 2008; Kong et al. 2009). Here, we investigate the case of ultra-relativistic dilute electron-positron beams, as appropriate for blazar-induced beams.

Zoom In Zoom Out Reset image size

For each choice of γ_b and α, we follow the beam-plasma system from the oblique phase until the quasi-linear relaxation (typically, up to ω_et  ∼  10⁵). We initialize a cold beam with thermal spread k_BT_b/m_ec²  ≃  10⁻⁴, so that the oblique instability initially proceeds in the reactive regime, for the range of γ_b and α covered by our simulations. In the reactive phase, we confirm that the fastest growing mode has a wavevector oriented at ∼45° to the beam direction of propagation. The growth rate is in excellent agreement with Equation (4). As described in Section 4.1.1, the exponential phase of the reactive oblique mode terminates due to self-heating of the beam in the transverse direction. At the end of the reactive phase, we find that the fractions of beam kinetic energy transferred to the plasma electrons, to the electric fields and to the magnetic fields scale, respectively, as _e∝δ_OBL, _{E, ⊥}  ∼  _{E, ∥}∝δ_OBL and $\epsilon _{B,\perp }\propto \delta _{\rm OBL}^3$, as in Section 4.1.1.

The reactive oblique phase is followed by the kinetic oblique phase. We find that the characteristic wave pattern in the kinetic oblique phase (see Figure 1(d) in 2D and Figure 2(b) in 3D) appears in the evolution of all the beam-plasma systems that we present in Figure 4. In the temporal evolution of the fraction of beam energy converted into plasma heating, the kinetic oblique mode is responsible for the additional increase that is seen in the most relativistic cases (γ_b ≳ 100) after the end of the reactive oblique phase (see the insets in the top row of Figure 4). Regardless of γ_b or α, we find that the kinetic oblique phase deposits only a fraction ∼δ_OBL of the beam kinetic energy into the background electrons, i.e., comparable to the reactive oblique phase (see Section 4.1.1).

The long-term evolution of the system is controlled by the quasi-longitudinal relaxation, which operates on a timescale τ_R  ≳  10² τ_OBL, where $\tau _{\rm OBL}=\omega _{\rm OBL}^{-1}$ is the characteristic e-folding time of the oblique mode. In Figure 4, the quasi-linear relaxation governs the growth in the plasma thermal energy (top row) and in the beam parallel momentum spread (middle row) occurring at ω_et ≳ 5000. In the regime γ_b ≫ 1, the quasi-linear relaxation terminates when the beam momentum spread in the longitudinal direction reaches Δp_{b, ∥}/γ_bm_ec  ∼  0.2, regardless of γ_b or α (middle row in Figure 4). Correspondingly, the fraction of beam energy transferred to the plasma saturates at _e  ∼  10% (top row in Figure 4).

Mildly relativistic beams with moderate density contrasts deviate from such simple scalings, for the following reason. The quasi-linear relaxation will not operate if the beam dispersion at the end of the oblique phase is already Δp_{b, ∥}/γ_bm_ec ≳ 0.2. According to Equation (7), the beam spread at the end of the oblique phase is Δp_{b, ∥}  ∼  Δp_{b, ⊥}  ∼  δ_OBLγ_bm_ec, so that the quasi-linear relaxation will be suppressed if δ_OBL  ∼  (α/γ_b)^1/3 ≳ 0.2, i.e., for mildly relativistic beams with moderate α.

A similar argument explains why a plateau distribution in the longitudinal momentum spectrum (bottom row in Figure 4) is established only for relatively small γ_b. As we have argued in Section 4.1.2, a transverse spread Δp_{b, ⊥}/m_ec ≳ 1 prevents the beam relaxation to the plateau distribution (shown as a dotted black line in the bottom row of Figure 4). At the end of the oblique phase, Δp_{b, ⊥}/m_ec  ∼  γ_b δ_OBL, so that the beam will relax to the plateau distribution only if $\gamma _b \delta _{\rm OBL}\,{\sim}\,(\gamma _b^2 \alpha)^{1/3}\ll 1$. Clearly, this constraint is hardest to satisfy for highly relativistic beams, which explains why, at fixed α, the momentum spectrum of more relativistic beams shows stronger deviations from the plateau distribution.

