Determination of the primordial magnetic field power spectrum by Faraday rotation correlations

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 RMs are available. This constraint is of the form B-rms less than or similar to 1 x (26 x 10(-7) cm(-3)/(n) over bar(b)]) h nG on similar to 10-50 h(-1) Mpc scales, with (n) over bar(b) the average baryon density and h the Hubble parameter in units of 100 km s(-1) Mpc The constraint is superior to and supersedes any other constraint that comes from either cosmic microwave background (CMB) fluctuations, baryonic nucleosynthesis, or the first two multipoles of the magnetic field expansion. The most adequate method for the constraint calculation uses the Bayesian approach to the maximum likelihood function. I demonstrate the ability to detect such magnetic fields 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 following an evaluation of their distribution. For a range of magnetic field power spectra with power indices -1 less than or equal to n less than or equal to 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 a similar to 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 similar or equal to 3(B-rms/1 nG)(n(src)/50 deg(-2))(2.6 x 10(-7) cm(-3)/(n) over bar(b)) h. An all-sky coverage, with a much sparser but carefully tailored sample of similar to 500 sources, yields Q similar or equal to 1 with the same scaling. An ideal combination of small densely sampled patches and sparse all-sky coverage yields Q similar or equal to 3 with better constraints for the power index. All of these estimates are corroborated by the simulations.