Pair Beam Instability in Cosmic Plasmas

• Pair beam instability is a plasma phenomenon where relativistic electron–positron pairs trigger rapid growth of Langmuir modes in the intergalactic medium.
• The analysis differentiates between reactive and kinetic regimes, with the kinetic mode yielding growth rates significantly higher than the reactive estimates.
• Implications include efficient energy transfer to plasma waves that suppress gamma-ray cascades, challenging previous intergalactic magnetic field interpretations.

Pair beam instability refers to the collective microinstabilities that develop when relativistic electron–positron beams, typically produced via photon-photon annihilation (e.g., TeV gamma rays from blazars interacting with the extragalactic background light), propagate through a plasma such as the intergalactic medium (IGM). These instabilities govern the transport, energy loss, and angular dynamics of the pair beam, and thus determine both the observational signatures of high-energy cosmic sources and the heating of cosmic voids. The mechanisms involve the interplay of kinetic effects (momentum spread, beam anisotropy) and plasma wave excitation (primarily electrostatic/Langmuir modes), and are instrumental in understanding why expected cascade signatures may be absent in observations.

1. Formation of Pair Beams and Distribution Functions

High-energy gamma-rays from sources such as distant blazars traverse the IGM and annihilate soft photons of the extragalactic background light (EBL) via $\gamma\gamma \rightarrow e^+e^-$. The resulting pair beams have

• Energy: Lorentz factors %%%%1%%%%,
• Directionality: Strongly anisotropic, beamed nearly along the axis of the parent gamma ray; opening angles $\Delta \theta \sim 10^{-6} - 10^{-5}$,
• Phase-Space Distribution: Parallel momentum distribution sharply peaked about $x = p/(m_ec) \sim x_c$, with a small but finite perpendicular (transverse) momentum spread. This distribution is well modeled with forms such as:

$n(x) = A_0 x^{-s}\exp(-x_c/x)$

and for transverse momentum, waterbag-like functions or Gaussian angular spreads (Schlickeiser et al., 2013, Vafin et al., 2018).

The finite angular spread, though extremely small, critically influences the coupling of the beam to plasma wave modes.

2. Linear Instability Regimes: Reactive vs. Kinetic

The stability of these relativistic pair beams is examined via linear instability analysis:

• Reactive (Hydrodynamic) Regime: Assumes all beam particles remain near the resonant phase velocity. The distribution is nearly delta-function, resulting in coherent, rapid growth of unstable modes. The maximum growth rate for parallel electrostatic (Langmuir) oscillations is

$$(\gamma_p)_{\text{reactive}} \approx 1.5 \times 10^{-10}\;\text{Hz}$$

for typical IGM parameters (Schlickeiser et al., 2013).

• Kinetic Regime: Incorporates the beam's finite momentum spread (both longitudinal and small perpendicular dispersion), leading to a more realistic dispersion relation:

$$1 - \frac{\omega_{p,t}^2}{\omega^2} - \frac{\omega_{p,b}^2}{\gamma^3(\omega - k_z v_b)^2}\left(\frac{\gamma^2 k_x^2 + k_z^2}{k_x^2 + k_z^2}\right) = 0$$

The kinetic regime applies when only a limited fraction of the beam participates in wave–particle resonance. Crucially, the kinetic maximum growth rate for $b=0$ (no perpendicular spread) is found numerically to be about an order of magnitude larger than the reactive value, contradicting simple expectations that kinetic spread strongly damps instability (Schlickeiser et al., 2013, Chang et al., 2016).

• Angular Spread Sensitivity: Including a small but finite perpendicular (transverse) spread reduces the growth rate by a factor $< 10^{-4}$ compared to the $b=0$ (purely parallel) case, and the effect is quantified by a correction factor $B(X)$, where $B(X) \approx 1$ for $b \lesssim 0.1$ [Eq. (f2) in (Schlickeiser et al., 2013), Table 1].

The kinetic regime is generally robust for typical blazar-induced beams due to the tiny intrinsic opening angles.

