Microwave Background Signatures of a Primordial Stochastic Magnetic Field

TL;DR

The paper develops a comprehensive analytic framework to predict CMB signatures from a primordial stochastic magnetic field with a power-law spectrum, focusing on vector and tensor perturbations and using the total angular momentum method. It derives the magnetic stress-energy spectra and their projection onto vector and tensor metric perturbations, then computes the resulting temperature and polarization power spectra, including cross-correlations, across a range of spectral indices $n$ and damping scales $k_D$. The results show that scale-invariant fields ($n\approx -3$) yield detectable signatures at current/future instruments, while causal fields ($n\ge 2$) produce much stronger constraints on the comoving mean-field amplitude $B_\lambda$, potentially reaching as low as $\sim 4\times10^{-13}$ G for $n\ge 2$, with B-type polarization offering the best sensitivity. These findings imply that forthcoming CMB observations, particularly of B-mode polarization, could decisively test primordial stochastic magnetic fields and distinguish their magnetogenesis scenarios. The analysis emphasizes that tensor and vector modes, damped scales, and the $k_D$-dependent behavior govern the amplitudes, and provides analytic predictions that are useful for constraining early-universe magnetism and informing related cosmological parameter surveys.

Abstract

A stochastic magnetic field in the early Universe will produce anisotropies in the temperature and polarization of the cosmic microwave background. We derive analytic expressions for the microwave background temperature and polarization power spectra induced by vector and tensor perturbations from a power-law magnetic field. For a scale-invariant stochastic magnetic field smoothed over a comoving scale of $1 {\rm Mpc}$, the MAP satellite has the potential to constrain the comoving mean-field amplitude to be no greater than approximately $2\times10^{-9}$ G. Limits improve as the power-law slope increases: for causally-generated power-law magnetic fields, the comoving mean-field amplitude has an upper bound of approximately $4\times10^{-13} {\rm G}$. Such constraints will surpass all current limits on galactic-scale primordial stochastic magnetic fields at decoupling.

Figures (3)

• Figure 1: The microwave background power spectra for vector (left panels) and tensor (right panels) perturbations from a power-law stochastic magnetic field with spectral index $n$. Solid line represents $\Theta\Theta$, dash-dot line represents EE, dotted line represents BB, and dash-dot-dot-dot line represents $\Theta\hbox{E}$. The magnetic comoving mean-field amplitude is chosen to be $B_\lambda=10^{-9}\,{\rm G}$, with a smoothing Gaussian sphere comoving radius of $\lambda=1\,{\rm Mpc}$. The magnetic damping cutoff wavenumbers for vector and tensor perturbations are given by Eqs. (\ref{['eq:V-damping']}) and (\ref{['eq:T-damping']}) respectively. The absolute values of the vector cross correlations are plotted. For the tensor perturbations, we assume $z_{\text{in}}/z_{\text{eq}}=10^9$.
• Figure 2: Same as in Fig. \ref{['fig:Figure1']}, except that the microwave background power spectra are for vector plus tensor perturbations.
• Figure 3: Each panel shows a single power spectrum for various values of $n$. Solid line represents $n=-2.99$, dashed line represents $n=-2.0$, dash-dot line represents $n=0.0$, dash-dot-dot-dot line represents $n=1.0$, and dotted line represents $n=2.0$.