==================== Explanation by words ==================== - 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 =================================== Explanation by the minimal equation =================================== 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. =============================== Dark energy => Potential decays =============================== 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"