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Magnetically-induced CMB $TT$ (\emph{top left}), $TE$ (\emph{top right}), $EE$ (\emph{bottom left}), and $BB$ (\emph{bottom right}) power spectra. The solid lines represent primary CMB anisotropies, the dotted lines represent magnetically-induced compensated scalar modes (except for the $BB$ panel, where it represents the lensing contributions and the solid line represents primary tensor modes with a tensor-to-scalar ratio of $r=0.1$), the dashed lines represent vector modes, whereas the dot-dashed lines represent magnetically-induced compensated tensor modes. We consider PMFs with $B_{1\,\mathrm{Mpc}}=4.1$\,nG and $n_B=-1$.

Magnetically-induced CMB $TT$ (\emph{top left}), $TE$ (\emph{top right}), $EE$ (\emph{bottom left}), and $BB$ (\emph{bottom right}) power spectra. The solid lines represent primary CMB anisotropies, the dotted lines represent magnetically-induced compensated scalar modes (except for the $BB$ panel, where it represents the lensing contributions and the solid line represents primary tensor modes with a tensor-to-scalar ratio of $r=0.1$), the dashed lines represent vector modes, whereas the dot-dashed lines represent magnetically-induced compensated tensor modes. We consider PMFs with $B_{1\,\mathrm{Mpc}}=4.5$\,nG and $n_B=-1$.

Magnetically-induced CMB $TT$ (\emph{top left}), $TE$ (\emph{top right}), $EE$ (\emph{bottom left}), and $BB$ (\emph{bottom right}) power spectra. The solid lines represent primary CMB anisotropies, the dotted lines represent magnetically-induced compensated scalar modes (except for the $BB$ panel, where it represents the lensing contributions and the solid line represents primary tensor modes with a tensor-to-scalar ratio of $r=0.1$), the dashed lines represent vector modes, whereas the dot-dashed lines represent magnetically-induced compensated tensor modes. We consider PMFs with $B_{1\,\mathrm{Mpc}}=4.1$\,nG and $n_B=-1$.

Magnetically-induced CMB $TT$ (\emph{top left}), $TE$ (\emph{top right}), $EE$ (\emph{bottom left}), and $BB$ (\emph{bottom right}) power spectra. The solid lines represent primary CMB anisotropies, the dotted lines represent magnetically-induced compensated scalar modes (except for the $BB$ panel, where it represents the lensing contributions and the solid line represents primary tensor modes with a tensor-to-scalar ratio of $r=0.1$), the dashed lines represent vector modes, whereas the dot-dashed lines represent magnetically-induced compensated tensor modes. We consider PMFs with $B_{1\,\mathrm{Mpc}}=4.5$\,nG and $n_B=-1$.

Dependence of the magnetically-induced CMB power spectrum on the spectral index. For all plotted cases, the amplitude is $B_{1\,\mathrm{Mpc}}=4.5$\,nG. The black lines show primary CMB anisotropies; for the other colours we refer to the legend. \emph{Left}: scalar contributions, \emph{right}: vector contributions.

Dependence of the magnetically-induced CMB power spectrum on the spectral index. For all plotted cases, the amplitude is $B_{1\,\mathrm{Mpc}}=4.1$\,nG. The black lines show primary CMB anisotropies; for the other colours we refer to the legend. \emph{Left}: scalar contributions, \emph{right}: vector contributions.

Magnetically-induced CMB $TT$ (\emph{top left}), $TE$ (\emph{top right}), $EE$ (\emph{bottom left}), and $BB$ (\emph{bottom right}) power spectra due to passive tensor modes, compared with the ones due to compensated modes. The solid lines represent primary CMB anisotropies, the dotted lines represent magnetically-induced compensated scalar modes (except for the $BB$ panel, where it represents the lensing contribution), the dashed lines represent vector modes, whereas dot-dashed lines represent magnetically-induced passive tensor modes. We consider PMFs with $B_{1\,\mathrm{Mpc}}=4.1$\,nG and $n_B=-2.9$.

Magnetically-induced CMB $TT$ (\emph{top left}), $TE$ (\emph{top right}), $EE$ (\emph{bottom left}), and $BB$ (\emph{bottom right}) power spectra due to passive tensor modes, compared with the ones due to compensated modes. The solid lines represent primary CMB anisotropies, the dotted lines represent magnetically-induced compensated scalar modes (except for the $BB$ panel, where it represents the lensing contribution), the dashed lines represent vector modes, whereas dot-dashed lines represent magnetically-induced passive tensor modes. We consider PMFs with $B_{1\,\mathrm{Mpc}}=4.5$\,nG and $n_B=-2.9$.

