Non-Gaussianity from cosmic magnetic fields

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 nonlinear theories and could prove to be an important probe of the early Universe.

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, nonlinearity 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 realizations 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 wave vectors for the three legs, while ${\mathbf{k}}^{\prime }$ is an integration mode.

Figure 6

Here we show the various collinear bispectra for the flat spectrum $n=0$ generated from a collection of realizations. Where the analytic results have been calculated, they agree well with the numerical results.

Figure 7

Here we show the collinear bispectra for a steep spectrum ($n=-2.5$). The scaling is as expected naively, $\propto k^{3n+3}=k^{-9/2}$ (dotted line).