  About 1/3 of X-ray-luminous clusters show smooth, Mpc-scale radio emission,
known as giant radio haloes. One promising model for radio haloes is Fermi-II
acceleration of seed relativistic electrons by compressible turbulence. The
origin of these seed electrons has never been fully explored. Here, we
integrate the Fokker-Planck equation of the cosmic ray (CR) electron and proton
distributions when post-processing cosmological simulations of cluster
formation, and confront them with radio surface brightness and spectral data of
Coma. For standard assumptions, structure formation shocks lead to a seed
electron population which produces too centrally concentrated radio emission.
Matching observations requires modifying properties of the CR population (rapid
streaming; enhanced CR electron acceleration at shocks) or turbulence
(increasing turbulent-to-thermal energy density with radius), but at the
expense of fine-tuning. In a parameter study, we find that radio properties are
exponentially sensitive to the amplitude of turbulence, which is inconsistent
with small scatter in scaling relations. This sensitivity is removed if we
relate the acceleration time to the turbulent dissipation time. In this case,
turbulence above a threshold value provides a fixed amount of amplification;
observations could thus potentially constrain the unknown CR seed population.
To obtain sufficient acceleration, the turbulent magneto-hydrodynamics cascade
has to terminate by transit time damping on CRs, i.e., thermal particles must
be scattered by plasma instabilities. Understanding the small scatter in radio
halo scaling relations may provide a rich source of insight on plasma processes
in clusters.