In the previous subsection, we have assumed that the beam is born with a negligible thermal spread. Here, we discuss how the results presented above for cold beams will be modified by temperature effects. We describe separately the role of thermal spreads in the development of the oblique instability, and in the relaxation phase.

As we have anticipated in Section 4.1.1, the oblique instability will proceed in the kinetic (rather than reactive) regime if the initial beam dispersion in the transverse direction is Δv_{0, ⊥}/c ≳ δ_OBL.¹⁷ The growth rate in the kinetic regime is reported in Equation (9). The plasma thermal energy grows exponentially, until the self-generated electric fields increase the transverse dispersion in beam velocity beyond the initial value Δv_{0, ⊥}. At this point, the exponential growth at the rate in Equation (9) will necessarily terminate. From the Lorentz force applied to the beam particles, we find that this will happen when

$$\begin{eqnarray} &&e E_{\perp }\sim \omega _k \gamma _b\, \Delta v_{0,\perp } m_e , \end{eqnarray} \tag{ 12 }$$

which results in a fraction _{E, ⊥}  ∼  δ_k of the beam energy transferred to the electric fields at the end of the kinetic phase (δ_k ≡ ω_k/ω_e is defined in Equation (9)). Since the kinetic oblique mode is an electrostatic instability, the fraction of beam energy converted into heat will also be _e  ∼  δ_k. Moreover, due to the fact that ω_k  ≲  ω_OBL—they are comparable only if Δv_{0, ⊥}/c  ∼  δ_OBL, i.e., at the boundary between reactive and kinetic regimes—we expect the kinetic oblique mode to be less efficient in heating the plasma electrons, as compared to the reactive phase. For beams with γ_b = 3–10 and α = 10⁻³–10⁻⁴, we have indeed verified with PIC simulations (not presented here) that the exponential growth of the kinetic oblique mode terminates at smaller _e for larger values of Δv_{0, ⊥}, in good agreement with the expected scaling $\epsilon _e\propto \Delta v_{0,\perp }^{-2}$ (at fixed γ_b and α).¹⁸

Regardless of the character of the oblique mode (reactive or kinetic), the long-term evolution of the beam-plasma system is controlled by the quasi-linear relaxation. In Figure 5, we show how the relaxation phase is affected by a finite beam temperature T_b. For the set of beam parameters employed in Figure 5 (γ_b = 300 and α = 10⁻²), the oblique phase is expected to occur in the reactive regime, as long as the beam comoving temperature is non-relativistic. In fact, for all the choices of T_b presented in Figure 5 (in the plot, T_b is in units of m_ec²/k_B), the early increase in the heating efficiency _e proceeds at the reactive rate ω_OBL (compare the curves in the inset of Figure 5(a) with the dotted red line, that scales with the oblique growth rate). Also, the kinetic oblique instability, which is responsible for the further growth in _e at ω_et  ∼  1200 (see the inset in Figure 5(a)), does not show any dependence on temperature, in the range 10⁻⁶ ≲ k_BT_b/m_ec² ≲ 0.3 explored in Figure 5.

Zoom In Zoom Out Reset image size

The beam temperature has profound effects on the quasi-linear relaxation phase for the beam parameters employed in Figure 5. For cold beams (k_BT_b/m_ec² ≲ 10⁻⁴, yellow and red lines in Figure 5), the relaxation phase does not depend on the beam temperature. In agreement with the results presented in Section 4.2.1, the longitudinal spread in the beam momentum increases during the relaxation stage until Δp_{b, ∥}/γ_bm_ec  ∼  0.2, as shown in Figure 5(b). This corresponds to a fraction _e  ∼  10% of the beam kinetic energy being converted into plasma heating (Figure 5(a)). Similar conclusions hold for moderate beam temperatures (k_BT_b/m_ec² = 3 × 10⁻², green line), whereas the quasi-linear relaxation is suppressed if the beam temperature at birth is such that the initial beam spread Δp_{b0, ∥}/γ_bm_ec ≳ 0.2 (cyan, blue, and black lines in Figure 5(b)). In this case, the quasi-linear phase does not mediate any further increase in the heating efficiency _e, beyond the early oblique phase (see the black line in Figure 5(a)).¹⁹ In short, if the initial momentum spread is Δp_{b0, ∥}/γ_bm_ec ≳ 0.2, the plasma heating efficiency _e stays fixed at the value _e  ∼  δ_OBL attained at the end of the reactive oblique phase—or _e  ∼  δ_k, if the oblique instability proceeds in the kinetic regime.

