An accurate description of noise levels is of crucial importance for the correct simulation of instability-driven processes, such as the density modulation of a long proton bunch traversing a plasma.

To insure that the correct instability develops, a seed field must be larger than the cumulative shot noise. We develop an analytical theory of the noise field and compare it with multidimensional simulations. We find that the natural noise wakefield generated in a plasma by the CERN Super Proton Synchrotron bunches is very low, at the level of 10 kV / m. This fortunate fact eases the requirements on the seed. Our three-dimensional simulations show that even a few tens MeV electron bunch precursor of a very moderate intensity is sufficient to seed the proton bunch self-modulation in plasma.