http://arxiv.org/abs/2110.05784

Analytical tools are valuable to study gravitational collapse. However, solutions are hard to find due to the highly non-linear nature. 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 tool based on an elementary step of mass cascade, i.e. a two-body collapse model. TBCM plays the same role as harmonic oscillator in dynamics and can be fundamental to understand structure evolution. For convenience, TBCM is formulated for gravity with any exponent $n$ in a static background with fixed damping. The competition between gravity, expanding background (or damping), and angular momentum classifies two-body collapse into: 1) free fall collapse for weak angular momentum, where free fall time is greater if same system starts to collapse at earlier time; 2) equilibrium collapse for weak damping that persists longer in time, where 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 n=(2-6m)/(1+3m)= -1,-10/7…for integer m. Critical density ratio ($18\pi^2$) is obtained for $n$=-1 that is consistent with SCM. TBCM predicts angular velocity $\propto Hr^{-3/2}$ with r. The isothermal density is a result of infinitesimal halo lifetime. TBCM is able to demonstrate SCP, i.e. mean pairwise velocity (first moment) $\langle\Delta u\rangle=-Hr$. A generalized SCH is developed for higher order moments $\langle\Delta u^{2m+1}\rangle=-(2m+1)\langle\Delta u^{2m}\rangle Hr$. Energy evolution in TBCM is independent of mass and energy equipartition does not apply. TBCM can be considered as a non-radial SCM. Both models predict same critical ratio, while TBCM contains much richer information.

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Z. Xu
Wed, 13 Oct 21
39/80

Comments: 6 figures, 1 table