Correlation of inflation-produced magnetic fields with scalar fluctuations

If the conformal invariance of electromagnetism is broken during inflation, then primordial magnetic fields may be produced. If this symmetry breaking is generated by the coupling between electromagnetism and a scalar field—e.g. the inflaton, curvaton, or Ricci scalar—then these magnetic fields may be correlated with primordial density perturbations, opening a new window to the study of non-Gaussianity in cosmology. In order to illustrate, we couple electromagnetism to an auxiliary scalar field in a de Sitter background. We calculate the power spectra for scalar-field perturbations and magnetic fields, showing how a scale-free magnetic-field spectrum with rms amplitude of $\sim \mathrm{nG}$ at Mpc scales may be achieved. We explore the Fourier-space dependence of the cross correlation between the scalar field and magnetic fields, showing that the dimensionless amplitude, measured in units of the power spectra, can grow as large as $\sim 500H_I/M$, where $H_I$ is the inflationary Hubble constant and $M$ is the effective mass scale of the coupling.

Figure 1

The magnetic field strength $B_{\mathrm{Mpc}}=(d⟨B^2⟩/d\ln k{)}^{1/2}$ at 1 Mpc is shown by the solid line as a function of the index $n_B$. Horizontal and vertical short-dashed lines indicate the nG field strength obtained for $n_B=0$. The effect of the $\Gamma$ function in Eq. (23) is negligible compared to the exponential factor ${10}^{-18-24.3n_B}$. The long-dashed lines rotated clockwise and counterclockwise show the magnetic field strength at Gpc and kpc scales, respectively.

Figure 2

The ratio $C_n(\cos \theta )/C_n(-1)$ is shown for $n=0$, 1, 2, and $-2$, as functions of $\cos \theta$. In the $n=-2$ panel, the dashed line indicates where the absolute value has been taken. In the $n=2$, $-2$ cases we have used $2\pi n_1|{\eta }_I/L|\sim {10}^{-27}$, corresponding approximately to a Gpc length scale. Note that the case of cosmological interest is $n=2$.

Figure 3

The ratio $C_n(1)/C(-1)$, the ratio of the discretized Fourier-space cross-correlation coefficients for the flattened triangle to that of the universal result for the squeezed triangle, is shown as a function of $n$. No amplification, $n=0$, yields zero cross correlation. Hence, the flattened triangle may be used as an indicator of an amplification mechanism. The ratio is negative along the dashed line, where we have taken the absolute value. We have set $2\pi n_1|{\eta }_I/L|\sim {10}^{-6}$ for ease of numerical computation; using $2\pi n_1|{\eta }_I/L|\sim {10}^{-27}$ to represent Gpc scales boosts the curve up to ${10}^3$ near $n=\pm 2$. Note that the case of cosmological interest is $n=2$.

Figure 4

The quantity $R$, defined in the text as the ratio of the Fourier-space cross-correlation coefficient to that of the universal result for the squeezed triangle, times a factor $x_{23}^2$, is shown as a function of the triangle side lengths. We have set $2\pi n_3|{\eta }_I/L|\sim {10}^{-27}$ for the cases $n=\pm 2$. Note that the case of cosmological interest is $n=2$.