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

TL;DR

This study analyzes the fractal properties of isothermal lines in the CMB, confirming their fractal structure and comparing real data with simulations to reveal non-Gaussian features.

Contribution

It provides the first detailed fractal analysis of CMB isothermal lines, estimating multiple fractal and Hurst exponents and comparing real and simulated sky maps.

Findings

CMB isothermal lines exhibit fractal structure.

Real CMB and $ m extbf{ extit{ extLambda}}$CDM simulations show non-Gaussian, antipersistent behavior.

Fractal and Hurst exponents differ between real data and models.

Abstract

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 $D_{\mathrm{e}}$ of the entire pattern of isothermal lines, the fractal dimension $D_{\mathrm{c}}$ of a single isothermal line, the exponent $\zeta$ in Kor\v{c}ak's law for the size distribution of isothermal loops, the two kind of Hurst exponents, $H_{\mathrm{e}}$ for the profile of the CMB radiation temperature, and $H_{\mathrm{c}}$ 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 $\Lambda$ Cold Dark Matter ($\Lambda$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 $\Lambda$CDM model are non-Gaussian, in the sense that the displacement of isothermal lines and loops has an antipersistent property indicated by $H_{\mathrm{e}} \simeq 0.23 < 1/2$.