In collisionless space and astrophysical plasmas, shock waves play an important role in particle acceleration and heating and involve a rich physics at many scales. While many theoretical and observational efforts have been devoted to the study of ions behavior at shocks, relatively less attention has been paid to electrons. The evolution of the electron distribution function through quasi-perpendicular collisionless shocks is believed to be dominated by the electron dynamics in the large-scale coherent and quasi-stationary (DC) magnetic and electric fields.

Here we investigate the electron distributions measured onboard Cluster by the PEACE instrument at the Earth's bow shock. Features of the observed distributions are compared with those predicted by conservation of phase-space density along particle trajectories, for the first time for all pitch-angles and all types of electron trajectories (passing, reflected or trapped). The agreement is generally found to be good, confirming the validity of electron "heating" theory based on DC-fields as a zeroth order approximation, although some deviations are pointed out for larger shock angle and lower plasma beta. This approach also provides a way to estimate the cross-shock electric potential profile across the shocks making full use of the electron measurements, and the results are compared to other estimates relying on the steady-state dissipationless electron fluid equations. Finally, we show how, in contrast to methods using electron velocity-moments, the technique can be used to produce higher-time resolution electric potentials.