Context. More than one hundred pulsars have been detected up to now at GeV energies by the Large Area Telescope (LAT) on the Fermi gamma-ray observatory. Current modelling proposes that the high-energy emission comes from outer magnetospheric gaps, but radiation from the equatorial current sheet that separates the two magnetic hemispheres outside the light cylinder has also been investigated. Aims. We discuss the region outside the light cylinder, the “near wind” zone. We investigate the possibility that synchrotron radiation emitted by thermal populations in the equatorial current sheet of the pulsar wind in this region can explain the lightcurves and spectra observed by Fermi/LAT. Methods. We used analytical estimates as well as detailed numerical computation to calculate the γ-ray luminosities, lightcurves, and spectra of γ-ray pulsars. Results. Many of the characteristics of the γ-ray pulsars observed by Fermi/LAT can be reproduced by our model, most notably the position of these objects in the P???? diagram, and the range of γ-ray luminosities. A testable result is a sub-exponential cutoff with an index b?=?0.35. We also predict the existence of a population of pulsars with cutoff energies in the MeV range. These have systematically lower spindown luminosities than the Fermi/LAT-detected pulsars. Conclusions. It is possible for relativistic populations of electrons and positrons in the current sheet of a pulsar’s wind immediately outside the light cylinder to emit synchrotron radiation that peaks in the sub-GeV to GeV regime, with γ-ray efficiencies similar to those observed for the Fermi/LAT pulsars.
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