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

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 $\propto Hr^{-3/2}$ 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) $\langle\Delta u\rangle=-Hr$. A generalized SCH is developed for higher order moments $\langle\Delta u^{2m+1}\rangle=-(2m+1)\langle\Delta u^{2m}\rangle Hr$ 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.