Crab pulsar: IXPE observations reveal unified polarization properties in the optical and soft X-ray bands

We present a phase-dependent analysis of the polarized emission from the Crab pulsar based on three sets of observations by the Imaging X-ray Polarimetry Explorer (IXPE). We found that a phenomenological model involving a simple linear transformation of the Stokes parameters adequately describes the IXPE data. This model enabled us to establish a connection between the polarization properties of the Crab pulsar in the optical and soft X-ray bands for the first time, which suggests a common underlying emission mechanism in these bands that likely is synchrotron radiation. In particular, the phase-dependent polarization degree in X-rays for the pure pulsar emission shows similar features, but is reduced by a factor approximate to(0.46 - 0.56) compared to the optical band (when we accounted for the contribution of the knot in the optical), which implies an energy-dependent polarized emission. Using this model, we also studied the polarization angle swing in the X-rays and identified a potentially variable phase shift at the interpulse relative to the optical band, alongside a phase shift that is marginally consistent with zero and persists at the main pulse. While the origin of this variability is unknown and presents a new challenge for the theoretical interpretation, our findings suggest that the emission mechanism for the main pulse is likely located far from the neutron star surface, perhaps near to or beyond the light cylinder, and that it does not operate in the inner magnetosphere, where vacuum birefringence is expected to be at work. Ignoring the phase shifts would result in identical phase-dependent polarization angles between the optical and X-ray bands for the pure pulsar emission.