Anderson's localization in a random metric: applications to cosmology

It is considered an equation for the Lyapunov exponent $$ 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($\gamma$)) is obtained as function of the metric-fluctuation-distance $\mathrm{\Delta }R$ (function of disorder) and the frequency $\omega$ of the wave. Explicitly, low-frequencies propagate longer than high, that is $L_{c}\omega ^{2}=C^{te}$. Direct applications with cosmological quantities like background radiation microwave ($\lambda \sim 1\mathrm{/}2\times {10}^{-3}$ [m]) and the Universe-length (`localization length’ $L_c\sim 1.6\times {10}^{25}$ [m]) permits to evaluate the metric-fluctuations-distance as $\mathrm{\Delta }R\sim {10}^{-35}$ [m], a number at order of the Planck’s length.