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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 for structure formation and evolution. This paper focus on the baryonic-to-halo mass ratio (BHMR) and its evolution. We first review unique properties of self-gravitating collisionless dark matter flow (SG-CFD), followed by their application to derive BHMR. To maximize system entropy, the long-range interaction requires a broad size of halos to be formed. These halos facilitate inverse mass and energy cascade from small to large scales with a constant rate of energy cascade $\varepsilon_u$. In addition, dark matter flow exhibits scale-dependent flow behaviors that is incompressible on small scale and irrotational on large scale. With these properties and considering a given halo with a total baryonic mass $m_b$, halo mass $m_h$, halo virial size $r_h$, and flat rotation speed $v_f$, BHMR can be analytically derived by combining the baryonic Tully-Fisher relation and constant $\varepsilon_u$ in small and large halos. A maximum BHMR ratio ~0.076 is found for halos with a critical mass $m_{hc}\sim 10^{12}M_{\odot}$ 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 $\varepsilon_u\propto v_f/r_h$, $v_f\propto r_h$, and $m_b\propto m_h^{4/3}$; ii) for large halos with mass $m_h>m_{hc}$, we have $\varepsilon_u\propto v_f^3/r_h$, $v_f\propto r_h^{1/3}$, and $m_b\propto m_h^{4/9}$. Combined with double-$λ$ halo mass function, the average BHMR ratio in all halos (~0.024 at z=0) can be analytically derived, along with its redshift evolution. The fraction of total baryons in all galaxies is ~7.6% at z=0 and increases with time $\propto t^{1/3}$. The SPARC (Spitzer Photometry \& Accurate Rotation Curves) data with 175 late-type galaxies were used for derivation and comparison.