Impact of beam far side-lobe knowledge in the presence of foregrounds for LiteBIRD

We present a study of the impact of a beam far side-lobe lack of knowledge on the measurement of the Cosmic Microwave Background B-mode signal at large scale. Beam far side-lobes induce a mismatch in the transfer function of Galactic foregrounds between the dipole and higher multipoles which degrads the performances of component separation methods. This leads to foreground residuals in the CMB map. It is expected to be one of the main source of systematic effects in future CMB polarization observations. Thus, it becomes crucial for all-sky survey missions to take into account the interplays between beam systematic effects and all the data analysis steps. LiteBIRD is the ISAS/JAXA second strategic large-class satellite mission and is dedicated to target the measurement of CMB primordial B modes by reaching a sensitivity on the tensor-to-scalar ratio r of σ (r) ≤ 10-3 assuming r = 0. The primary goal of this paper is to provide the methodology and develop the framework to carry out the end-to-end study of beam far side-lobe effects for a space-borne CMB experiment. We introduce uncertainties in the beam model, and propagate the beam effects through all the steps of the analysis pipeline, most importantly including component separation, up to the cosmological results in the form of a bias δr. As a demonstration of our framework, we derive requirements on the calibration and modeling for the LiteBIRD's beams under given assumptions on design, simulation, component separation method and allocated error budget. In particular, we assume a parametric method of component separation with no mitigation of the far side-lobes effect at any stage of the analysis pipeline. We show that δr is mostly due to the integrated fractional power difference between the estimated beams and the true beams in the far side-lobes region, with little dependence on the actual shape of the beams, for low enough δr. Under our set of assumptions, in particular considering the specific foreground cleaning method we used, we find that the integrated fractional power in the far side-lobes should be known at the level of ∼ 10-4, to achieve the required limit on the bias δr < 1.9 × 10-5. The framework and tools developed for this study can be easily adapted to provide requirements under different design, data analysis frameworks and for other future space-borne experiments, such as PICO or CMB-Bharat. We further discuss the limitations of this framework and potential extensions to circumvent them.