  Recent ab-initio studies of electron transport in GaAs have reported that
electron-phonon (e-ph) interactions beyond the lowest order play a fundamental
role in charge transport and noise phenomena. Inclusion of the
next-leading-order process in which an electron scatters with two phonons was
found to yield good agreement for the high-field drift velocity, but the
characteristic non-monotonic trend of the power spectral density of current
fluctuations (PSD) with electric field was not predicted. The high
computational cost of the ab-initio approach necessitated various
approximations to the two-phonon scattering term, which were suggested as
possible origins of the discrepancy. Here, we report a semi-analytical
transport model of two-phonon electron scattering via the Fr\"ohlich mechanism,
allowing a number of the approximations in the ab-initio treatment to be lifted
while retaining the accuracy to within a few percent. We compare the calculated
and experimental transport and noise properties as well as scattering rates
measured by photoluminescence experiments. We find quantitative agreement
within 15% for the drift velocity and 25% for the $\Gamma$ valley scattering
rates, and agreement with the $\Gamma-L$ intervalley scattering rates within a
factor of two. Considering these results and prior studies of current noise in
GaAs, we conclude that the most probable origin of the non-monotonic PSD trend
versus electric field is the formation of space charge domains rather than
intervalley scattering as has been assumed.
