Primordial Magnetic Field Limits from Cosmic Microwave Background Bispectrum of Magnetic Passive Scalar Modes

TL;DR

This paper investigates how primordial magnetic fields imprint non-Gaussian signals in the CMB via the passive scalar mode, which is sourced by uncompensated magnetic anisotropic stress before neutrino decoupling. The authors compute the CMB reduced bispectrum $b_{l_1 l_2 l_3}$ on large scales by modeling the Sachs-Wolfe contribution and applying a detailed angular decomposition of the magnetic stress correlations, including two limiting configurations (Case I with $s$-independent terms and Case II with squeezed collinear terms). They find that the passive scalar mode generates a bispectrum magnitude $l_1(l_1+1)l_3(l_3+1) b_{l_1 l_2 l_3}$ of order $-10^{-16}$ to $-10^{-16}$ (depending on configuration and $n$) and that this signal is about $10^6$ times larger than the compensated mode, enabling strong constraints on $B_0$. By comparing with the WMAP7 non-Gaussianity bounds through an $f_{NL}$-like framework, they obtain an upper limit $B_0 \lesssim 2$ nG for scale-invariant spectra, with modest dependence on the generation epoch $\tau_B$. The work highlights magnetically induced non-Gaussian signals as powerful probes of cosmic magnetism and anticipates stronger bounds from Planck data and inclusion of ISW, tensor, and vector contributions.

Abstract

Primordial magnetic fields lead to non-Gaussian signals in the cosmic microwave background (CMB) even at the lowest order, as magnetic stresses and the temperature anisotropy they induce depend quadratically on the magnetic field. In contrast, CMB non-Gaussianity due to inflationary scalar perturbations arises only as a higher order effect. Apart from a compensated scalar mode, stochastic primordial magnetic fields also produce scalar anisotropic stress that remains uncompensated till neutrino decoupling. This gives rise to an adiabatic-like scalar perturbation mode that evolves passively thereafter (called the passive mode). We compute the CMB reduced bispectrum ($b_{l_{_1}l_{_2}l_{_3}}$) induced by this passive mode, sourced via the Sachs-Wolfe effect, on large angular scales. For any configuration of bispectrum, taking a partial sum over mode-coupling terms, we find a typical value of $l_1(l_1+1)l_3(l_3+1) b_{l_{_1}l_{_2}l_{_3}} \sim 6-9 \times 10^{-16}$, for a magnetic field of $B_0 \sim 3$ nG, assuming a nearly scale-invariant magnetic spectrum . We also evaluate, in full, the bispectrum for the squeezed collinear configuration over all angular mode-coupling terms and find $l_1(l_1+1)l_3(l_3+1) b_{l_{_1}l_{_2}l_{_3}} \approx -1.4 \times 10^{-16}$. These values are more than $\sim 10^6$ times larger than the previously calculated magnetic compensated scalar mode CMB bispectrum. Observational limits on the bispectrum from WMAP7 data allow us to set upper limits of $B_0 \sim 2$ nG on the present value of the cosmic magnetic field of primordial origin. This is over 10 times more stringent than earlier limits on $B_0$ based on the compensated mode bispectrum.