Two-body collapse model for self-gravitating flow of dark matter and generalized stable clustering hypothesis for pairwise velocity

Abstract

Analytical tools are extremely hard to find for non-linear gravitational collpase. Only a few simple but powerful tools exist so far. Two examples are the spherical collapse model (SCM) and stable clustering hypothesis (SCH). We present a new analytical tool, a two-body collapse model (TBCM), that plays the same fundamental role as harmonic oscillator in dynamics. For convenience, TBCM is formulated for gravity with any potential exponent $n$ in a static background with a fixed damping ($n$=-1 for Newtonian gravity). The competition between gravity, expanding background (or damping), and angular momentum classifies two-body collapse into: 1) free fall collapse, where free fall time is greater if same system starts to collapse at earlier time; 2) equilibrium collapse that persists longer in time, whose perturbative solutions lead to power-law evolution of system energy and momentum. Two critical values ${\beta }_{s1}=1$ and ${\beta }_{s2}=1/3\pi$ are identified that quantifies the competition between damping and gravity. Value ${\beta }_{s2}$ only exists for discrete values of potential exponent $n=(2-6m)/(1+3m)=$ -1,-10/7... for integer $m$. Critical density ratio (${\Delta }_c=18{\pi }^2$) is obtained for $n$=-1 that is consistent with SCM. TBCM predicts angular velocity $\propto H{r}^{-3/2}$ for two-body system of size $r$. The isothermal density is a result of extremely fast mass accretion. TBCM is able to demonstrate SCH, i.e. mean pairwise velocity (first moment) $⟨\Delta u⟩=-Hr$. A generalized SCH is developed for higher order moments $⟨\Delta u^{2m+1}⟩=-(2m+1)⟨\Delta u^{2m}⟩Hr$ that is validated by N-body simulation. Energy evolution in TBCM is independent of particle mass and energy equipartition does not apply. TBCM can be considered as a non-radial SCM. Both models predict the same critical density ratio, while TBCM contains much richer information.

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Cited by in corpus (1)

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