  The relaxation of high-energy "hot" carriers in semiconductors is known to
involve the redistribution of energy between (i) hot and cold carriers and (ii)
hot carriers and phonons. Over the past few years, these two processes have
been identified in lead-halide perovskites (LHPs) using ultrafast pump-probe
experiments, but the interplay between these processes is not fully understood.
Here we present a comprehensive kinetic model to elucidate the individual
effects of the hot and cold carriers in bulk and nanocrystal $CsPbBr_{3}$ films
obtained from "pump-push-probe" measurements. In accordance with our previous
work, we observe that the cooling dynamics in the materials decelerate as the
number of hot carriers increases, which we explain through a "hot-phonon
bottleneck" mechanism. On the other hand, as the number of cold carriers
increases, we observe an acceleration of the cooling kinetics in the samples.
We describe the interplay of these opposing effects using our model, and by
using series of natural approximations, reduce this model to a simple form
containing terms for the carrier-carrier and carrier-phonon interactions. The
model can be instrumental for evaluating the details of carrier cooling and
electron-phonon couplings in a broad range of LHP optoelectronic materials.