The longitudinal momentum spectrum in Figure 5(c) clarifies why the quasi-linear relaxation is suppressed for hot beams. As we have discussed in Section 4.1.2, the relaxation is mediated by Langmuir waves excited by the beam. The beam particles that are slightly faster than the wave tend to transfer energy to the wave, and thus excite it, while those that are slightly slower than the wave tend to receive energy from it, and thus damp the wave. It follows that a mode is unstable if the number of beam particles moving slightly faster than the wave exceeds that of those moving slightly slower. More precisely, the instability will be stronger for a harder slope dlog N/dlog p_∥ of the longitudinal momentum spectrum at momenta ≲ γ_bm_ec (this is the relevant momentum scale, since the phase velocity of the most unstable mode is slightly smaller than the beam speed). For a sharply peaked beam (i.e., with k_BT_b/m_ec² ≪ 1), the slope dlog N/dlog p_∥ will be extremely hard, and the excitation of the Langmuir modes that mediate the relaxation process will be most efficient. As a result of the quasi-linear relaxation, the beam particles will be scattered down to lower energies, and when the beam momentum spread exceeds Δp_{b, ∥}/γ_bm_ec ≳ 0.2 the slope of the momentum spectrum below the peak becomes too shallow (see the red and yellow curves in Figure 5(c) at p_∥/m_ec ≲ 300), and the quasi-linear relaxation terminates. If the beam temperature at birth is too hot (i.e., such that Δp_{b0, ∥}/γ_bm_ec  ≳  0.2), the initial slope of the momentum spectrum at p_∥ ≲ γ_bm_ec is already too shallow to trigger efficient excitation of Langmuir waves, and the quasi-relaxation process is suppressed.

In Figure 5(c), we further support this argument by showing that in the latest stages of evolution, the shape of the beam momentum spectrum below γ_bm_ec (in Figure 5(c), at 30 ≲ p_∥/m_ec ≲ 300), is nearly independent of the beam temperature at birth. For initially cold beams (yellow and red lines), the quasi-linear relaxation increases the beam momentum spread over time (see Figure 5(b)), until the beam spectrum relaxes to the broad bump shown in Figure 5(c). At this point, the beam is stable, although it has not relaxed to the so-called plateau distribution (as we explain in Section 4.1.2, the relaxation to the plateau distribution requires Δp_{b, ⊥}/m_ec ≪ 1, which is not satisfied here). For initially hot beams, the initial momentum spectrum below γ_bm_ec resembles the final momentum distribution of initially cold beams. This explains why hot beams starting with a longitudinal momentum spread Δp_{b0, ∥}/γ_bm_ec ≳ 0.2 will not experience the quasi-linear relaxation phase.

So far, we have discussed the effect of thermal spreads on the beam relaxation, assuming that the beam spectrum at birth is a drifting Maxwellian. However, the conclusions derived above hold for more complicated beam distributions. In particular, we have performed a set of PIC simulations, assuming that the beam spectrum at birth is a power law ${dN}/{dp}_\parallel \propto p_\parallel ^{-2}$ for p_∥ ⩾ p_min ≫ m_ec. For blazar environments, such a beam spectrum is expected as a result of intense IC cooling off the CMB.

