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 } | 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 { Mpc } } \la 6 \times 10 ^ { -9 } G while the homogeneous limit is B _ { 50 { 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 .