Constraining the magnetic field structure in collisionless relativistic shocks with a radio afterglow polarization upper limit in GW 170817

Gamma-ray burst (GRB) afterglow arises from a relativistic shock driven into the ambient medium, which generates tangled magnetic fields and accelerates relativistic electrons that radiate the observed synchrotron emission. In relativistic collisionless shocks the post-shock magnetic field B is produced by the two-stream and/or Weibel instabilities on plasma skin-depth scales (c/ω_p), and is oriented predominantly within the shock plane (B_⊥; transverse to the shock normal, \hat{n}_{sh}), and is often approximated to be completely within it (B_\parallel ≡ \hat{n}_{sh} \cdot B=0). Current 2D/3D particle-in-cell simulations are limited to short time-scales and box sizes ≲10⁴(c/ω_p) ≪ R/Γ_sh much smaller than the shocked region's comoving width, and hence cannot probe the asymptotic downstream B structure. We constrain the latter using the linear polarization upper limit, \vert Π \vert < 12 per cent, on the radio afterglow of GW 170817 / GRB 170817A. Afterglow polarization depends on the jet's angular structure, our viewing angle, and the B structure. In GW 170817 / GRB 170817A the latter can be tightly constrained since the former two are well-constrained by its exquisite observations. We model B as an isotropic field in 3D that is stretched along \hat{n}_{sh} by a factor ξ ≡ B_\parallel /B_⊥, whose initial value ξ_f ≡ B_\parallel,_f/B_{⊥, f} describes the field that survives downstream on plasma scales ≪R/Γ_sh. We calculate Π(ξ_f) by integrating over the entire shocked volume for a Gaussian or power-law core-dominated structured jet, with a local Blandford-McKee self-similar radial profile (used for evolving ξ downstream). We find that independent of the exact jet structure, B has a finite, but initially sub-dominant, parallel component: 0.57 ≲ ξ_f ≲ 0.89, making it less anisotropic. While this motivates numerical studies of the asymptotic B structure in relativistic collisionless shocks, it may be consistent with turbulence amplified magnetic field.