  A concise explanation of studies on distortion of space-time dimension is
briefly introduced. Second we obtain the limits (i.e., bounded values) of the
dimensionless chemical potential $\mu$, the Sunyaev--Zeldovich (SZ) effect y
and distortion of the space-dimension $\varepsilon$ by Monte Carlo (MC)
analysis of the parameter set (T, $d=3+\varepsilon$, $\mu$, and $y$) in cosmic
microwave data assuming that the SZ effect is positive (y>0). In this analysis,
the magnitude of the space-dimension d with distortion of the space-dimension
$\varepsilon$ is defined by $d=3+\varepsilon$. The limits of $\mu$ and $y$ are
determined as $|\mu| < 9\times 10^{-5} (2\sigma$) ($\mu = (-3.9\pm 2.6)\times
10^{-5} (\sigma$)), $|y| < 5\times 10^{-6} (2\sigma$) ($y = (2.0\pm 1.4)\times
10^{-6} (\sigma$)), while the distortion of the space-dimension is
$|\varepsilon| < 6\times 10^{-5} (2\sigma$) ($\varepsilon = (-0.78\pm
2.50)\times 10^{-5} (\sigma$)). The magnitudes of these three estimated limits
are ordered as $|\mu| \ge |\varepsilon| > |y|$. The estimated limit of $|y| <
5\times 10^{-6}$ appears to be related to re-ionization processes occurring at
redshift $z_{ri}\sim 10$. We also present data analysis assuming a relativistic
SZ effect.
