Primordial magnetic helicity constraints from WMAP nine-year data

If a primordial magnetic field in the Universe has nonzero helicity, the violation of parity symmetry results in nonzero 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 $10{\mathrm{nG}}^2\mathrm{Gpc}$ 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.

Figure 1

A comparison between the temperature-B-polarization cross correlation model for $B_{\text{eff}}=1\mathrm{nG}$, $n_B=-2.99$; the solid red line is for a helicity amplitude of ${\mathcal{H}}_B={10}^5{\mathrm{nG}}^2\mathrm{Mpc}$ and helicity spectral index $n_H=-1.9$ while the dotted blue line is for ${\mathcal{H}}_{\text{B}}={10}^8{\mathrm{nG}}^2\mathrm{Mpc}$ and $n_H=-0.6$. Also shown are the nine-year WMAP data (solid gray dots with bars indicating uncertainties).

Figure 2

The 95% upper limits on ${\mathcal{H}}_B$ as a function of $n_H$ for $n_B=-2.99$ (black solid), $n_B=-2.0$ (red short dash), and $n_B=n_H-1$ (blue long dash). For all three cases, $B_{\text{eff}}=1\mathrm{nG}$.

Figure 3

The 95% upper limits on ${\mathcal{H}}_{\lambda }$ for a smoothing scale $\lambda =1\mathrm{Mpc}$, as a function of $n_H$ for $n_B=-2.99$ (black solid), $n_B=-2.0$ (red short dash), and $n_B=n_H-1$ (blue long dash). For all three cases, $B_{\text{eff}}=1\mathrm{nG}$.