Evolution of energy, momentum, and spin parameter in dark matter flow and integral constants of motion

N-body equations of motion in comoving system and expanding background are reformulated in a transformed system with static background and fixed damping. The energy and momentum evolution in dark matter flow are rigorously formulated for both systems. The energy evolution in transformed system has a simple form that is identical to the damped harmonic oscillator. The cosmic energy equation can be easily derived in both systems. For entire N-body system, 1) combined with the two-body collapse model (TBCM), kinetic and potential energy increase linearly with time $t$ such that $K_p=\varepsilon_ut$ and $P_y=-7\varepsilon_ut/5$, where $\varepsilon_u$ is a constant rate of energy cascade; 2) an effective gravitational potential exponent $n_e=-10/7\ne-1$ ($n_e=-1.38$ from simulation) can be identified due to surface energy of fast growing halos; 3) the radial momentum $G\propto a^{3/2}$ and angular momentum $H\propto a^{5/2}$, where $a$ is the scale factor. On halo scale, 1) halo kinetic and potential energy can be modelled by two dimensionless constants $α_s^*$ and $β_s^*$. Both constants are independent of time and halo mass; 2) both halo radial and angular momentum $\propto a^{3/2}$ and can be modeled by two mass-dependent coefficients $τ_s^*$ and $η_s^*$; 3) halo spin parameter is determined by $α_s^*$ and $η_s^*$ and decreases with halo mass with derived values of 0.09 and 0.031 for small and large halos. Finally, the radial and angular momentum are closely related to the integral constants of motion $I_m$, i.e. the integral of velocity correlation or the $m$th derivative of energy spectrum at long wavelength limit. On large scale, angular momentum is negligible, $I_2$=0 reflects the conservation of linear momentum, while $I_4$ reflects the fluctuation of radial momentum $G$. On halo scale, $I_4$ is determined by both momentum that are comparable with each other.

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