3. Dispersion Relations and Growth Rate Characterization

The full kinetic calculation leads to an analytic and numerical computation of growth rates for electrostatic modes:

• The imaginary part of the plasma dielectric function, responsible for instability, remains largely insensitive to both the longitudinal and small perpendicular momentum spreads.
• Maximum kinetic growth rate (for negligible perpendicular spread) is given by

$$\gamma_p^{\text{max}}(b=0) = \frac{\gamma_p^0\kappa_0 x_c}{e^3}$$

where $\gamma_p^0$ is set by $n_b/N_e$ and $\kappa_0$ is the wavenumber of peak growth [Eq. (d15), (Schlickeiser et al., 2013)].

• With perpendicular spread included, the correction to this rate remains minuscule—always less than $10^{-4}$ for the physical parameters of interest (e.g., $b \le 0.1$, $s=2$) [Eq. (g3), Table 1, (Schlickeiser et al., 2013)].

Contrary to arguments that momentum spread stabilizes the beam, the analysis clearly shows negligible impact from realistic spreads.

4. Implications for Intergalactic Magnetic Fields and Energy Dissipation

The rapid growth rate of electrostatic instabilities means that the bulk kinetic energy of the beam is efficiently converted into plasma turbulence—primarily Langmuir waves—on short timescales compared to inverse Compton cooling. This process occurs in an unmagnetized IGM and thus does not require any pre-existing IGM magnetic field for significant beam energy dissipation.

This finding has direct implications:

• The hypothesis that the FERMI non-detection of inverse Compton-scattered GeV gamma rays from blazars could be solely attributed to magnetic deflection of the pairs is unnecessary. Plasma instabilities alone, via rapid kinetic energy transfer to waves, can suppress the cascade signal.
• Therefore, lower bounds previously derived for IGMF strengths (from the absence of cascades) are invalid in this context (Schlickeiser et al., 2013).
• The explanation of “missing” GeV cascades as a consequence of the beam-plasma instability is robust, challenging earlier criticisms that asserted the need for finite IGMFs for beam scattering.

5. Extensions, Limitations, and Future Directions

The kinetic instability framework provides several avenues for further paper:

• Geometry Extensions: The present analysis focuses on parallel-propagating electrostatic waves. Significant motivation exists to generalize to oblique and transverse modes, which may also participate in nonlinear coupling and saturation [see conclusions in (Schlickeiser et al., 2013)].
• Nonlinear Saturation: The ultimate fate of the beam energy (plasma heating, further electromagnetic mode coupling) depends on saturation mechanisms, nonlinear wave–particle interactions, and possible feedback on beam momentum distributions.
• Observational Signatures: Correlating rapid beam energy dissipation with IGM heating—e.g., via thermal imprints or secondary emission—remains a frontier, requiring high-resolution simulations and indirect observation.
• Plasma Composition Effects: The inclusion of finite temperature corrections or weak magnetic fields in the background IGM plasma could introduce additional modification to the instability boundaries or wave–particle coupling, warranting further investigation.

A broader implication is the necessity to incorporate full kinetic beam-plasma instability physics in interpreting high-energy astrophysical observations, as simplified models may incorrectly estimate the efficiency or even the presence of energy dissipation channels.

6. Summary Table: Key Parameters and Influence on Growth Rate

Parameter	Influence on Instability Maximum Growth Rate	Typical Magnitude
Longitudinal Spread	Negligible impact (no significant reduction vs. reactive)	$\sim$narrow (delta)
Perpendicular Spread ($b$)	Correction factor $B(X) < 10^{-4}$ for $b \le 0.1$	$b \approx 0.1$
Density Ratio ($n_b/N_e$)	Sets amplitude of growth rate	$n_b/N_e \ll 1$
Magnetic Field ($B_\text{IGM}$)	Not required for beam instability; no influence in unmagnetized limit	N/A

7. Conclusion

The pair beam instability in unmagnetized intergalactic plasma, specifically for relativistic beams resulting from blazar–EBL interactions, is robust against realistic beam momentum spreads. The kinetic growth rates not only remain undamped, but in the absence of perpendicular spread are enhanced by an order of magnitude relative to reactive estimates. As such, plasma effects alone can explain both the absence of cascade GeV emission and invalidate the need for a significant lower bound on IGMF inferred from this non-detection, compelling a revisitation of cascade modeling and IGMF diagnostics (Schlickeiser et al., 2013). Further progress depends on detailed kinetic analyses, multi-angle generalizations, and improved nonlinear modeling to completely characterize beam-plasma energy dissipation in cosmic environments.