Bandpass mismatch error for satellite CMB experiments I: estimating the spurious signal

Future Cosmic Microwave Background (CMB) satellite missions aim to use the $B$ mode polarization to measure the tensor-to-scalar ratio $r$ with a sensitivity of about $10^{-3}$. Achieving this goal will not only require sufficient detector array sensitivity but also unprecedented control of all systematic errors inherent to CMB polarization measurements. Since polarization measurements derive from differences between observations at different times and from different sensors, detector response mismatches introduce leakages from intensity to polarization and thus lead to a spurious $B$ mode signal. Because the expected primordial $B$ mode polarization signal is dwarfed by the known unpolarized intensity signal, such leakages could contribute substantially to the final error budget for measuring $r.$ Using simulations we estimate the magnitude and angular spectrum of the spurious $B$ mode signal resulting from bandpass mismatch between different detectors. It is assumed here that the detectors are calibrated, for example using the CMB dipole, so that their sensitivity to the primordial CMB signal has been perfectly matched. Consequently the mismatch in the frequency bandpass shape between detectors introduces difference in the relative calibration of galactic emission components. We simulate using a range of scanning patterns being considered for future satellite missions. We find that the spurious contribution to $r$ from reionization bump on large angular scales ($\ell < 10$) is $\approx 10^{-3}$ assuming large detector arrays and 20 percent of the sky masked. We show how the amplitude of the leakage depends on the angular coverage per pixels that results from the scan pattern.