Fractal Structure of Isothermal Lines and Loops on the Cosmic Microwave Background

The statistics of isothermal lines and loops of the cosmic microwave background (CMB) radiation on the sky map is studied and the fractal structure is confirmed in the radiation temperature fluctuation. We estimate the fractal exponents, such as the fractal dimension De of the entire pattern of isothermal lines, the fractal dimension Dc of a single isothermal line, the exponent in Korčak’s law for the size distribution of isothermal loops, the two kind of Hurst exponents, He for the profile of the CMB radiation temperature, and Hc for a single isothermal line. We also perform fractal analysis of two artificial sky maps simulated by a standard model in physical cosmology, the WMAP best-fit cold dark matter ( CDM) model, and by the Gaussian free model of rough surfaces. The temperature fluctuations of the real CMB radiation and in the simulation using the CDM model are non-Gaussian, in the sense that the displacement of isothermal lines and loops has an antipersistent property indicated by He ’ 0:23 < 1=2.