Deterministic chaos in the large-scale universe: data versus speculations

Abstract

It is shown, that the both angular CMB Doppler spectrum: $C_l$ (Planck space telescope - cosmic microwave background [1]) and the 3D galaxy-galaxy power spectrum: $P(k)$ (Sloan Digital Sky Survey SDSS-II [2]), exhibit a considerable range with an exponential decay: $370 < l < 2500$ and $0.05 < k < 0.27~~(h/Mpc)$, respectively. The rates of the exponential decay are $l_c \simeq 300$ for $C_l \sim \exp-(l/l_c)$ and $k_c \simeq 0.09~~(h/Mpc)$ for $P(k) \sim \exp-(k/k_c)$. A waviness is observed along a straight line representing the exponential decay in the log-linear scale graphs of these spectra. In both cases the waviness has period (distance between peaks) equal to the same $l_c$ and $k_c$ as for the exponential decay. It means that the waviness is generated by the same, presumably chaotic, mechanism that generates the exponential decay. A more complex, distributed, chaos is observed in the Baryon Oscillation Spectroscopic Survey (the largest component of the SDSS-III containing nearly one million galaxies) [3]. In this case the $P(k)$ spectrum is a weighted superposition of the exponentials. At assumption that dynamics of the dispersive waves, driving the distributed chaos, is dominated by the effects of a surface tension this weighted superposition of the exponentials is converged to a compact form of a stretched exponential $P(k) \sim \exp-(k/k_b)^{1/2}$, in good agreement with the data.