Cosmological magnetic field limits in an inhomogeneous universe

We study the effect of inhomogeneities in the matter distribution of the universe on the Faraday rotation of light from distant QSOs and derive new limits on the cosmological magnetic field. The matter distribution in the universe is far from homogeneous, and for the redshifts of interest in relation to rotation measures (RMs), it is well described by the observed Ly alpha forest. We use a lognormal distribution to model the Lya forest, and we assume that a cosmological magnetic field is frozen into the plasma and therefore is a function of the density inhomogeneities. The Ly alpha forest results are much less sensitive to the cosmological magnetic field coherence length than the results for a homogeneous universe, and they show an increase in the magnitude of the expected RM for a given field by over an order of magnitude. The forest also introduces a large scatter in RMs for different lines of sight, with a highly non-Gaussian tail that renders the variance and the mean RM impractical for setting limits. The median \RM\ is a better statistical indicator that we use to derive the following limits using the observed RMs for QSOs between z = 0 and z = 2.5. We set Omega(b)h(2) = 0.02 and get, for cosmological fields that are coherent across the present horizon, BH0-1 less than or similar to 10(-9) G in the case of a Ly alpha forest that is stronger than the limit for a homogeneous universe, B-H0-(h)(1) less than or similar to 2 x 10(-s) G; for a 50 Mpc coherence length, the inhomogeneous case gives B-50Mpc less than or similar to 6 x 10(-9) G, while the homogeneous limit is B-50Mpc(h) less than or similar to 10(-7) G; and for coherence length equal to the Jeans length, B-lambda J less than or similar to 10(-8) G for the Ly alpha case, while B-lambda J(h) less than or similar to 10(-6) G.