results. We have separated out a factor $f_e$, which is the efficiency of converting the spin-down power of the pulsar into $e^+ e^-$ pairs (after an age of $10^5 ~{\rm yr}$), and the factor $\dot{N}_{100}$, which is the number of pulsars created every century.

The normalization $Q_0$ is fixed by the spin down power of all the pulsars, that is $W_0 = N_p \langle I \Omega^2 \dot{P}/P \rangle$ where $N_p$ is the total number of pulsars and the brackets indicate averaging over the galactic population. We set $N_p = 0.01
\dot{N}_{100} (T/{\rm yr}-10^5) \simeq 1000 \dot{N}_{100} T/10^5{\rm
  yr}$, where $T$ is the typical age of the pulsar contributing the $e^+ e^-$ pairs. The normalization condition for $Q_0$ is given by $$\begin{aligned}
\int \dot{N}_{100} Q_0 E^{-\alpha} e^{-E/{E_{\rm cut}}} 
E dE & = & W_0 = N_p\times 6 \times 10^{38} \frac{\rm GeV}{\rm s}
\langle \left(\frac{P}{0.1 {\rm s}}\right)^{-4}
\left(\frac{B}{10^{12}{\rm G}}\right)^2\rangle\,,\\ 
\Rightarrow Q_0 & \approx & 
5 \times 10^{40} {\rm GeV}^{-1}{\rm s}^{-1}
\frac{100^{\alpha-1.6}\Gamma(0.4)}{\Gamma(2-\alpha)}
\left(\frac{100 {\rm GeV}}{{E_{\rm cut}}}\right)^{2-\alpha} 
\,, \label{eq:Q0def}\end{aligned}$$ where $E$ is the electron or positron energy in GeV, and we have assumed median values for the pulsar mass of $1.4M_\odot$, radius of 15 km, initial period of 20 ms, and surface magnetic field 