Carrier scattering processes are studied in CH$_3$NH$_3$PbI$_3$ using temperature-dependent four-wave mixing experiments. Our results indicate that scattering by ionized impurities limits the interband dephasing time (T$_2$) below 30~K, with strong electron-phonon scattering dominating at higher temperatures (with a timescale of 125 fs at 100 K). Our theoretical simulations provide quantitative agreement with the measured carrier scattering rate and show that the rate of acoustic phonon scattering is enhanced by strong spin-orbit coupling, which modifies the band-edge density of states. The Rashba coefficient extracted from fitting the experimental results ($gamma_c=2$ eV angstrom) is in agreement with calculations of the surface Rashba effect and recent experiments using the photogalvanic effect on thin films.
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