Title: Impact of half-wave plate systematics on the measurement of CMB $B$-mode polarization

Abstract: Polarization of the cosmic microwave background (CMB) can help probe the fundamental physics behind cosmic inflation via the measurement of primordial $B$ modes. As this requires exquisite control over instrumental systematics, some next-generation CMB experiments plan to use a rotating half-wave plate (HWP) as polarization modulator. However, the HWP non-idealities, if not properly treated in the analysis, can result in additional systematics. In this paper, we present a simple, semi-analytical end-to-end model to propagate the HWP non-idealities through the macro-steps that make up any CMB experiment (observation of multi-frequency maps, foreground cleaning, and power spectra estimation) and compute the HWP-induced bias on the estimated tensor-to-scalar ratio, $r$. We find that the effective polarization efficiency of the HWP suppresses the polarization signal, leading to an underestimation of $r$. Laboratory measurements of the properties of the HWP can be used to calibrate this effect, but we show how gain calibration of the CMB temperature can also be used to partially mitigate it. On the basis of our findings, we present a set of recommendations for the HWP design that can help maximize the benefits of gain calibration.