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We present a study of the impact of an uncertainty in the beam far side-lobeknowledge on the measurement of the Cosmic Microwave Background $B$-mode signalat large scale. It is expected to be one of the main source of systematiceffects in future CMB observations. Because it is crucial for all-sky surveymissions to take into account the interplays between beam systematic effectsand all the data analysis steps, the primary goal of this paper is to providethe methodology to carry out the end-to-end study of their effect for aspace-borne CMB polarization experiment, up to the cosmological results in theform of a bias $\delta r$ on the tensor-to-scalar ratio $r$. LiteBIRD isdedicated to target the measurement of CMB primordial $B$ modes by reaching asensitivity of $\sigma \left( r \right) \leq 10^{-3}$ assuming $r=0$. As ademonstration of our framework, we derive the relationship between theknowledge of the beam far side-lobes and the tentatively allocated error budgetunder given assumptions on design, simulation and component separation method.We assume no mitigation of the far side-lobes effect at any stage of theanalysis pipeline. We show that $\delta r$ is mostly due to the integratedfractional power difference between the estimated beams and the true beams inthe far side-lobes region, with little dependence on the actual shape of thebeams, for low enough $\delta r$. Under our set of assumptions, in particularconsidering the specific foreground cleaning method we used, we find that theintegrated fractional power in the far side-lobes should be known at a level astight as $\sim 10^{-4}$, to achieve the required limit on the bias $\delta r <1.9 \times 10^{-5}$. The framework and tools developed for this study can beeasily adapted to provide requirements under different design, data analysisframeworks and for other future space-borne experiments beyond LiteBIRD.

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