The baryonic-to-halo mass relation from mass and energy cascade in
  self-gravitating collisionless dark matter flow
  The relation between properties of galaxies and dark matter halos they reside
in can be valuable to understand structure formation and evolution,
particularly the baryonic-to-halo mass ratio (BHMR) and its dynamic evolution.
We first review some unique properties of self-gravitating collisionless dark
matter flow (SG-CFD), followed by their implications in derivation of BHMR. To
maximize system entropy, the long-range interaction requires a broad size of
halos to be formed. These halos facilitate an inverse mass and energy cascade
from small to large mass scales that involves a constant rate of energy
transfer $\epsilon_u$=$-4.6\times10^{-7}m^2/s^3$. Considering galaxies and
halos with a total baryonic mass $m_b$, halo mass $m_h$, halo virial size
$r_h$, and flat rotation velocity $v_f$, the baryonic-to-halo mass relation can
be analytically derived by combining the baryonic Tully-Fisher relation and
constant rate of energy cascade $\epsilon_u$. We found a maximum BHMR ratio
~0.076 for halos with a critical mass $m_{hc}$~$10^{12}M_{sun}$ at z=0. That
ratio is much lower for both smaller and larger halos such that two regimes can
be identified: i) for incompressible small halos with mass $m_h$<$m_{hc}$, we
have $\epsilon_u$$\propto$$v_f/r_h$, $v_f$$\propto$$r_h$, and
$m_b$$\propto$$m_h^{4/3}$; ii) for large halos with mass $m_h$>$m_{hc}$, we
have $\epsilon_u$$\propto$$v_f^3/r_h$, $v_f$$\propto$$r_h^{1/3}$, and
$m_b$$\propto$$m_h^{4/9}$. Combined with proposed double-$\lambda$ halo mass
function, the average BHMR ratio for all halos (~0.024 at z=0) can be
analytically derived, along with its redshift evolution. The fraction of
baryons in galaxy is ~7.6% at z=0 and increases with time $\propto$$t^{1/3}$.
Most (~92.4%) of the baryons are not in galaxies. The SPARC (Spitzer Photometry
& Accurate Rotation Curves) data with ~175 late-type galaxies were used for the
derivation and comparison.
