A Fermi-LAT Study of Globular Cluster Dynamical Evolution in the Milky Way: Millisecond Pulsars as the Probe

Using archival {\it Fermi}-LAT data with a time span of $\sim12$ years, we study the population of Millisecond Pulsars (MSPs) in Globular Clusters (GlCs) and investigate their dependence on cluster dynamical evolution in the Milky Way Galaxy. We show that the $\gamma$-ray luminosity ($L_{\gamma}$) and emissivity ($\epsilon_{\gamma}=L_{\gamma}/M$) are good indicators of the population and abundance of MSPs in GlCs, and they are highly dependent on the dynamical evolution history of the host clusters. Specifically speaking, the dynamically older GlCs with more compact structures are more likely to have larger $L_{\gamma}$ and $\epsilon_{\gamma}$, and these trends can be summarized as strong correlations with cluster stellar encounter rate $\Gamma$ and the specific encounter rate ($\Lambda=\Gamma/M$), with $L_{\gamma}\propto \Gamma^{0.70\pm0.11}$ and $\epsilon_{\gamma}\propto \Lambda^{0.73\pm0.13}$ for dynamically normal GlCs. However, as GlCs evolve into deep core collapse, these trends are found to be reversed, implying that strong encounters may have lead to the ejection of MSPs from core-collapsed Systems. Besides, the GlCs are found to exhibit larger $\epsilon_{\gamma}$ with increasing stellar mass function slope, decreasing tidal radius and distances from the Galactic Center (GC). These correlations indicate that, as GlCs losing kinetic energy and spiral in towards GC, tidal stripping and mass segregation have a preference in leading to the loss of normal stars from GlCs, while MSPs are more likely to concentrate to cluster center and be deposited into the GC. Moreover, we gauge $\epsilon_{\gamma}$ of GlCs is $\sim10-1000$ times larger than the Galactic bulge, the latter is thought to reside thousands of unresolved MSPs and may responsible for the GC $\gamma$-ray excess, which support that GlCs are generous contributors to the population of MSPs in the GC.