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

TL;DR

The paper introduces a cascade framework for self-gravitating collisionless dark matter flow, revealing a universal transition range around a characteristic halo mass $m_h^*$ where gravity dominates and memory of the primordial spectrum is erased. It provides a dual-scale picture: a mass cascade in halo space producing a universal double-λ halo mass function and an energy cascade in halo interiors yielding a universal -4/3 density slope near $r_s$, both supported by N-body simulations (e.g., Illustris, Virgo). The global inverse mass cascade and local energy cascade combine to form a statistically steady state with scale-independent fluxes, maximizing entropy while continuing energy transfer. From this framework, the authors derive constraints on dark matter particle mass (suggesting heavy, potentially superheavy candidates) and connect observable halo statistics to the microphysics of dark matter, offering a testable, first-principles route to understanding nonlinear structure formation.

Abstract

Between the linear and nonlinear regimes, we identify a universal transition range centered on a characteristic halo mass $m_h^*\propto t$, 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 $m_h^*$ 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 $τ_g\propto m_h^{-λ}$; A Fokker-Planck equation gives mass function $f_M\propto m_h^{-λ}$ and $λ=2/3$ for the gravity-dominant transition range. Locally, a radially directed cascade governs particle migration with waiting time $τ_{gr}\propto r^{-γ}$, yielding density $ρ_r\propto r^{-2γ}$ and $γ=2/3$ on scales near $m_h^*$. 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-$λ$ mass functions and double-$γ$ density in excellent agreement with simulations. Using Illustris and Virgo, we measure an inverse kinetic-energy cascade from small to large scales at $\varepsilon_u \approx -10^{-7}$m$^2$/s$^3$, a direct potential-energy cascade of $-1.4\varepsilon_u$, and a net dissipation of -0.4$\varepsilon_u$ 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.

Figures (23)

• Figure 1: The density power spectrum $P(k,z)$ with comoving wavenumber $k$ at different redshifts z from Virgo SCDM simulations. The dotted line denotes the linear power spectrum $P_L(k)$. The figure presents three distinct regimes and two critical scales: 1) a linear regime on scales $k<k_{NL1}$, where scale $k_{NL1}$ denotes the deviation from the linear theory; 2) a universal transition range (thick solid lines) on scales $k_{NL1}<k<k_{NL2}$ with $P(k)\propto k^{-1}$, where the gravitational dynamics dominate over the the initial spectrum; 3) a fully nonlinear range on scales $k>k_{NL2}$ that retains the memory of the initial spectrum. While both scales decrease with time, the ratio $k_{NL2}/k_{NL1}$ increases with time, reflecting the expanding universal range. The scale $k_h^*\approx$ 1h/Mpc corresponds to the size of haloes of characteristic mass $m^*_h$ at z=0, such that these haloes are primarily dominated by the gravitational dynamics, not the initial conditions. This paper focuses on the universal range and associated mass and energy cascade across scales, and their effects on structure formation and evolution.
• Figure 2: The power spectrum $P(k,z)$ at $z$=0 from different Virgo simulations. Symbols plot the power spectrum with various cosmologies, exhibiting a universal transition range $P(k)\propto k^{-1}$ on scales $k_{NL1}<k<k_{NL2}$, where the gravitational dynamics dominate over the effect of the initial spectrum. Solid lines present the power spectrum from a power-law initial spectrum $P_{ini}(k)\propto k^{n_{i}}$, with $n_{i}$= -2, -1.5, -1, and 0. In the fully nonlinear range on scales $k>k_{NL2}$, the spectrum $P(k)$ retains the memory of the initial spectrum and develops an asymptotic slope $n_{a}$ as a function of the initial spectrum index $n_{i}$ (See Fig. \ref{['fig:777']}).
• Figure 3: The variation of asymptotic slope $n_a$ of nonlinear spectrum $P(k)$ with the slope $n_{i}$ of initial power-law spectrum. Symbols plot the simulation results. The dashed line plots Peebles' prediction without involving a universe transition range ($k_{NL1}=k_{NL2}$ and $\alpha_x=0$). The discrepancy with simulation data suggests a missing piece in that model. We propose that the existence of a universal range also impacts the asymptotic slope $n_a$, leading to a better agreement with simulations (Eq. \ref{['eq:2-2-2-9']}). Solid lines plots the prediction with a ratio $k_{NL2}/k_{NL1}=a^{\alpha_x}$, where $\alpha_x=1/2$ is preferred.
• Figure 4: Schematic plot for the halo random walk and mass and energy cascade in halo mass space. Haloes of mass $M$ merging with a single merger (free particles of mass $m$) leads to a mass flux to a larger scale $M+m$ (smooth accretion), that is, the halo of mass $M$ walking into the next mass scale $M+m$ after merging. For a given halo of mass $m_h$, a merge occurs with an average waiting time $\tau_g(m_h,z)$. Because haloes have finite mass and kinetic energy, continuous merging facilitates an inverse mass and kinetic energy cascade from small to large scales and a direct cascade of potential energy from large to small scales. The scale-independent mass and energy flux ($\varepsilon_m$ and $\varepsilon_u$ are independent of mass scale $m_h$) are expected in the propagation range ($m=m_p<m_h<m_h^*$), a typical feature of a statistical steady state in a nonequilibrium system. The rate of cascade becomes scale-dependent in the deposition range ($m_h>m_h^*$) (see Figs. \ref{['fig:3-2']}, \ref{['fig:3-3']} for $\varepsilon_m$ and Fig. \ref{['fig:3-7']} for $\varepsilon_u$).
• Figure 5: Comparison between different mass functions (log(f($\nu$))) in Appendix \ref{['sec:A3']} and cosmological simulations. The black square represents the mass function from the Virgo SCDM simulation at z=0. The color symbols represent the mass function from the Illustris simulation at different redshifts $z$. The PS mass function underestimates the mass in large haloes. The fitted JK mass function matches the simulation for a given range of halo size, but not the entire range. The double-$\lambda$ mass function (Eq. \ref{['eq:3-20']}) matches both the simulation and ST mass function for the entire range.