Weibel Instability in Collisionless Plasmas Across Astrophysical and Laboratory Shocks

We present a cold-fluid analysis of the purely transverse Weibel (current-filamentation) instability across four regimes: non-relativistic (NR) single-species, NR multi-species, relativistic single-species, and relativistic multi-species (electron--positron and electron--proton). Beginning from linearized fluid equations, we derive the dispersion relations in each regime and extract scaling laws for the maximum growth rate $\gamma_{\rm max}$ and characteristic unstable wavenumber $k_{\rm max} = \omega_{pi}/c$. Relativistic corrections suppress $\gamma_{\rm max}$ by up to 40 per cent above $v_0 \approx 0.2c$, peaking near $v_0 \approx 0.9c$. Multi-species effects are significant only for $m_e/m_i \gtrsim 1/500$. For the tabletop laser experiment of Bai et al., Nat.Commun., 16, 3770 (2025), the cold-fluid prediction gives $d_i = c/\omega_{pi} \approx 31.7\,\mu{\rm m}$, within 2 per cent of the measured filament spacing $\lambda_F \approx 31\,\mu{\rm m}$. The saturation field estimate $B_{\rm sat} \approx 2.3\times10^4$ T is an upper bound, consistent with the measured $\approx 5000$ T under kinetic suppression. Two MMS burst-mode bow shock crossings (October 16, 2015 and November 25, 2017) confirm $k_{\rm max} d_i = 1$ from FGM/FPI data. A multi-environment scatter plot spans 21 orders of magnitude in $n_i$, with all points within a factor of 3 of the 1:1 line.