The Cosmic Microwave Background and Helical Magnetic Fields: the tensor mode

TL;DR

This work analyzes how a primordial magnetic field with nonzero helicity can source tensor perturbations that imprint on the CMB. By modeling the field with symmetric and helical power spectra and applying the total angular momentum formalism, the authors derive analytic approximations for the parity-even and parity-odd CMB signals, showing helicity induces a parity-odd gravity-wave component and nonzero $\Theta B$ and $EB$ correlations. They find parity-odd signals are generically small and require near scale-invariant spectra with near-maximal helicity to be observable, and that causal generation scenarios suppress these effects dramatically. The results provide a framework to constrain or detect primordial magnetic helicity via parity-violating CMB observables, offering insight into parity-violating physics in the early universe and potential connections to inflationary mechanisms.

Abstract

We study the effect of a possible helicity component of a primordial magnetic field on the tensor part of the cosmic microwave background temperature anisotropies and polarization. We give analytical approximations for the tensor contributions induced by helicity, discussing their amplitude and spectral index in dependence of the power spectrum of the primordial magnetic field. We find that an helical magnetic field creates a parity odd component of gravity waves inducing parity odd polarization signals. However, only if the magnetic field is close to scale invariant and if its helical part is close to maximal, the effect is sufficiently large to be observable. We also discuss the implications of causality on the magnetic field spectrum.

Figures (4)

• Figure 1: On the top panel we show the amplitudes of the parity even correlators, $\ell^2C^{(\theta\theta)}_{(A)\ell}$ (solid, black), $\ell^2C^{(EE)}_{(A)\ell}$ (dotted, red) and $\ell^2C^{(\theta E)}_{(A)\ell}$ (dashed, blue) as a function of the spectral index $n_A$ for $\ell=50$. The logarithm of the absolute value of $\ell^2C_{(A)\ell}^{(XY)}$ is shown in units of $(\Omega_A/\Omega_r)^2\ln^2(z_{\rm in}/z_{\rm eq})$. We do not plot $\ell^2C^{(BB)}_{(A)\ell}$ which equals $\ell^2C^{(EE)}_{(A)\ell}$ within our approximation. The spikes at $n_A=-2$ for $\ell^2C^{(EE)}_\ell$ and at $n_A=-3/2$ are not real. They are artefacts due to the break-down of our approximations at these values. On the bottom panel we show the corresponding parity odd correlators, $\ell^2C^{(\theta B)}_{(A)\ell}$ (solid, black), $\ell^2C^{(EB)}_{(A)\ell}$ (dashed, red) in units of $(\Omega_A\Omega_S/\Omega_r^2)\ln^2(z_{\rm in}/z_{\rm eq})$ for $n_S=-2.99$ and $n_S=2$. In this last case, only the allowed range $n_A\geq n_S=2$ is plotted. Again the spike at $n_A=1$ for $n_S=-2.99$ and the precipitous drop at $n_A=-1$ in $\ell^2C^{(EB)}_{(A)\ell}$, are due to the limitation of our approximation close to the transition indices.
• Figure 2: We show the ratio of the correlators, $C^{(\theta B)}_\ell/C^{(\theta E)}_\ell$ (solid, black), and $C^{(E B)}_\ell/C^{(EE)}_\ell$ for $n_S=-3$ as functions of the spectral index $n_A$ for $\ell=50$. The logarithm of the absolute value is shown in units of $\Omega_A/\Omega_S\le 1$. The spikes visible at certain values of the spectral index $n_A$ are mainly due to our relatively crude approximations.
• Figure 3: In both panels, as a function of $x_0$: the green dotted line shows the numerical value of the integral in (\ref{['approx1']}), the blue, long dashed line shows the analytic approximation (right hand side of Eq. (\ref{['approx1']})), and the red, solid line shows the numerical value of integral (\ref{['approx1']}) if $x_{\rm dec}$ is not put to zero. All these functions are squared, and multiplied by $x_0^3$: this gives us an indication of the result, after the integration over $x_0$, as stated in Eq. (\ref{['int-appendix']}). In the left panel $\ell=50$, in the right panel $\ell=200$. First of all, we note that it appears clearly that the value of the integrals is dominated at $x_0\simeq\ell$, and that the function goes to zero quicker than $x_0^{-3}$, which justifies our approximation $x_D\rightarrow\infty$ and the use of formula \ref{['eq:GR-6.574.2']}. Secondly, we note that for $\ell=50$ and $x_0\sim \ell$, our approximation (blue, long-dashed) is good for both the integrals. However, if $\ell=200$, the approximation overestimate the correct numerical result by about a factor of ten.
• Figure 4: We plot the value of integral (\ref{['divergent-approx']}) squared and multiplied by $x_0^3$ as function of $x_0$, for $\ell=30$. The green, dotted line represents again the numerical result ($x_{\rm dec}\rightarrow 0$), and the blue, long dashed line is the analytic approximation. In this case the slope is positive, and hence the integral $dx_0/x_0$ of this function is dominated by the upper cutoff.