X iv : g r - qc / 0 50 10 12 v 1 5 J an 2 00 5 Anderson ’ s localization in a random metric : applications to cosmology

It is considered an equation for the Lyapunov exponent γ in a random metric for a scalar propagating wave field. At first order in frequency this equation is solved explicitly. The localization length L c (reciprocal of Re(γ)) is obtained as function of the metric-fluctuation-distance ∆R (function of disorder) and the frequency ω of the wave. Explicitly, low-frequencies propagate longer than high, that is L c ω 2 = C te. Direct applications with cos-mological quantities like background radiation microwave (λ ∼ 1/2 × 10 −3 [m]) and the Universe-length ('localization length' L c ∼ 1.6 × 10 25 [m]) permits to evaluate the metric-fluctuations-distance as ∆R ∼ 10 −35 [m], a number at order of the Planck's length. PACS: 04.30.-W (Waves in general relativity) 72.15.Rn (Anderson's localization) 98.80.-k (Cosmology) 04.40.-b (Classical fields in general relativity)