Cosmological Magnetic Fields 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 being homogeneous and, for the redshifts of interest to rotation measures (RM), it is well described by the observed Ly-$\alpha$ forest. We use a log-normal distribution to model the Ly-$\alpha$ forest and assume that a cosmological magnetic field is frozen into the plasma and is therefore a function of the density inhomogeneities. The Ly-$\alpha$ forest results are much less sensitive to the cosmological magnetic field coherence length than those for a homogeneous universe and 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 RM for different lines-of-sight with a highly non-gaussian tail that renders the variance and the mean RM impractical for setting limits. The $median|{\rm RM}|$ is a better statistical indicator which we use to derive the following limits using the observed RM for QSOs between $z = 0$ and $z = 2.5$. We set $\Omega_b h^2 = 0.02$ and get for cosmological fields coherent accross the present horizon, $B_{H_0^{-1}} \la 10^{-9}$ G in the case of a Ly-$\alpha$ forest which is stronger than the limit for a homogeneous universe, $B_{H_0^{-1}}^h \la 2 \times 10^{-8}$ G; while for 50 Mpc coherence length, the inhomogeneous case gives $B_{50 {\rm Mpc}} \la 6 \times 10^{-9}$ G while the homogeneous limit is $B_{50 {\rm Mpc}}^h \la 10^{-7}$ G and for coherence length equal to the Jeans length, $B_{\lambda_J} \la 10^{-8}$ G for the Ly-$\alpha$ case while $B_{\lambda_J}^h \la 10^{-6}$ G.