Implementing and Verifying a Fourier Domain Approach to Fast Stochastic X-ray Polarimetry Timing

The launch of the Imaging X-ray Polarimetry Explorer (IXPE), the first space-based polarimeter since 1978, offers a two order of magnitude improvement to the measurement of X-ray polarisation than its predecessor OSO-8, offering unprecedented precision for the measurement of polarisation degree and polarisation angle of X-ray sources. This advancement lends itself to the birth of a number of contemporary techniques to study Galactic compact objects, including X-ray polarimetry-timing, the study of how polarisation properties evolve over short timescales. However, the statistical nature of polarisation measurements poses a challenge for studies on arbitrarily short timescales, as a large number of photons are required to achieve statistically significant measurements of polarisation degree and angle for time-resolved analyses. Furthermore, if the polarisation variability is stochastic, then phase-folding techniques introduce systematic errors in the phase assignment of photons. Ingram and Maccarone presented a model independent Fourier-based technique that circumvents these issues. It can be used on arbitrarily short timescales for any kind of variability, whether aperiodic, quasi-periodic or purely periodic. Here we implement this method on real IXPE data. We address several instrumental effects and test the technique on X-ray pulsars, RX-J0440.9+4431 and Hercules X-1 . We verify that our technique recovers the polarisation variability signal that we already know to be there from typical phase-folding techniques. It will now be possible to study fast stochastic polarisation variability of X-ray sources, with applications including quasi-periodic oscillations, mass accretion rate fluctuations, and reverberation mapping.