A non-radial two-body collapse model (TBCM) for gravitational collapse
  of dark matter in expanding background and generalized stable clustering
  hypothesis (GSCP)
  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.