Magnetically-induced CMB $TT$ (\emph{top left}), $TE$ (\emph{top right}), $EE$ (\emph{bottom left}), and $BB$ (\emph{bottom right}) power spectra due to passive tensor modes, compared with the ones due to compensated modes. The solid lines represent primary CMB anisotropies, the dotted lines represent magnetically-induced compensated scalar modes (except for the $BB$ panel, where it represents the lensing contribution), the dashed lines represent vector modes, whereas dot-dashed lines represent magnetically-induced passive tensor modes. We consider PMFs with $B_{1\,\mathrm{Mpc}}=4.5$\,nG and $n_B=-2.9$.

Magnetically-induced CMB $TT$ (\emph{top left}), $TE$ (\emph{top right}), $EE$ (\emph{bottom left}), and $BB$ (\emph{bottom right}) power spectra due to passive tensor modes, compared with the ones due to compensated modes. The solid lines represent primary CMB anisotropies, the dotted lines represent magnetically-induced compensated scalar modes (except for the $BB$ panel, where it represents the lensing contribution), the dashed lines represent vector modes, whereas dot-dashed lines represent magnetically-induced passive tensor modes. We consider PMFs with $B_{1\,\mathrm{Mpc}}=4.1$\,nG and $n_B=-2.9$.

Dependence of the magnetically-induced CMB power spectrum due to passive tensor modes on the spectral index for a GUT-scale PMF (\emph{left}) and comparison between the two extremes for the time ratio $\tau_\nu/\tau_B$ (\emph{right}). The black lines show the primary CMB anisotropies; for the other colours we refer to the legend. Solid lines represent PMFs generated at the GUT scale, whereas dashed lines represent PMFs generated at late times.

Dependence of the magnetically-induced CMB power spectrum due to passive tensor modes on the spectral index for a GUT-scale PMF (\emph{left}) and comparison between the two extremes for the time ratio $\tau_\nu/\tau_B$ (\emph{right}). The black lines show the primary CMB anisotropies; for the other colours we refer to the legend. Solid lines represent PMFs generated at the GUT scale, $\tau_\nu/\tau_B=10^{17}$, whereas dashed lines represent PMFs generated at late times, $\tau_\nu/\tau_B=10^6$.

CMB $TT$ (\emph{top left}), $TE$ (\emph{top right}), $EE$ (\emph{bottom left}), and $BB$ (\emph{bottom right}) power spectra due to helical PMFs compared to the ones due to non-helical PMFs. Solid lines are non-helical predictions, while dashed lines are helical predictions. Blue are the scalar modes, green the vector, and red the compensated tensor modes. We consider PMFs with $B_{1\,\mathrm{Mpc}}=5$\,nG and $n_B=-1$.

CMB $TT$ (\emph{top left}), $TE$ (\emph{top right}), $EE$ (\emph{bottom left}), and $BB$ (\emph{bottom right}) power spectra due to helical PMFs compared to the ones due to non-helical PMFs. Solid lines are non-helical predictions, while dashed lines are helical predictions. Blue are the scalar modes, green the vector, and red the compensated tensor modes. We consider PMFs with $B_{1\,\mathrm{Mpc}}=4.5$\,nG and $n_B=-1$.

CMB $TT$ (\emph{top left}), $TE$ (\emph{top right}), $EE$ (\emph{bottom left}), and $BB$ (\emph{bottom right}) power spectra due to helical PMFs compared to the ones due to non-helical PMFs. Solid lines are non-helical predictions, while dashed lines are helical predictions. Blue are the scalar modes, green the vector, and red the compensated tensor modes. We consider PMFs with $B_{1\,\mathrm{Mpc}}=4.5$\,nG and $n_B=-1$.

CMB $TT$ (\emph{top left}), $TE$ (\emph{top right}), $EE$ (\emph{bottom left}), and $BB$ (\emph{bottom right}) power spectra due to helical PMFs compared to the ones due to non-helical PMFs. Solid lines are non-helical predictions, while dashed lines are helical predictions. Blue are the scalar modes, green the vector, and red the compensated tensor modes. We consider PMFs with $B_{1\,\mathrm{Mpc}}=5$\,nG and $n_B=-1$.

