  The Galactic Center Excess (GCE) is an extended gamma-ray source in the
central region of the Galaxy found in Fermi Large Area Telescope data. In
recent years it has become apparent that the GCE may not be spherically
symmetric, but may be spatially correlated with the distribution of stellar
mass in the Galactic bulge, potentially favoring an unresolved population of
millisecond pulsars (MSPs) scenario. In this thesis, we perform detailed
modelling of the Galactic MSP population. Including in our model the spin down
between formation and observation, we find a model in which luminosity $L
\propto E_{\rm cut}^{1.2 \pm 0.3} B^{0.1 \pm 0.4} \dot{E}^{0.5 \pm 0.1}$
provides the best fit to the data, where $E_{\rm cut}$ is spectral energy
cutoff, $B$ is magnetic field strength, and $\dot{E}$ is the spin-down power.
Due to differing star formation histories it is expected that the MSPs in the
Galactic bulge are older and therefore dimmer than those in the Galactic disk.
Our results demonstrate that we do not require that there is anything
systematically different about the inner Galaxy MSPs to explain the GCE. In the
"recycling" channel of MSP formation the neutron star forms from a core
collapse supernovae that undergoes a random "kick" due to the asymmetry of the
explosion. This would imply a smoothing out of the spatial distribution of the
MSPs. We use N-body simulations to model how the MSP spatial distribution
changes. We estimate the probability distribution of natal kick velocities
using the resolved gamma-ray MSP proper motions, where MSPs have velocities
relative to circular motion of 77 +/- 6 km/s, as determined as part of our
Galactic MSP population model. We find that, due to the natal kicks, there is
an approximately 10% increase in each of the bulge MSP spatial distribution
dimensions and also the bulge MSP distribution becomes less boxy but is still
far from being spherical.
