With the large sample of young gamma-ray pulsars discovered by the Fermi Large Area Telescope (LAT), population synthesis has become a powerful tool for comparing their collective properties with model predictions. We synthesised a pulsar population based on a radio emission model and four gamma-ray gap models (Polar Cap, Slot Gap, Outer Gap, and One Pole Caustic) normalizing to the number of detected radio pulsars in select group of surveys. The luminosity and the wide beams from the outer gaps can easily account for the number of Fermi detections in 2 years of observations. The wide slot-gap beams requires an increase by a factor of ~10 of the predicted luminosity to produce a reasonable number of gamma-ray pulsars. Such large increases in the luminosity may be accommodated by implementing offset polar caps. The narrow polar-cap beams contribute at most only a handful of LAT pulsars. Standard distributions in birth location and pulsar spin-down power (Edot) fail to reproduce the LAT findings: all models under-predict the number of LAT pulsars with high Edot, and they cannot explain the high probability of detecting both the radio and gamma-ray beams at high Edot. The beaming factor remains close to 1 over 4 decades in Edot evolution for the slot gap whereas it significantly decreases with increasing age for the outer gaps. The evolution of the slot-gap luminosity with Edot is compatible with the large dispersion of gamma-ray luminosity seen in the LAT data. The stronger evolution predicted for the outer gap, which is linked to the polar cap heating by the return current, is apparently not supported by the LAT data. The LAT sample of gamma-ray pulsars therefore provides a fresh perspective on the early evolution of the luminosity and beam width of the gamma-ray emission from young pulsars, calling for thin and more luminous gaps.
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