Kinetic Simulations of Electron-Positron Induced Streaming Instability in the Context of Gamma-Ray Halos Around Pulsars

The possibility of slow diffusion regions as the origin for extended TeV emission halos around some pulsars (such as PSR J0633+1746 and PSR B0656+14) challenges the standard scaling of the electron diffusion coefficient in the interstellar medium. Self-generated turbulence by electron-positron pairs streaming out of the pulsar wind nebula was proposed as a possible mechanism to produce the enhanced turbulence required to explain the morphology and brightness of these TeV halos. We perform fully kinetic 1D3V particle-in-cell simulations of this instability, considering the case where streaming electrons and positrons have the same density. This implies purely resonant instability as the beam does not carry any current. We compare the linear phase of the instability with analytical theory and find very reasonable agreement. The non-linear phase of the instability is also studied, which reveals that the intensity of saturated waves is consistent with a momentum exchange criterion between a decelerating beam and growing magnetic waves. With the adopted parameters, the instability-driven wavemodes cover both the Alfv\'enic (fluid) and kinetic scales. The spectrum of the produced waves is non-symmetric, with left-handed circular polarisation waves being strongly damped when entering the ion-cyclotron branch, while right-handed waves are suppressed at smaller wavelength when entering the Whistler branch. The low-wavenumber part of the spectrum remains symmetric when in the Alfv\'enic branch. As a result, positrons behave dynamically differently compared to electrons. The final drift velocity of positrons can maintain a larger value than the ambient Alfv\'en speed $V_A$ while the drift of electrons can drop below $V_A$. We also observed a second harmonic plasma emission in the wave spectrum. An MHD-PIC approach is warranted to probe hotter beams and investigate the Alfv\'en branch physics. We provide a few such test simulations to support this assertion. This work confirms that the self-confinement scenario develops essentially according to analytical expectations, but some of the adopted approximations (like the distribution of non-thermal particles in the beam) need to be revised and other complementary numerical techniques should be used to get closer to more realistic configuration.