Implications of primordial power spectra with statistical anisotropy on CMB temperature fluctuation and polarizations

Both the Wilkinson Microwave Anisotropy Probe (WMAP) and Planck observations reported the hemispherical asymmetry of the cosmic microwave background (CMB) temperature fluctuation. The hemispherical asymmetry might be stemmed from the primordial statistical anisotropy during the inflationary era of the universe. In this paper, we study possible implications of the primordial power spectra with dipolar anisotropy on the CMB temperature fluctuation and polarizations. We explicitly show that the statistical dipolar anisotropy may induce the off-diagonal ($\ell'\neq\ell$) $TT$, $EE$, $BB$, and $TE$ correlations, as well as the diagonal ($\ell'=\ell$) $TB$ and $EB$ spectra. In particular, these correlation coefficients are expected to be $m$-dependent generically. These signals of statistical anisotropy might be tested by CMB observations in future.