A possible evolutionary connection between high-mass X-ray binary pulsars and isolated rotation-powered pulsars

We investigate the long-term spin evolution of high-mass X-ray binary (HMXB) pulsars in the Small Magellanic Cloud (SMC) using archival measurements of spin periods and period derivatives from \textit{XMM-Newton}, \textit{Chandra}, and \textit{RXTE}. In the $P$--$|\dot P|$ plane, the HMXB pulsars follow an approximately linear relation in log--log space with slope \rev{$\hmxbslope\pm\hmxbslopeerr$}, \rev{similar to the trend shown by the youngest} isolated rotation-powered pulsars \rev{(RPPs)} \citep[e.g.,][]{Manchester2005,Lorimer2005}. \rev{For comparison, young RPPs from the ATNF Pulsar Catalogue give slopes that depend strongly on the characteristic-age cut ($\rppAslope\pm\rppAslopeerr$ for $\tau_c<10^4$\,yr and $\rppslope\pm\rppslopeerr$ for $\tau_c<10^5$\,yr), whereas the full non-recycled population shows no significant correlation. The HMXB pulsars lie in the same range of $\tau_c$ ($\sim1$--$10$\,kyr) as the youngest RPPs, but at spin periods two to three orders of magnitude longer.} Reference characteristic-age contours in the HMXB $P$--$|\dot P|$ plane are roughly parallel to this relation, consistent with a continuous evolutionary sequence. \rev{If the measured long-term spin-up rates were sustained, slowly rotating accreting pulsars would reach spin periods typical of isolated radio pulsars within only $\sim10^{2}$--$10^{4}$\,yr; for example, SXP\,101 ($P\approx101$\,s, $|\dot P|\approx1.9\times10^{-9}$\,s\,s$^{-1}$) would reach $P\sim1$\,s after $\sim2\times10^{3}$\,yr. Because these timescales are far shorter than the lifetimes of the Be donors, the spin at the end of the accretion phase is more likely set by torque equilibrium than by uninterrupted spin-up.} We suggest that a subset of isolated neutron stars may descend from accretion-powered HMXB pulsars following binary disruption\rev{, most plausibly at the supernova of the donor star} \citep[e.g.,][]{Bhattacharya1991,TaurisTakens1998}, providing an observational link between accretion-powered and rotation-powered neutron-star populations.