Electron spectra following their injection from a single discrete source (normalised to the injection level at the minimum energy indicated by the dashed red line, n_e, 0), accounting for variations in the energy densities of the radiation fields governing inverse-Compton cooling as well as electron diffusion. In all panels, the minimum energy at which CR secondary electrons may be injected is indicated by the dashed red vertical line – this would correspond to the secondaries produced when a 0.28 GeV CR proton primary undergoes a pp interaction. The injection spectral index would be approximately Γ = −2.1, as indicated in blue in the upper panel. Indices of subsequent steady-state spectra are steeper than this due to the energy dependence of the electron cooling rate. Upper panel: spectral variation with distance: dashed light grey line shows the steady-state spectrum expected at the injection point (we note that this is not the injection spectrum); remaining lines (solid black through to solid grey) denote the spectrum at 0.1–0.5 kpc in 0.1 kpc increments when adopting a CMB energy density at z = 7 and no stellar radiation field. The steepening of the high-end of the spectrum with distance is due to the energy-dependence of the radiative cooling function and of electron diffusion (with higher energy particles diffusing and undergoing radiative cooling more quickly). Middle panel: as above, but spectra calculated at 0.2 kpc (black lines) and 0.5 kpc (blue lines) from their injection location, with CMB at z = 7 and the spectral lines indicating evolution according to: no stellar radiation field (solid lines), stellar radiation field consistent with ℛ_SF = 3.5 M_⊙ yr⁻¹ (dashed lines), ℛ_SF = 35 M_⊙ yr⁻¹ (dot-dashed lines) and ℛ_SF = 350 M_⊙ yr⁻¹ (dotted lines) where star-formation is concentrated in a 1 kpc volume. Lower panel: as above, but with all lines calculated at 0.2 kpc from their injection source location, lines in black corresponding to a CMB-only radiation field and lines in blue corresponding to a strong radiation field with ℛ_SF = 350 M_⊙ yr⁻¹, shown at different redshifts over a similar range to our high-redshift galaxy sample: z = 6 (solid line), 7 (dashed line), 8 (dot-dashed line) and 9 (dotted line).