Determination of the Primordial Magnetic Field Power Spectrum by Faraday Rotation Correlations

Abstract

(Abridged) This paper introduces the formalism which connects between rotation measure ($\RM$) measurements for extragalactic sources and the cosmological magnetic field power spectrum. It is shown that the amplitude and shape of the cosmological magnetic field power spectrum can be constrained by using a few hundred radio sources, for which Faraday $\RM$s are available. This constraint is of the form $B_{rms} \simless 1 \times [2.6\times10^{-7} cm^{-3}/ \bar n_b] h $ nano-Gauss (nG) on $\sim 10-50 \hmpc$ scales. The constraint is superior to and supersedes any other constraint which come from either CMB fluctuations, Baryonic nucleosyn thesis, or the first two multipoles of the magnetic field expansion. Demonstration of the ability to detect such magnetic fields, using Bayesian statistics, is carried out by constructing simulations of the field and mimicking observations. This procedure also provides error estimates for the derived quantities. The two main noise contributions due to the Galactic RM and the internal RM are treated in a statistical way. For power indices -1\leq n \leq 1 in a flat cosmology (Omega_m=1) we estimate the signal-to-noise ratio, Q, for limits on the magnetic field B_{rms} on ~50 h^{-1}Mpc scale. Employing one patch of a few square degrees on the sky with source number density n_{src}, an approximate estimate yields Q\simeq 3 \times (B_{rms}/1 {nG})(n_{src}/50 {deg}^{-2}) (2.6\times10^{-7} {cm}^{-3}/ \bar n_b) h $. An all sky coverage, with much sparser, but carefully tailored sample of ~500 sources, yields Q \simeq 1 with the same scaling. An ideal combination of small densely sampled patches and sparse all-sky coverage yields Q\simeq 3 with better constraints for the power index. All of these estimates are corroborated by the simulations.