Universal acceleration a _0 emerges in various empirical laws, yet its fundamental nature remains unclear. Using Illustris and Virgo N -body
Universal acceleration a _0 emerges in various empirical laws, yet its fundamental nature remains unclear. Using Illustris and Virgo N -body simulations, we focus on the velocity and acceleration fluctuations in collisionless dark matter involving long-range gravity. For comparison, in the kinetic theory of gases, molecules undergo random elastic collisions involving short-range interactions, where only velocity fluctuations are relevant. Hierarchical structure formation proceeds through the merging of smaller halos to form larger halos, which facilitates a continuous energy cascade from small to large halos at a constant rate ε _u ≈ −10 ^−7 m ^2 s ^−3 . Velocity fluctuations involve a critical velocity u _c ∝ (1 + z ) ^−3/4 . Acceleration fluctuations involve a critical acceleration a _c ∝ (1 + z ) ^3/4 . Two critical quantities are related by the rate of energy cascade ε _u ≈ − a _c u _c /[2(3 π ) ^2 ], where factor 3 π is from the angle of incidence during merging. With critical velocity u _c on the order of 300 km s ^−1 at z = 0, the critical acceleration is determined to be a _c _0 ≡ a _c ( z = 0) ≈ 10 ^−10 m s ^−2 , suggesting a _c might explain the universal acceleration a _0 ≈ 10 ^−10 m s ^−2 in the empirical Tully–Fisher relation or modified Newtonian dynamics. The redshift evolution a _c ∝ (1 + z ) ^3/4 is in good agreement with Magneticum and EAGLE simulations and in reasonable agreement with limited observations. This suggests a larger a _0 at a higher redshift such that galaxies of fixed mass rotate faster at a higher redshift. Note that for dark energy (DE) density ${\rho }_{\mathrm{DE}0}\approx {a}_{c0}^{2}/G=1{0}^{-10}$ J m ^−3 , we postulate an entropic origin of the DE from acceleration fluctuations of dark matter, analogous to the gas pressure from velocity fluctuations. This leads to a dynamical DE coupled to the structure evolution involving a relatively constant DE density followed by a slow weakening phase, suggesting possible deviations from the standard ΛCDM paradigm.
