Scattering and Hawking Radiation from Einstein--Euler--Heisenberg--de Sitter Black Holes
We compute neutral scalar and neutral massless Dirac greybody factors and Hawking spectra for the positive-cosmological-constant branch of the Einstein--Euler--Heisenberg black hole. The finite region between the black-hole and cosmological horizons is treated as a two-sided scattering problem, with direct numerical integrations providing the transmission coefficients and a sixth-order WKB calculation used as a local check near the barrier top. Along the reference family studied here, increasing the Euler--Heisenberg coupling raises the dominant scalar and Dirac barriers and shifts the half-transmission frequencies upward. At fixed charge and nonlinear coupling, increasing the cosmological constant contracts the static patch and lowers the dominant greybody thresholds. The luminosity is controlled as much by the de Sitter temperature prescription as by the greybody factors: event-horizon prescriptions brighten the emission as the nonlinear correction grows, whereas effective static-patch temperatures give much smaller rates and can reverse the trend. Thus the evaporation interpretation is not unique unless the temperature convention is specified explicitly.