Non-Gaussianity from Cosmic Magnetic Fields

TL;DR

This work demonstrates that even Gaussian primordial magnetic fields can generate strong non-Gaussian signatures in the cosmic stress-energy through its quadratic dependence on the field, producing observable 1-, 2-, and 3-point statistics across scalar, vector, and tensor sectors. By combining analytic calculations with numerical realizations, the authors derive the full set of one-, two-, and three-point correlators for the magnetic stress-energy tensor, validating them against simulations for both flat and steep power spectra and highlighting notable cross-correlations between different perturbation types. The results reveal robust non-Gaussian features, including scalar–tensor–tensor and other cross-moments, which could serve as distinctive probes of early-universe magnetic fields when mapped onto CMB observables with appropriate transfer functions. The methodology and findings provide a framework to constrain or detect primordial magnetic fields via higher-order statistics, with potential extensions to other non-linear source models such as topological defects.

Abstract

Magnetic fields in the early universe could have played an important role in sourcing cosmological perturbations. While not the dominant source, even a small contribution might be traceable through its intrinsic non-Gaussianity. Here we calculate analytically the one, two and three point statistics of the magnetic stress energy resulting from tangled Gaussian fields, and confirm these with numerical realizations of the fields. We find significant non-Gaussianity, and importantly predict higher order moments that will appear between the scalar, vector and tensor parts of the stress energy (e.g. scalar-tensor-tensor moments). Such higher order cross correlations are a generic feature of non-linear theories and could prove to be an important probe of the early universe.

Figures (7)

• Figure 1: Sample realizations of the magnetic field for a spectrum $n=0$. Left: Gaussian magnetic field slice, $B_{x}|_{z=0}$; center: the (very non-Gaussian) isotropic pressure $\tau|_{z=0}$; right: the (slightly non-Gaussian) anisotropic pressure $\tau^S|_{z=0}$. Compared to the magnetic field, non-linearity transfers power to smaller scales in the sources.
• Figure 2: The left figure shows the probability distribution of the isotropic and anisotropic pressures for a spectrum $n=0$, with a Gaussian distribution shown for comparison. The isotropic distribution is well fit by a $\chi^2$ curve. The right figure compares the anisotropic pressure distribution for different spectral indices. The $x$-axis is in units of the root mean square amplitude of the relevant field.
• Figure 3: Magnetic stress-energy power spectra ($n=0$) -- the realizations agree well with the analytic predictions. The error bars from the realisations are small except at very low $k$.
• Figure 4: Magnetic stress-energy power spectra ($n=-2.5$). The dashed line shows the spectral dependence expected naively, $\propto k^{2n+3}$. The turnover at low $k$ reflects the finite size of the grid.
• Figure 5: The geometry for the bispectra calculations; $\mathbf{k},\mathbf{p},\mathbf{q}$ are the wavevectors for the three legs, while $\mathbf{k'}$ is an integration mode.