The Galactic population of canonical pulsars II
Pulsars are highly magnetized rotating neutron stars, emitting in a broad electromagnetic energy range. Reproducing the observed pulsars population refines our understanding of their formation and evolution scenarios as well as their radiation processes and geometry. In this paper, we improve our previous population synthesis by focusing on both the radio and $\gamma$-ray pulsar populations, investigating the impact of the Galactic gravitational potential and of the radio emission death line. In order to elucidate the necessity of a death line, refined initial distributions of spin period and spacial position at birth were implemented, elevating the sophistication of our simulations to the most recent state-of-the-art. The motion of each individual pulsar is tracked in the Galactic potential by a fourth order symplectic integration scheme. Our pulsar population synthesis takes into account the secular evolution of the force-free magnetosphere and magnetic field decay simultaneously and self-consistently. Each pulsar is evolved from its birth up to the present time. The radio and $\gamma$-ray emission locations are modelled respectively by the polar cap geometry and the striped wind model. By simulating ten million pulsars we found that including a death line better reproduces the observational trend. However, when simulating one million pulsars, we obtain an even more realistic $P-\dot{P}$ diagram, whether or not a death line is included. This suggests that the ages of the detected pulsars might be overestimated, therefore questioning the real need for a death line in pulsar population studies. Kolmogorov-Smirnov tests confirm the statistical similarity between the observed and simulated $P-\dot{P}$ diagram. Additionally, simulations with increased $\gamma$-ray telescope sensitivities hint to a significant contribution of $\gamma$-ray pulsars to the GeV excess in the Galactic centre.