Signatures of kinetic and magnetic helicity in the cosmic microwave background radiation

$P$ and $\mathrm{CP}$ violation in cosmology can be manifested as large-scale helical velocity flows in the ambient plasma and as primordial helical magnetic fields. We show that kinetic helicity at last scattering leads to temperature-polarization correlations ${(C}_l^{\mathrm{TB}}$ and $C_l^{\mathrm{EB}})$ in the cosmic microwave background radiation (CMBR) and calculate the magnitude of the effect. Helical primordial magnetic fields, expected from cosmic events such as electroweak baryogenesis, can lead to helical velocity flows and hence to nonvanishing correlations of the temperature and B-type polarization. However, we show that the magnitude of the induced helical flow is unobservably small because the helical component of a magnetic field is almost force-free. We discuss an alternate scheme for extracting the helicity of a stochastically homogeneous and isotropic primordial magnetic field using observations of the CMBR. The scheme involves constructing Faraday rotation measure maps of the CMBR and thus determining the sum of the helical and nonhelical components of the primordial magnetic field. The power spectrum of B-type polarization fluctuations, on the other hand, are sensitive only to the nonhelical component of the primordial magnetic field. The primordial magnetic helicity can then be derived by combining these two sets of observations.