Importance of resolving the spectral support of beam-plasma instabilities in simulations

Many astrophysical plasmas are prone to beam-plasma instabilities. For relativistic and dilute beams, the {\it spectral} support of the beam-plasma instabilities is narrow, i.e., the linearly unstable modes that grow with rates comparable to the maximum growth rate occupy a narrow range of wave numbers. This places stringent requirements on the box-sizes when simulating the evolution of the instabilities. We identify the implied lower limits on the box size imposed by the longitudinal beam plasma instability, i.e., typically the most stringent condition required to correctly capture the linear evolution of the instabilities in multidimensional simulations. We find that sizes many orders of magnitude larger than the resonant wavelength are typically required. Using one-dimensional particle-in-cell simulations, we show that the failure to sufficiently resolve the spectral support of the longitudinal instability yields slower growth and lower levels of saturation, potentially leading to erroneous physical conclusion.