Inverse mass cascade of self-gravitating collisionless flow and effects
  on halo mass functions
  The inverse mass cascade is proposed for statistically steady state of
self-gravitating collisionless flow (SG-CFD). The continuous mass transfer from
small to large mass scales (inverse cascade) is formulated. Direct effect on
mass functions is discussed. Mass cascade is local, two-way, and asymmetric in
mass space. Halos inherit/pass their mass from/to halos of similar size. Two
distinct ranges are identified: a propagation range with a scale-independent
rate of mass transfer and a deposition range with cascaded mass consumed to
grow halos. Dimensional analysis leads to a power-law mass function in
propagation range with a geometry exponent (${\lambda}$). A fundamental merging
frequency $f_0{\sim}m_p^{\lambda-1}a^{-1}$ is identified, where $a$ is the
scale factor. The particle mass $m_p$ can be determined if frequency is known.
The rate of mass transfer ${\epsilon}_m{\sim}a^{-1}$ is independent of halo
mass (key feature of propagation range). Typical halos grow as
$m_h{\sim}a^{3/2}$ and the waiting time (halo lifespan) scales as
${\sim}m_h^{-\lambda}$. Chain reaction of mass cascade provides non-equilibrium
system (SG-CFD) a mechanism to continuously release energy and maximize
entropy. A continuous injection of mass ("free radicals") at the smallest scale
is required to sustain the everlasting inverse mass cascade such that the total
halo mass $M_h$ increases as $a^{1/2}$. These "radicals" might be directly
generated at the smallest Planck scale or by a direct cascade from large to
small scales. The entire mass cascade can be formulated by random walk in mass
space, where halos migrate with an exponential distribution of waiting time.
This results in a heterogeneous diffusion model, where mass function can be
fully derived without relying on specific collapse models. A double-$\lambda$
mass function is proposed with different $\lambda$ for two ranges and compared
to simulations.
