Optimal limits on primordial magnetic fields from CMB temperature bispectrum of passive modes

TL;DR

This work derives a comprehensive framework to bound primordial magnetic fields from the CMB temperature bispectrum generated by PMF-induced passive scalar, vector, and tensor perturbations. It provides an exact-form bispectrum and a computationally efficient pole-approximation that preserves the shape, revealing a local-type non-Gaussian signature. Using a signal-to-noise approach with WMAP/Planck-like noise models, the authors obtain 95% CL bounds on the smoothed PMF strength of $B_{1\mathrm{Mpc}}$ in the few-nanogauss range for a nearly scale-invariant spectrum, with Planck yielding tighter limits than WMAP. The results demonstrate the viability of bispectrum-based PMF constraints, though they are slightly weaker than power-spectrum bounds and can be extended to polarization and higher-order correlators in future work.

Abstract

We investigate bounds on the strength of the primordial magnetic field (PMF) from the cosmic microwave background (CMB) bispectra of the intensity (temperature) modes induced from the auto- and cross-correlated bispectra of the scalar and tensor components of the PMF anisotropic stress. At first, we construct a general formula for the CMB intensity and polarization bispectra from PMFs composed of any type of perturbation. Then we derive an approximate expression which traces the exact shape of the CMB bispectrum in order to reduce the computation time with respect to a large number of the multipole configurations, and also show that the non-Gaussian structure coming from PMFs is classified as the local-type configuration. Computing the signal-to-noise ratio on the basis of the approximate formula with the information of the instrumental noises and resolutions, we find expected upper bounds on the magnetic field strength, when the magnetic spectrum is nearly scale invariant ($n_B = -2.9$), smoothed on $1 {\rm Mpc}$ scale at 95% confidence level from the WMAP and PLANCK experiments as $B_{1 \rm Mpc} < 4.0 - 6.7 {\rm nG}$ and $3.8 - 6.5 {\rm nG}$, respectively, depending on the energy scale of the magnetic field production from $10^{14} {\rm GeV}$ to $10^3 {\rm GeV}$. Our new consequences imply slight overestimations by the previous rough discussions.

Figures (4)

• Figure 1: Diagrams with respect to the multipoles. The left panel corresponds to the multipole configuration of equation (\ref{['eq:CMB_bis_PMF_complete']}). Due to the Wigner-$6j$ symbol originated with the sextuplicate dependence on the Gaussian PMFs, $\ell_1, \ell_2$ and $\ell_3$ are linked with the dummy ones $L, L'$, and $L"$, and the one-loop structure is realized. The right panel represents the tree-structure diagram, which arises from the CMB bispectrum induced by the four-point function of the Gaussian seed fields.
• Figure 2: Absolute values of the CMB bispectra of the intensity fluctuations generated from the magnetic $SSS, SST, STT, TTT$ and $VVV$ modes, and the local-type non-Gaussianity of curvature perturbations (\ref{['eq:cmb_bis_local']}) with $f_{\rm NL}^{\rm local} = 5$ for $\ell_1 = \ell_2 = \ell_3 \equiv \ell$. The circle points and lines of the magnetic $SSS, SST, STT, TTT$ and $VVV$ modes correspond to the exact bispectra (\ref{['eq:CMB_bis_PMF_numerical']}) and the approximate ones (\ref{['eq:cmb_bis_PMF_pole_app']}), respectively. The PMF parameters are fixed to $B_{1 \rm Mpc} = 4.7 {\rm nG}, n_B = -2.9$ 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 3: Noise-free signal-to-noise ratios. The "total" line denotes $S/N$ obtained from the total spectrum of the $SSS, SST, STS, TSS, STT, TST, TTS$ and $TTT$ spectra, and the others correspond to $S/N$'s coming from each mode. Here, we fix the PMF parameters as $n_B = -2.9$. The other parameters are identical to the mean values obtained from the WMAP-7yr data Komatsu:2010fb.
• Figure 4: Signal-to-noise ratios with zero, PLANCK- and WMAP-level noises obtained from the auto- and cross-correlated bispectra between the scalar and tensor modes. The instrumental noises and resolutions of the WMAP and PLANCK experiments are given by $(\theta_{\rm FWHM}{[\rm arcmin]}, \Delta T{[\rm \mu K]}) = (13,18.3), (5.0, 13.1)$, respectively Smith:2006udXia:2007gzHu:2001fb. Here, we fix the PMF parameters to the values presented in figure \ref{['fig:SN_III_S+T_B4.7_n-2.9_tntb1e17.eps']} and the other parameters are identical to the mean values obtained from the WMAP-7yr data Komatsu:2010fb.