  High-energy standard model (SM) particles in the early Universe are generated
by the decay of heavy long-lived particles. The subsequent thermalization
occurs through the splitting of high-energy primary particles into lower-energy
daughters in primordial thermal plasma. The principal example of such processes
is reheating after inflation caused by the decay of inflatons into SM
particles. Understanding of the thermalization at reheating is extremely
important as it reveals the origin of the hot Universe, and could open up new
mechanisms for generating dark matter and/or baryon asymmetry. In this paper,
we investigate the thermalization of high-energy SM particles in thermal
plasma, taking into account the Landau--Pomeranchuk--Migdal effect in the
leading-log approximation. The whole SM particle content and all the relevant
SM interactions are included for the first time, i.e., the full gauge
interactions of SU(3)$_c\times$SU(2)$_L\times$U(1)$_Y$ and the top Yukawa
interaction. The distribution function of each SM species is computed both
numerically and analytically. We have analytically obtained the distribution
function of each SM species after the first few splittings. Furthermore, we
demonstrate that, after a sufficient number of splittings, the particle
distributions are asymptotic to certain values at low momentum, independent of
the high-energy particles injected by inflaton decay. The results are useful to
calculate the DM abundance produced during the pre-thermal phase. An example is
provided to illustrate a way to calculate the DM abundance from the scattering
between the thermal plasma and high-energy particles in the cascade.