Comparison of the constraints on the smoothed PMF amplitude (\emph{top}) and the spectral index (\emph{bottom}) from the 2015 temperature and temperature plus polarization data with the 2013 results for magnetically-induced compensated initial conditions only.

Magnetically-induced odd-parity CMB cross-correlations for a helical PMF. The solid lines show $TB$, the dashed lines show $EB$. Green are the vector modes, red the compensated tensor modes, and yellow the passive tensor modes. We consider a PMF with $B_{1\,\mathrm{Mpc}}=5$\,nG and $n_B=-1$.

Comparison of the constraints on the smoothed PMF amplitude (\emph{top}) and the spectral index (\emph{bottom}) from the 2015 temperature and temperature plus polarization data with the 2013 results for magnetically-induced compensated initial conditions only.

Constraints on the smoothed PMF amplitude (\emph{top}) and spectral index (\emph{bottom}) from {\Planck} temperature data with and without the passive tensor contribution. Constraints including both compensated and passive modes are indicated with C+P in the legend, constraints using only compensated modes are marked with C.

Constraints on the smoothed PMF amplitude (\emph{top}) and spectral index (\emph{bottom}) from {\Planck} temperature data with and without the passive tensor contribution. Constraints including both compensated and passive modes are indicated with C+P in the legend, constraints using only compensated modes are marked with C.

Constraints on the smoothed PMF amplitude (\emph{top}) and spectral index (\emph{bottom}) from {\Planck} temperature data with and without the passive tensor contribution. Constraints including both compensated and passive modes are indicated with C+P in the legend, constraints using only compensated modes are marked with C.

PMF amplitude versus the spectral index for the baseline {\Planck} 2015 case. C+P denotes the case where both compensated and passive modes are considered, whereas C indicates the case with only compensated modes.The two contours represent the 68\,\% and 95\,\% confidence levels.

Constraints on the smoothed PMF amplitude (\emph{top}) and spectral index (\emph{bottom}) from {\Planck} temperature data with and without the passive tensor contribution. Constraints including both compensated and passive modes are indicated with C+P in the legend, constraints using only compensated modes are marked with C.

Two-dimensional posterior distributions of the PMF amplitude versus the parameter describing the Poissonian term of unresolved point sources for the three frequencies considered in the likelihood. The two contours represent the 68\,\% and 95\,\% confidence levels.

PMF amplitude versus the spectral index for the baseline {\Planck} 2015 case. C+P denotes the case where both compensated and passive modes are considered, whereas C indicates the case with only compensated modes.The two contours represent the 68\,\% and 95\,\% confidence levels.

Probability distributions for the PMF amplitude including the BICEP2/\textit{Keck}-\Planck\ cross-correlation, compared with the one based only on \Planck\ data. \emph{Top}: the case in which the spectral index is free to vary, \emph{bottom}: the case with $n_B = -2.9$.

Two-dimensional posterior distributions of the PMF amplitude versus the parameter describing the Poissonian term of unresolved point sources for the three frequencies considered in the likelihood. The two contours represent the 68\,\% and 95\,\% confidence levels.

Probability distributions for the PMF amplitude including the BICEP2/\textit{Keck}-\Planck\ cross-correlation, compared with the one based only on \Planck\ data. \emph{Top}: the case in which the spectral index is free to vary, \emph{bottom}: the case with $n_B = -2.9$.

PMF amplitude constraint for the helical case (solid black) compared with the non-helical case (dashed red). The dotted blue line shows the constraint on the amplitude of the helical component as an alternative interpretation of the constraints on the amplitude of PMFs with a helical component.

Probability contours of PMF strength vs.\ spectral index of the PMF power spectrum as constrained by the 70\,GHz observations.

Power spectra of the primordial Faraday depth $\Phi$ for magnetic field strengths of 100 (solid line), 10 (dotted line), and 1\,nG (dashed line), the Galactic Faraday depth from the all-sky Faraday map of \cite{oppermann14} (blue dashed line), the Galactic Faraday depth from \cite{oppermann14} at 45\deg\ latitude (green dashed line), the Galactic Faraday depth derived from Galactic emission at 1.4 and 23\,GHz (red circles), and the Galactic Faraday depth for a Galactic magnetic field model (black crosses).

Probability contours for the PMF amplitude and the foreground parameters for the \Planck 2013 likelihood.

Probability contours for the PMF amplitude and the foreground parameters.

Probability contours for the PMF amplitude and the foreground parameters.