nebulae. The younger pulsars are fewer in number and could contribute significantly to the higher energy end of the $e^+e^-$ spectrum. However, the bulk of the synchrotron radiation in the WMAP bands comes from $e^+e^-$ with energies much less than 100 GeV, which justifies our focus on mature pulsars.

To demonstrate the feasibility of our assertion that pulsars could explain most of the visible WMAP haze, we follow the Cheng and Zhang 2001 model (CZ01 from here on) [@ChengZhang], which relies on the production of highly energetic radiation in the outer magnetosphere gap of a rapidly spinning pulsar [@Cheng:1986qt; @Chiang:1994si; @zhang97]. In the CZ01 model, the mean energy of $e^+ e^-$ injected into the inter-stellar medium ${\overline{E}}$ is set by its period, $P$, which increases with time. For a rotating magnetic dipole (in vacuum) this spin-down is given by $$\begin{aligned}
P(t) & = & P_0 \left( 1+\Delta \! t/\tau_0 \right)^{1/2}\,,
\label{eq:dipole}\\
\tau_0 & = & 1.35\times 10^4 {\rm yr} 
\left(\frac{P_0}{30 {\rm ms}}\right)^2 
\left(\frac{M}{1.4M_\odot}\right) 
\left(\frac{R}{15 {\rm km}}\right)^4 
\left(\frac{B}{10^{12} {\rm G}}\right)^{-2}\,, 
\label{eq:tau0}\end{aligned}$$ where $\Delta \! t$ is time since birth of the pulsar, $P_0$ is the initial period, $M$ is the mass of the pulsar, $R$ its radius and $B$ is the 