Neutron stars and millisecond pulsars in star clusters: implications for the diffuse $\gamma$-radiation from the Galactic Centre

Globular clusters (GCs) are the ideal environment for the formation of neutron stars (NSs) and millisecond pulsars (MSPs). NSs origin and evolution provide a useful information on stellar dynamics and evolution in star clusters, and are among the most interesting astrophysical objects, being precursors of several high-energy phenomena such as gravitational waves and gamma-ray bursts. Due to a large velocity kick that they receive at birth, most of the NSs escape the local field, affecting the evolution and dynamics of their parent cluster. In this paper, we study the origin and dynamical evolution of NSs within GCs with different initial masses, metallicities and primordial binary fractions. We find that the radial profile of NSs is shaped by the BH content of the cluster, which partially quenches the NS segregation until most of the BHs are ejected from the system. Independently on the cluster mass and initial configuration, the NSs map the average stellar population, as their average radial distance is $\approx 60-80\%$ of the cluster half-mass radius. Finally, by assuming a recycling fraction of $f_\mathrm{rec}=0.1$ and an average MSP gamma-ray emission of $L_\gamma=2\times 10^{33}$ erg s$^{-1}$, we show that the typical gamma-ray emission from our GCs agrees with observations and supports the MSP origin of the gamma-ray excess signal observed by the Fermi-LAT telescope in the Galactic Centre.