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

(Abridged) This paper introduces the formalism which connects betweenrotation measure ($\RM$) measurements for extragalactic sources and thecosmological magnetic field power spectrum. It is shown that the amplitude andshape of the cosmological magnetic field power spectrum can be constrained byusing a few hundred radio sources, for which Faraday $\RM$s are available. Thisconstraint 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 issuperior to and supersedes any other constraint which come from either CMBfluctuations, Baryonic nucleosyn thesis, or the first two multipoles of themagnetic field expansion. Demonstration of the ability to detect such magneticfields, using Bayesian statistics, is carried out by constructing simulationsof the field and mimicking observations. This procedure also provides errorestimates for the derived quantities. The two main noise contributions due to the Galactic RM and the internal RMare treated in a statistical way. For power indices -1\leq n \leq 1 in a flatcosmology (Omega_m=1) we estimate the signal-to-noise ratio, Q, for limits onthe magnetic field B_{rms} on ~50 h^{-1}Mpc scale. Employing one patch of a fewsquare degrees on the sky with source number density n_{src}, an approximateestimate 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 muchsparser, but carefully tailored sample of ~500 sources, yields Q \simeq 1 withthe same scaling. An ideal combination of small densely sampled patches andsparse all-sky coverage yields Q\simeq 3 with better constraints for the powerindex. All of these estimates are corroborated by the simulations.Comment: 30 pages, figures included, To appear in the Ap.