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Analysis of distribution of cosmic microwave background photon in terms of non-extensive st.pdf

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Analysis of distribution of cosmic microwave background photon in terms of non-extensive st

a r X i v : c o n d - m a t / 0 4 0 7 7 9 2 v 1 [ c o n d - m a t .s t a t - m e c h ] 3 0 J u l 2 0 0 4 Analysis of distribution of cosmic microwave background photon in terms of non-extensive statistics and formulas with temperature fluctuation M. Biyajima, M. Kaneyama, Y. Kurashima, and T. Yamashita Department of physics, Shinshu University, Matsumoto 390-0821, Japan T. Mizoguchi Toba National College of Maritime Technology, Toba 517-8501, Japan (Dated: February 2, 2008) Abstract To take into account the temperature fluctuation in the Planck distribution, we calculate con- volution integral with several probability distributions. Using these formula as well the Planck distribution and a formula in the non-extensive statistics, we analyze the data measured by the Cosmic Background Explorer (COBE). Our analysis reveals that the derivation from the Planck distribution is estimated as |q?1| = 4.4×10?5, where q means the magnitude of the non-extensivity or the temperature fluctuation, provided that the dimensionless chemical potential proposed by Zeldovich and Sunyaev exists. Comparisons of new formulas and the Planck distribution including the Sunyaev-Zeldovich (S-Z) effect are made. PACS numbers: 05.10.Gg, 05.30.Jp, 98.70.Vc 1 I. INTRODUCTION One of interesting subjects in thermodynamics is relating to the non-extensive statistics [1, 2, 3]. Several years ago, authors of Refs. [4, 5] showed that formulas in the non-extensive statistics, i.e., the temperature fluctuations, are calculated by the convolution integral with the gamma distribution [4] and a calculation with a distribution described by exp(?2|u|α), where α is fractional number [5]. In this report, we calculate the temperature fluctuation in the Planck distribution DP lanck(β, ν, μ) = CBν 3 eβω+μ ? 1 , (1) where CB = 8πh/c 3, ω = hν and β = 1/kBT . μ denotes the dimensionless chemical potential introduced by Zeldovich and Sunyaev [6]. Equation (1) is applied to the data measured by the Cosmic Backgr