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Explanation by words
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- Expansion time scale = 1/H [where H = \dot{a}/a]
- Free-fall time = 1/sqrt(G \rho_matter)
During the matter-dominated era: The Friedmann equation gives
H^2 = (8\pi G/3)\rho_matter
Thus,
1/H ~ 1/sqrt(G \rho_matter) ~ Free-fall time
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Explanation by the minimal equation
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During this period, the density fluctuation defined by
(\rho - \rho_mean)/\rho_mean
is proportional to the scale factor.
From this, and using the Poisson equation, you can derive
\Phi = constant.
- Derivation -
\Phi is proportional to a^2(\rho - \rho_mean), which is proportional to
\Phi ~ a^2\rho_mean * a
Since \rho_mean ~ 1/a^3,
i.e., \Phi ~ constant.
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Dark energy => Potential decays
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By words:
DE makes the Universe expand faster. Faster than the gravity attracts. Namely, the expansion time scale is shorter than the free-fall time. Thus, matter can't cluster anymore, making \Phi decay.
The Friedmann equation is
H^2 = (8\pi G/3)[\rho_matter + \rho_darkenergy] ~ (8\pi G/3)\rho_darkenergy
=> 1/H ~ 1/sqrt(G \rho_darkenergy)
Free-fall time ~ 1/sqrt(G \rho_matter)
During the DE era, \rho_darkenergy > \rho_matter. Thus,
1/H < Free-fall time => Expansion is "too fast"