and cut-off 100 GeV, we have $W_0=\dot{N}_{100}(1/2)I\Omega_0^2/100 {\rm yr}$, which works out to $Q_0=2\times 10^{41}$ GeV$^{-1}$ s$^{-1}$ if we take (conservatively) $(1/2)I\Omega_0^2 = 10^{52} {\rm GeV}$. The efficiency required then to get the same normalization as the CZ01 model is a factor of 4 less.

The origin of the $E^{-1.6}$ spectrum in the CZ01 model are the scalings of ${\overline{E}}$ and the spin down power with period $P$. We note that $dn/dE \propto T /P^4/E^2 \propto 1/P^2/E^2$ where we have used the fact that the number of pulsars is proportional to the age and we have used the approximation $P^2 \propto T$. We include the ${\overline{E}}$ dependence on $P$ to obtain $dn/dE \propto E^{2/3.6-2}
\propto E^{-1.4}$, somewhat different from the $E^{-1.6}$ scaling because of the approximations we have made.

The cut-off in the spectrum around 100 GeV is related to our assumption that the pulsars have to be approximately 100 kyr or older to contribute significantly to the haze, and that the mean energy of pairs injected into the ISM depends on the pulsar period in the CZ01 model (see Eq. \[eq:Emean\]). This estimate of the cut-off is uncertain both because of our blanket assumption that pulsars younger than 100 