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pa(Ω)=ca g(u),g(u)=24 ue2πu−1,u=cΩap_a(\Omega) = \frac{c}{a}\, g(u), \qquad g(u) = \frac{24\,u}{e^{2\pi u}-1}, \qquad u = \frac{c\Omega}{a}

Why this formula appears here

Write the normalized accelerated spectrum in terms of the dimensionless ratio u ≡\equiv cΩ\Omega/a , using the exact Bose-Einstein shape derived above. Carrying out the normalization integral, ∫0∞\int_0^\infty Ω\Omega\,(e2πcΩ/ae^{2\pi c\Omega/a}-1)^{-1}\,dΩ\Omega = a2a^2/(24c2c^2) , using the standard result ∫0∞\int_0^\infty x(exe^x-1)^{-1}dx=π2\pi^2/6 , gives the scale-family form pa(Ω)=ca g(u),g(u)=24 ue2πu−1,u=cΩap_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+δ\delta a , standard calculus for a scale family gives, to leading order in δ\delta a ,

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e2πue^{2\pi u}

Symbol e^2pi u

e2e^2pi u occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

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e2πu−1e^{2\pi u}-1

Denominator: e^2pi u-1

The complete quantity below the fraction bar; it must be nonzero for this division.

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With a fixed numerator, increasing a nonzero denominator reduces the fraction. Read it with the definitions, units, and assumptions supplied by the article.

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Published contexts (1)

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pa(Ω)=ca g(u),g(u)=24 ue2πu−1,u=cΩa.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}.

Equation 76 · Evolutionary Physics

No Particle Without a Cosigner

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

Write the normalized accelerated spectrum in terms of the dimensionless ratio u ≡\equiv cΩ\Omega/a , using the exact Bose-Einstein shape derived above. Carrying out the normalization integral, ∫0∞\int_0^\infty Ω\Omega\,(e2πcΩ/ae^{2\pi c\Omega/a}-1)^{-1}\,dΩ\Omega = a2a^2/(24c2c^2) , using the standard result ∫0∞\int_0^\infty x(exe^x-1)^{-1}dx=π2\pi^2/6 , gives the scale-family form pa(Ω)=ca g(u),g(u)=24 ue2πu−1,u=cΩap_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+δ\delta a , standard calculus for a scale family gives, to leading order in δ\delta a ,

Meanings in this article

  • uu: exactly twice that maximum.
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