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The death of massive stars leaves behind exotic compact objects that are neutron stars and black holes. Among the neutron stars, especially
interesting are the Anomalous X-ray Pulsars and Soft Gamma-Ray Repeaters (SGRs), which are young neutron stars characterized by high X-ray
quiescent luminosities, outbursts, and in the case of SGRs, sporadic giant flares. They are believed to be magnetars, that is neutron
stars powered by ultra-strong magnetic fields. However, the diversity of their behaviours, and especially the observation of magnetar-like
bursts from 'low-field' neutron stars, has been a theoretical puzzle.
In the first part of the talk, I will discuss results of long-term MHD simulations which by following the evolution of magnetic stresses
within the neutron star crust have allowed us to relate the observed magnetar phenomenology to the physical properties of the neutron
stars, and in particular to their age and magnetic field strength and topology. The dichotomy of 'high-B' field pulsars versus magnetars is
naturally explained, and occasional outbursts from old, low B-field neutron stars are predicted.
In the second part of the talk, I will discuss how observations of highly magnetized neutron stars can be handy tools in the cosmological domain, and in particular as a way to set constraints on the hypothetical particle 'axion'.