kyr do not contribute pairs, and also because in framing the arguments above we have assumed all pulsars are born with spin period of 30 ms. We certainly expect scatter about both these parameters. Including such scatter will change the details of the cut-off significantly but not the main result of the paper. In addition, a small change in the strong dependence of the mean energy on the period would affect the cut-off significantly. This steep dependence arises from processes that accelerate the pairs into the ISM and these processes are not well-understood. We note that the energy spectrum at lower energies ($\sim 10 {\rm GeV}$) is much less sensitive to the above uncertainties.

At even lower energies, $E < 1$ GeV, pair emission luminosity is influenced by the modeling of a gamma-ray pulsar. This requirement, discussed above, imposes a magnetic field dependent upper limit on the period, which translates to a cut-off in the luminosity at low energies. These model assumptions therefore introduce uncertainties in the GeV range pair emission luminosity, which must be kept in mind when comparing to experiments like HEAT.

The estimates in this section assuming a vacuum dipole rotator model for the mature pulsars have 