Systematic effects induced by Half Wave Plate precession into Cosmic
  Microwave Background polarization measurements
  The most accessible method to measure polarization features of the CMB
radiation is by means of a Stokes Polarimeter based on the rotation of an Half
Wave Plate. The current observational cosmology is starting to be limited by
the presence of systematic effects. The Stokes polarimeter with a rotating Half
Wave Plate (HWP) has the advantage of mitigating a long list of potential
systematics, by modulation of the linearly polarized component of the
radiation, but the presence of the rotating HWP can by itself introduce new
systematic effects, which must be under control, representing one of the most
critical part in the design of a B-Modes experiment. In this paper we present,
simulate and analyse the spurious signal arising from the precession of a
rotating HWP. We first find an analytical formula for the impact of the
systematic effect induced by the HWP precession on the propagating radiation,
using the 3D generalization of the Muller formalism. We then perform several
numerical simulations, showing the effect induced on the Stokes parameters by
this systematic. We also derive and discuss the impact into B-modes measured by
a satellite experiment. We find the analytical formula for the Stokes
parameters from a Stokes polarimeter where the HWP follows a precessional
motion with an angle $\theta_0$. We show the result depending on the HWP
inertia tensor, spinning speed and on $\theta_0$. The result of numerical
simulations is reported as a simple timeline of the electric fields. Finally,
assuming to observe all the sky with a satellite mission, we analyze the effect
on B-modes measurements. The effect is not negligible giving the current
B-modes experiments sensitivity, therefore it is a systematic which needs to be
carefully considered for future experiments.
