From Sachs-Wolfe Acoustic Theorem to Fractal Laplace-Beltrami Operator

Abstract

According to Czaja et. al. [2], if one considers the acoustic field propagating in the radiation-dominated (p=e/3) universe of arbitrary space curvature (K=0, ±1), then the field equations are reduced to the d’Alembert equation in an auxiliary static Robertson-Walker spacetime. This is related to the so-called Sachs-Wolfe acoustic theorem, which can be found useful in the observation and analysis of Cosmic Microwave Background anisotropies. In this paper, I will discuss what Laplace-Beltrami operator for curved space is and how this operator may be extended further to become fractal Laplace-Beltrami Operator. I will also discuss possible implications for dark energy observation.