Primordial magnetic helicity constraints from WMAP nine-year data

If a primordial magnetic field in the universe has non-zero helicity, the violation of parity symmetry results in non-zero correlations between cosmic microwave background temperature and B-mode polarization. In this paper we derive approximations to the relevant microwave background power spectra arising from a helical magnetic field. Using the cross-power spectrum between temperature and B-mode polarization from the WMAP nine-year data, we set a 95\% confidence level upper limit on the helicity amplitude to be 10 nG$^2$ Gpc for helicity spectral index $n_H = -1.9$, for a cosmological magnetic field with effective field strength of 3 nG and a power-law index $n_B = -2.9$ near the scale-invariant value. Future microwave background polarization maps with greater sensitivity will be able to detect the helicity of an inflationary magnetic field well below the maximum value allowed by microwave background constraints on the magnetic field amplitude.