Tangled magnetic fields and CMBR signal from reionization epoch

We compute the secondary cosmic microwave background radiation (CMBR) anisotropy signal from the reionization of the Universe in the presence of tangled magnetic fields. We consider the tangled-magnetic-field-induced scalar, vector, and tensor modes for our analysis. The most interesting signal for $\ell \lesssim 100$ arises from tensor perturbations. In particular, we show that the enhancement observed by Wilkinson microwave anisotropy probe (WMAP) in the TE cross-correlation signal for $\ell \lesssim 10$ could be explained by tensor TE cross correlation from tangled magnetic fields generated during the inflationary epoch for magnetic field strength $B_0\simeq 4.5\times {10}^{-9}\mathrm{G}$ and magnetic field power spectrum spectral index $n\simeq -2.9$. Alternatively, a mixture of tensor mode signal with primordial scalar modes gives weaker bounds on the value of the optical depth to the reionization surface, ${\tau }_{\mathrm{reion}}$: ${\tau }_{\mathrm{reion}}=0.11\pm 0.02$. This analysis can also be translated to a limit on magnetic field strength of $\simeq 5\times {10}^{-9}\mathrm{G}$ for wave numbers $\lesssim 0.05\mathrm{Mp}{\mathrm{c}}^{-1}$.

Figure 1

The secondary temperature angular power spectrum from vector modes is shown. The solid and the dashed lines correspond to the contribution from vorticity and the total signal, respectively (see text for details). The power spectrum is plotted for $B_0=3\times {10}^{-9}$ and $n=-2.9$ (Eq. (2)).

Figure 2

The secondary polarization angular power spectrum from vector modes is shown. The solid and the dot-dashed lines correspond, respectively, to the $B$- and $E$-mode contribution from the free-streaming quadrupole (Eq. (15)). The dotted and dashed curves $B$- and $E$-mode signals that arise from the source term given by Eq. (16). The power spectra are plotted for $B_0=3\times {10}^{-9}$ and $n=-2.9$ (Eq. (2)).

Figure 3

The contribution of tensor modes to the temperature power spectrum is shown. The solid and dashed lines give, respectively, the power spectra without and with reionization. The thick solid line, shown here for comparison, corresponds to the temperature power spectrum from scalar modes, for the best-fit parameters from WMAP ([30]). The power spectra are plotted for $B_0=3\times {10}^{-9}$, $n=-2.9$ (Eq. (2)), and ${\eta }_{\star }/{\eta }_{\mathrm{in}}={10}^{18}$ (Eq. (b10)).

Figure 4

The tensor secondary and primary polarization power spectra are shown along with the expected signals from primordial scalar and tensor modes. The dot-dot-dot-dashed and thin dotted lines correspond to the secondary $E$- and $B$-mode power spectra, respectively. The dot-dashed and dashed lines give the primary $E$- and $B$-mode power spectra. The two top solid lines, shown here for comparison, correspond to the $E$- and $B$-mode power spectra from primordial scalar modes, for the best-fit parameters from WMAP ([30]). For $B$-mode signal we assume the ratio of tensor to scalar quadrupole $T/S=0.7$ and the tensor spectral index $n_t=0$. The bottom solid lines show the $B$-mode signal expected from gravitational lensing. The thick dashed line shows the $1\mathrm{\text{−}}\sigma$ errors expected from the future CMBR experiment Planck surveyor for one year of integration (Eq. (26)). The power spectra are plotted for $B_0=3\times {10}^{-9}$, $n=-2.9$ (Eq. (2)), and ${\eta }_{\star }/{\eta }_{\mathrm{in}}={10}^{18}$ (Eq. (b10)).

Figure 5

The tensor TE cross correlation power spectra are shown along with the expected signal from primordial scalar modes. The dot-dashed line shows the TE cross correlation (secondary plus primary) from tangled magnetic fields. The thick solid line, shown here for comparison, corresponds to the (absolute value of) TE cross correlation power spectrum from primordial scalar modes, for the best-fit parameters from WMAP ([30]). The power spectrum is plotted for $B_0=3\times {10}^{-9}$, $n=-2.9$ (Eq. (2)), and ${\eta }_{\star }/{\eta }_{\mathrm{in}}={10}^{18}$ (Eq. (b10)).

Figure 6

The secondary temperature power spectrum from scalar mode perturbations, seeded by tangled magnetic fields, is shown. The power spectrum is plotted for $B_0=3\times {10}^{-9}$ and $n=-2.9$ (Eq. (2)).