Influence of the (inverse) beam temperature ρ₁: Panel a: maximum growth rates ω_i for different beam inverse temperatures ρ₁ obtained from the analytical solution of the linear dispersion relation (dots) and from simulations (crosses) with fit function (solid line). Panel b: integrated growth rates from solutions of the dispersion relation (dots) and from simulations (crosses) with fit function (solid line). Panel c: fractional bandwidth Δω_r at the point where ω_i is at its maximum, normalised to the corresponding real part of the frequency ω_r obtained by linear theory (dots) and simulation (crosses). All three figures are for ρ₀ = 1, γ_b = 26 and r_n = 10⁻³. All fit parameters are included in Table 1.