Parity violation of primordial magnetic fields in the CMB bispectrum

TL;DR

The study addresses parity violation in the CMB bispectrum arising from primordial magnetic fields (PMFs) by deriving a general bispectrum formula that incorporates both non-helical and helical components and applying a pole approximation to reduce the one-loop complexity to a tree-level form. It shows that helicity induces parity-odd signals in the $III$ configuration (with $\sum_{n=1}^3 \ell_n = {\rm odd}$) and that PMF anisotropic stress yields local-type non-Gaussianity, enabling efficient computation and interpretation. Numerical analysis using CAMB indicates parity-even and parity-odd signals, with parity-odd signals offering a direct probe of PMF helicity; detectability with current data requires $B_{1 \rm Mpc}^{2/3} {\cal B}_{1 \rm Mpc}^{1/3} \gtrsim 2.7$–$4.5$ nG for $\tau_\nu/\tau_B \sim 10^{17}$, and Planck-level data should improve sensitivity. Overall, the work provides a practical framework to test parity-violating PMFs through CMB bispectra and motivates including helicity in non-Gaussianity analyses.

Abstract

We study the parity violation in the cosmic microwave background (CMB) bispectrum induced by primordial magnetic fields (PMFs). Deriving a general formula for the CMB bispectrum generated from not only non-helical but also helical PMFs, we find that helical PMFs produce characteristic signals, which disappear in parity-conserving cases, such as the intensity-intensity-intensity bispectra arising from $\sum_{n=1}^3 \ell_n = {\rm odd}$. For fast numerical calculation of the CMB bispectrum, we reduce the one-loop formula to the tree-level one by using the so-called pole approximation. Then, we show that the magnetic anisotropic stress, which depends quadratically on non-helical and helical PMFs and acts as a source of the CMB fluctuation, produces the local-type non-Gaussianity. Comparing the CMB bispectra composed of the scalar and tensor modes with the noise spectra, we find that assuming the generation of the nearly scale-invariant non-helical and helical PMFs from the grand unification energy scale ($10^{14} {\rm GeV}$) to the electroweak one ($10^{3} {\rm GeV}$), the intensity-intensity-intensity bispectrum for $\sum_{n=1}^3 \ell_n = {\rm odd}$ can be observed by the WMAP experiment under the condition that $B_{1 \rm Mpc}^{2/3} {\cal B}_{1 \rm Mpc}^{1/3} > 2.7 - 4.5 {\rm nG}$ with $B_{1 \rm Mpc}$ and ${\cal B}_{1 \rm Mpc}$ being the non-helical and helical PMF strengths smoothed on 1 Mpc, respectively.

Figures (2)

• Figure 1: Absolute values of the CMB $III$ bispectra generated from the $TTT$ (red solid line), $STT+TST+TTS$ (blue dashed one) and $SST+STS+TSS$ (magenta dotted one) modes as the function in terms of $\ell_3$ when the two multipoles are fixed as $(\ell_1, \ell_2) = (100, 105)$. The curves in the left and right panels correspond to the parity-even and -odd bispectra arising from $\sum_{n=1}^3 \ell_n = {\rm even}$ and $= {\rm odd}$, respectively. The PMF parameters are fixed to $B_{1 \rm Mpc} = 4.7 {\rm nG}, {\cal B}_{1 \rm Mpc} = 1.35 {\rm nG}, n_B = n_{\cal B} = -2.9$ and $\tau_\nu / \tau_B = 10^{17}$, and other parameters are identical to the mean values derived from the WMAP 7-yr data Komatsu:2010fb
• Figure 2: Noise-free signal-to-noise ratios from the parity-even (left panel) and -odd (right one) CMB $III$ bispectra coming from $\sum_{n=1}^3 \ell_n = {\rm even}$ and $= {\rm odd}$, respectively. The "total" line denotes $S/N$ obtained from the total spectrum of the $TTT, STT, TST, TTS, SST, STS$ and $TSS$ modes, and the others correspond to $S/N$'s coming from each mode. Here, we fix the PMF parameters as $B_{\rm 1Mpc} = 1.0 {\rm nG}, {\cal B}_{\rm 1Mpc} = 0.287 {\rm nG}, n_B = n_{\cal B} = -2.9$ and $\tau_\nu / \tau_B = 10^{17}$. The other parameters are identical to the mean values obtained from the WMAP-7yr data Komatsu:2010fb.