  Electron-phonon coupling in graphene is extensively modeled and simulated
from first principles. We find that using an accurate model for the
polarizations of the acoustic phonon modes is crucial to obtain correct
numerical results. The interactions between electrons and acoustic phonon
modes, the gauge field and deformation potential, are calculated at the DFT
level in the framework of linear response. The zero-momentum limit of acoustic
phonons is interpreted as a strain pattern, allowing the calculation of the
acoustic gauge field parameter in the GW approximation. The role of electronic
screening on the electron-phonon matrix elements is investigated. We then solve
the Boltzmann equation semi-analytically in graphene, including both acoustic
and optical phonon scattering. We show that, in the Bloch-Gr\"uneisen and
equipartition regimes, the electronic transport is mainly ruled by the
unscreened acoustic gauge field, while the contribution due to the deformation
potential is negligible and strongly screened. By comparing with experimental
data, we show that the contribution of acoustic phonons to resistivity is
doping- and substrate-independent. The DFT+GW approach underestimates this
contribution to resistivity by about 30 %. Above 270K, the calculated
resistivity underestimates the experimental one more severely, the
underestimation being larger at lower doping. We show that, beside remote
phonon scattering, a possible explanation for this disagreement is the
electron-electron interaction that strongly renormalizes the coupling to
intrinsic optical-phonon modes. Finally, after discussing the validity of the
Matthiessen rule in graphene, we derive simplified analytical solutions of the
Boltzmann equation to extract the coupling to acoustic phonons, related to the
strain-induced gauge field, directly from experimental data.
