Universal spectrum and scaling laws for halo mass function, structure, and dark matter mass constraints

Abstract

Between the linear and nonlinear regimes, we identify a universal transition range centered on a characteristic halo mass $m_h^{\ast }\propto t$, within which gravitational dynamics self-organize the matter field toward an effective spectral index n=-1. In a bottom-up hierarchy, early collapse of low-mass halos preserves imprints of the primordial spectrum, whereas prolonged assembly of halos near $m_h^{\ast }$ erases that memory and establishes universality. We formulate a scale-to-scale cascade, the redistribution of mass and energy across scales, that yields universal scaling laws for the halo mass function and internal structure. Globally, the cascade drives a random walk of halos with mass-dependent waiting time $\ddot{I}{„}_g\propto m_h^{-\hat{I}»}$; A Fokker-Planck equation gives mass function $f_M\propto m_h^{-\hat{I}»}$ and $\hat{I}»=2/3$ for the gravity-dominant transition range. Locally, a radially directed cascade governs particle migration with waiting time $\ddot{I}{„}_{gr}\propto r^{-{\hat{I}}^3}$, yielding density $\ddot{I}{}_r\propto r^{-2{\hat{I}}^3}$ and ${\hat{I}}^3=2/3$ on scales near $m_h^{\ast }$. The cascade drives the system toward a statistically steady state that continuously releases energy and maximizes entropy, characterized by scale-independent rates, preventing mass or energy buildup at intermediate scales. Scale-dependent dominance of the primordial spectrum versus gravity implies two effective exponents, producing double-$\hat{I}»$ mass functions and double-${\hat{I}}^3$ density in excellent agreement with simulations. Using Illustris and Virgo, we measure an inverse kinetic-energy cascade from small to large scales at ${\epsilon }_u\approx -10^{-7}$m${}^2$/s${}^3$, a direct potential-energy cascade of $-1.4{\epsilon }_u$, and a net dissipation of -0.4${\epsilon }_u$ via halo mergers and particle migration. The dependence of waiting time and step length on the particle mass suggests new constraints near $10^{12}$GeV.

Updated version