The cosmic microwave background temperature bispectrum from scalar perturbations induced by primordial magnetic fields

TL;DR

This paper addresses how a stochastic primordial magnetic field (PMF) imprints a large-scale temperature bispectrum in the CMB. It develops a framework to compute the PMF energy-density bispectrum via Wick contractions of a Gaussian magnetic field, propagates it to the CMB using the Sachs-Wolfe relation, and analyzes collinear, squeezed, and equilateral configurations across representative spectral indices $n=2$ and $n=-2$. The study provides analytic and semi-analytic estimates of the reduced CMB bispectrum $b_{\ell_1\ell_2\ell_3}$, derives an effective local NG parameter $f^{\rm eff}_{\rm NL}$, and translates existing CMB NG bounds into constraints on the PMF amplitude, finding bounds of order 9–25 nG on the PMF strength for various $n$. The results highlight the potential of CMB non-Gaussianity to constrain early-universe magnetism and discuss Planck-level improvements and caveats due to using the Sachs-Wolfe approximation and neglecting transfer-function effects and vector/tensor PMF modes.

Abstract

We evaluate the angular bispectrum of the CMB temperature anisotropy at large angular scale due to a stochastic background of primordial magnetic fields. The shape of non-Gaussianity depends on the spectral index of the magnetic field power spectrum and is peaked in the squeezed configuration for a scale-invariant magnetic spectrum. By using the large angular part of the bispectrum generated by magnetic fields, the present bounds on non-Gaussianity set a limit on the amplitude of the primordial magnetic field of the order of 10 nGauss for the scale-invariant case and 20 nGauss for the other spectral indexes.

Figures (4)

• Figure 1: The geometrical configuration used to perform the integration: ${\mathbf k},~{\mathbf q}$ and ${\mathbf p}$ are free, while ${\bf\tilde{k}}$ is the integration wave-vector (see Brown:2005kr).
• Figure 2: The magnetic field bispectrum in the collinear configuration ${\mathbf p}={\mathbf q}=-{\mathbf k}/2$, normalised by the quantity $A^3 k_D^{3n+3} / (576\pi^3)$, as a function of $k/k_D$, for $n=2$ (left plot) and $n=-2$ (right plot). We only show the infrared region $k\leq k_D/2$. The blue, solid line is the exact result, while the red, dashed line the approximation given in Eq. (\ref{['approxcoll']}).
• Figure 3: The magnetic field bispectrum in the collinear configuration ${\mathbf p}={\mathbf q}=-{\mathbf k}/2$ (blue, solid) and the magnetic field spectrum to the $3/2$ (red, dashed), both multiplied by the phase space density $k^3$, as a function of $k/k_D$ for $n=2$ and $n=-2$. Note that in the $n=-2$ case, we only calculated the bispectrum up to $k=k_D/2$, while the spectrum is known up to $k=k_D$.
• Figure 4: The approximations for the integral in Eq. (\ref{['exintegral']}). Upper left plot, for $m=2$: the integral (solid) and the approximation $y/2$ (dashed) are shown for $\ell=20$, $\ell=100$, $\ell=500$ as a function of $y$. Upper right plot, for $m<1$: the approximations for $y\gg\ell$ (solid) and for $\ell\gg 1$ (dashed) given in Eq. (\ref{['approxmin1']}) are shown as a function of $\ell$ for $m=0$, $m=-1$ and $m=-2$. Lower plots, for $m=1$: the integral (solid) and the approximation in Eq. (\ref{['approx1']}) (dashed) are shown as a function of $y$ for $\ell=20$ (left plot) and as a function of $\ell$ for $y=100$ (right plot).