distribution of pulsars in the inner galactic region [@Lorimer:2006qs]. We note that most of the pulsars in the distribution assumed in Eq. \[eq:rho\] are at $r \sim r_0$. This is consistent with one of the two models advocated to explain the observed pulsar distribution in the Parkes multibeam pulsar survey [@Lorimer:2006qs]. We do not consider a secondary source of pairs from the pulsars in the bulge given that the bulge is thought to be old and have a low star formation rate (compared to the disk).

The CZ01 Monte Carlo predicts that the energy spectrum of the $e^+
e^-$ pairs should be $E^{-1.6}$ above about a GeV up to tens of GeV. The spectrum drops sharply above ${E_{\rm cut}}\sim 100$ GeV. Both these features (the slope and cut-off energy) are model dependent and we discuss the effect of changing these later on. To keep the discussion more general, we therefore adopt an energy spectrum (number of $e^+
e^-$ pairs per unit time per unit energy) given by $$Q(E) = 
\dot{N}_{100}Q_0 f_e 
\left(\frac{E}{\rm GeV}\right)^{-\alpha}e^{-E/{E_{\rm cut}}}\,, 
\label{eq:source}$$ where our baseline model has $\alpha=1.6$ and $E_{\rm cut} = 100$ GeV, and we allow it to vary later to see how it changes our 