If the beam power-law distribution has a negligible spread in the transverse direction (which is typically not the case for blazar-induced beams, see Section 4.3), then the oblique instability will proceed in the reactive regime. Apart from factors of the order of unity, we find that the exponential growth rate is ∼ (α m_ec/p_min)^1/3, similar to the case of mono-energetic beams. As we have discussed above, the effectiveness of the quasi-linear relaxation—which ultimately determines whether the heating efficiency _e can reach ∼10%—is determined by the spread in the beam longitudinal spectrum below the peak, i.e., at p_∥ ≲ p_min. If the beam spectrum has a sharp low-energy cutoff at p_min, then the quasi-linear relaxation does operate, and the beam deposits ∼10% of its energy into the background electrons. However, if the low-energy end of the beam distribution is broader, the quasi-linear relaxation will be inhibited, and the amount of beam energy transferred to the plasma will be much smaller, in agreement with the results in Figure 5. We find that if the beam longitudinal spectrum below p_min can be modeled as a power law ${dN}/{dp}_\parallel \propto p_{\parallel }^s$, the quasi-linear relaxation is suppressed if s ≲ 3, with the case s = 0 corresponding to the plateau distribution.

We now compare our numerical work with earlier analytical studies of the relaxation of blazar-induced beams in the IGM. As we have emphasized in Section 3, our PIC simulations are the first to address the evolution of dilute ultra-relativistic electron–positron beams, as appropriate for blazar-induced beams in the IGM. All of the previous PIC studies (e.g., Dieckmann et al. 2006b; Gremillet et al. 2007; Bret et al. 2008; Kong et al. 2009) have focused on mildly relativistic electron beams.

In agreement with Broderick et al. (2012) and Schlickeiser et al. (2012a), we find that the fastest growing instability for blazar-induced beams propagating through the IGM is the oblique mode. If the pair beam were to be initially cold, the oblique instability would evolve at the reactive rate in Equation (4), with wavevector oriented at ∼45° relative to the beam (see Schlickeiser et al. 2012a). However, the initial transverse spread in beam momentum is large enough such that the oblique mode evolves at the kinetic rate in Equation (9), as argued by Broderick et al. (2012) and Miniati & Elyiv (2013). We remark that the transverse beam spread does affect the growth of the oblique mode, but it will not impact the evolution of longitudinal waves, as found by Schlickeiser et al. (2013). However, since the early evolution of blazar-induced beams is controlled by the oblique (rather than longitudinal) mode, transverse thermal effects are indeed important for the beam-plasma interaction at early times.

The evolution of blazar-induced beams at late times is governed by nonlinear plasma processes, whose efficiency depends sensitively on the assumed beam distribution. Schlickeiser et al. (2012b) and Miniati & Elyiv (2013) attempted to describe the nonlinear relaxation of blazar-induced beams, reaching opposite conclusions regarding the ultimate fate of the beam energy. Schlickeiser et al. (2012a) assumed that the beam momentum distribution can be modeled as a delta function (i.e., they approximated the beam as mono-energetic and uni-directional), and they found that the relaxation phase occurs much faster than the IC cooling losses. From this, they argued that more than 50% of the beam energy can be transferred to the IGM plasma. Miniati & Elyiv (2013) reached the opposite conclusion, when accounting for the effect of the finite transverse momentum spread of blazar-induced beams, as derived from a detailed Monte Carlo model of the electromagnetic cascade. They found that the beam energy is radiated by IC off the CMB well before the relaxation phase.

With our PIC simulations, we find that the relaxation phase occurs on a much longer timescale than the exponential oblique growth, at least by two orders of magnitude. However, it might be delayed even more for more extreme beam parameters, as suggested by both Schlickeiser et al. (2012b) and Miniati & Elyiv (2013). Even under the conservative assumption that the relaxation phase is faster than the IC cooling time, this does not imply that all of the beam energy is ultimately deposited into the IGM plasma. In short, the relaxation process being faster than the IC losses does not guarantee that it will also be efficient in heating the IGM electrons. Under the unrealistic assumption that blazar-driven beams are born with a small longitudinal momentum spread Δp_{b0, ∥}/γ_bm_ec ≲ 0.2, we find that only ∼10% (rather than ∼50%–100%, as assumed by Broderick et al. (2012)) of the beam energy is transferred to the plasma. This should be compared to the heating efficiency of ∼50% predicted by 1D models, that cannot properly account for the transverse broadening of the beam. For the realistic spread Δp_{b0, ∥}/γ_bm_ec  ∼  1 of blazar-induced beams, the coupling between the beam and the plasma will be much less efficient, with the heating efficiency as low as _e  ≃  10⁻⁹(γ_b/10⁷)(α/10⁻¹⁶).