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.
