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.Comment: 7 pages, 9 figures, accepted by Ap
