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 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
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