Correlation of inflation-produced magnetic fields with scalar fluctuations

TL;DR

The paper investigates whether inflation-produced magnetic fields can correlate with scalar density perturbations by coupling electromagnetism to a spectator scalar field in a de Sitter background. Using a toy model with $V(\phi)=-3 n M H_I^2 \phi$ and $W(\phi)=e^{2\phi/M}$, it derives a scale-free magnetic spectrum for $n=2$ and computes both scalar and magnetic spectra as well as the cross-correlation via the in-in formalism. A key result is that the dimensionless cross-correlation can be enhanced by large logarithmic factors to reach ${\cal O}(500)$ in certain regimes, but is ultimately bounded by $H_I/M \lesssim 5\times 10^{-4}$ to respect normalization, with the strongest observable signatures arising for flattened Fourier triangles. The findings offer a potential link between inflationary magnetogenesis and density perturbations, suggesting observable imprints in the CMB and large-scale structure and guiding future searches with radio surveys like SKA and LOFAR.

Abstract

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 the 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 ~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 ~500 H_I/M, where H_I is the inflationary Hubble constant and M is the effective mass scale of the coupling.

Figures (3)

• Figure 1: 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 2: 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 3: 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 $n=2$.