subscript

The passage around this formula

Write the normalized accelerated spectrum in terms of the dimensionless ratio u $\equiv$ c$\Omega$/a , using the exact Bose-Einstein shape derived above. Carrying out the normalization integral, $\int_0^\infty$ $\Omega$\,($e^{2\pi c\Omega/a}$-1)^{-1}\,d$\Omega$ = $a^2$/(24$c^2$) , using the standard result $\int_0^\infty$ x($e^x$-1)^{-1}dx=$\pi^2$/6 , gives the scale-family form $p_a(\Omega) = \frac{c}{a}\, g(u), \qquad g(u) = \frac{24\,u}{e^{2\pi u}-1}, \qquad u = \frac{c\Omega}{a}$. Every accelerated detector’s normalized spectrum is the same universal curve g , only rescaled by a/c . For two accelerations separated by a small increment, a and a+$\delta$ a , standard calculus for a scale family gives, to leading order in $\delta$ a ,

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