On a critical acceleration scale of dark matter in Lambda-CDM and dynamical dark energy

Abstract: Universal acceleration $a_0$ emerges in various empirical laws, yet its fundamental nature remains unclear. Using Illustris and Virgo N-body simulations, we propose $a_0$ is the scale of acceleration fluctuations in collisionless dark matter involving long-range gravity. In contrast, in the kinetic theory of gases, molecules undergo random elastic collisions involving short-range interactions, where only velocity fluctuations are relevant. We identify the redshift evolution $a_0\propto (1+z)^{3/4}$ that 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 baryonic mass rotate faster at a higher redshift. The velocity fluctuations involve a critical velocity $u_c\propto (1+z)^{-3/4}$. The acceleration fluctuations involve a critical acceleration $a_c\propto (1+z)^{3/4}$. Two critical quantities are related by the rate of energy cascade $\varepsilon_{u}\approx -{a_c u_c/[2(3\pi)^2]}$, where factor $3\pi$ is from the angle of incidence and $\varepsilon_u\approx -10^{-7}$m$^2$/s$^3$. With critical velocity $u_c$ on the order of 300 km/s at $z=0$, the critical acceleration is determined to be $a_{c0}\equiv a_c(z=0) \approx 10^{-10}$m/s$^2$, suggesting $a_c$ might explain the universal acceleration $a_0\approx 10^{-10}$m/s$^2$ in the empirical Tully-Fisher relation or modified Newtonian dynamics (MOND). Note that dark energy (DE) density $\rho_{DE0}\approx {a_{c0}^{2}/G}=10^{-10}$J/m$^3$, we postulate an entropic origin of the dark energy from acceleration fluctuations of dark matter, in analogy to the gas pressure from velocity fluctuations. This leads to a dynamical dark energy coupled to the structure evolution involving a relatively constant DE density followed by a slow weakening phase, suggesting possible deviations from the standard $\Lambda$CDM.