分类： 天文学 >> 天文学 提交时间： 2023-02-19
摘要： 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 $\alpha_s^*$ and $\beta_s^*$. 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 $\tau_s^*$ and $\eta_s^*$; 3) halo spin parameter is determined by $\alpha_s^*$ and $\eta_s^*$ 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.
分类： 天文学 >> 天文学 提交时间： 2023-02-19
摘要： 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.
分类： 天文学 >> 天文学 提交时间： 2023-02-19
摘要： The halo-mediated inverse cascade is a key feature of the intermediate
statistically steady state for self-gravitating collisionless dark matter flow
(SG-CFD). How the inverse mass and energy cascade maximize system entropy and
develop limiting velocity/energy distributions are fundamental questions to
answer. We present a statistical theory concerning the maximum entropy
distributions of velocity, speed, and energy for system involving a power-law
interaction with an arbitrary exponent $n$. For $-2
分类： 天文学 >> 天文学 提交时间： 2023-02-19
摘要： By decomposing velocity dispersion into non-spin and spin-induced, mean flow and dispersion are analytically solved for axisymmetric rotating and growing halos. The polar flow can be neglected and azimuthal flow is directly related to dispersion. The fictitious ("Reynolds") stress acts on mean flow to enable energy transfer from mean flow to random motion and maximize system entropy. For large halos (high peak height $\nu$ at early stage of halo life) with constant concentration, there exists a self-similar radial flow (outward in core and inward in outer region). Halo mass, size and specific angular momentum increase linearly with time via fast mass accretion. Halo core spins faster than outer region. Large halos rotate with an angular velocity proportional to Hubble parameter and spin-induced dispersion is dominant. All specific energies (radial/rotational/kinetic/potential) are time-invariant. Both halo spin ($\sim$0.031) and anisotropic parameters can be analytically derived. For "small" halos with stable core and slow mass accretion (low peak height $\nu$ at late stage of halo life), radial flow vanishes. Small halos rotate with constant angular velocity and non-spin axial dispersion is dominant. Small halos are spherical in shape, incompressible, and isotropic. Radial and azimuthal dispersion are comparable and greater than polar dispersion. Due to finite spin, kinetic energy is not equipartitioned with the greatest energy along azimuthal direction. Different from normal matter, small halos are hotter with faster spin. Halo relaxation from early to late stage involves variation of shape, density, mean flow, momentum, and energy. During relaxation, halo isotopically "stretches" with conserved specific rotational kinetic energy, increasing concentration and momentum of inertial. Halo "stretching" leads to decreasing angular velocity, increasing angular momentum and spin parameter.
分类： 天文学 >> 天文学 提交时间： 2023-02-19
摘要： The halo-mediated inverse mass cascade is a key feature of the intermediate statistically steady state for self-gravitating collisionless dark matter flow (SG-CFD). A broad spectrum of halos and halo groups are necessary to form from inverse mass cascade for long-range interaction system to maximize its entropy. The limiting velocity ($\textbf X$), speed ($\textbf Z$), and energy ($\textbf E$) distributions of collisionless particles can be obtained analytically from a maximum entropy principle. Halo mass function, the distribution of total mass in halos, is a fundamental quantity for structure formation and evolution. Instead of basing mass functions on simplified spherical/elliptical collapse models, it is possible to reformulate mass function as an intrinsic distribution to maximize system entropy during the everlasting statistically steady state. Starting from halo-based description of non-equilibrium dark matter flow, distributions of particle virial dispersion ($\textbf H$), square of particle velocity ($\textbf P$), and number of halos ($\textbf J$) are proposed. Their statistical properties and connections with velocity distribution ($\textbf X$) are well studied and established. With $\textbf H$ being essentially the halo mass function, two limiting cases of $\textbf H$ distribution are analyzed for large halos ($\textbf H_\infty$) and small halos ($\textbf H_s$), respectively. For large halos, $\textbf H_\infty$ is shown to also be a maximum entropy distribution. For small halos, $\textbf H_s$ approximates the $\textbf P$ distribution and recovers the Press-Schechter mass function. The full solution of $\textbf H$ distribution is determined by the velocity distribution ($\textbf X$) that maximizes system entropy and the exact model of halo velocity dispersion.
分类： 天文学 >> 天文学 提交时间： 2023-02-19
摘要： Statistical theory for self-gravitating collisionless dark matter flow is not fully developed because of 1) intrinsic complexity involving constant divergence flow on small scale and irrotational flow on large scale; 2) lack of self-closed description for peculiar velocity; and 3) mathematically challenging. To better understand dark matter flow, kinematic and dynamic relations must be developed for different types of flow. In this paper, a compact derivation is presented to formulate general kinematic relations of any order for incompressible, constant divergence, and irrotational flow. Results are validated by N-body simulation. Dynamic relations can only be determined from self-closed description of velocity evolution. On large scale, we found i) third order velocity correlation can be related to density correlation or pairwise velocity; ii) effective viscosity in adhesion model originates from velocity fluctuations; iii) negative viscosity is due to inverse energy cascade; iv) $q$th order velocity correlations follow $\propto a^{(q+2)/2}$ for odd $q$ and $\propto a^{q/2}$ for even $q$; v) overdensity is proportional to density correlation on the same scale, $\langle\delta\rangle\propto\langle\delta\delta'\rangle$; vi) (reduced) velocity dispersion is proportional to density correlation on the same scale. On small scale, self-closed description for velocity evolution is developed by decomposing velocity into motion in halo and motion of halos. Vorticity, enstrophy, and energy evolution can all be derived subsequently. Dynamic relation is derived to relate second and third order correlations. Third moment of pairwise velocity is determined by energy cascade rate $\epsilon_u$ or $\langle(\Delta u_L)^3\rangle\propto\epsilon_uar$. Combined kinematic and dynamic relations determines the exponential and one-fourth power law velocity correlations on large and small scales, respectively.
分类： 天文学 >> 天文学 提交时间： 2023-02-19
摘要： A halo-based non-projection approach is proposed to study the scale and redshift dependence of density and velocity distributions (PDF) in dark matter flow. All particles are divided into halo and out-of-halo particles such that PDF can be studied separately. Without projecting particle fields onto grid, scale dependence is analyzed by counting all pairs on different scales $r$. Redshift dependence is studied via generalized kurtosis. From this analysis, we can demonstrate: i) Delaunay tessellation can be used to reconstruct density field. Density correlations/spectrum are obtained, modeled and compared with theory; ii) $m$th moment of pairwise velocity can be analytically modelled. On small scale, even order moments can be modelled by a two-thirds law $\langle(\Delta u_L)^{2n}\rangle\propto{(-\epsilon_ur)}^{2/3}$, while odd order moments $\langle(\Delta u_L)^{2n+1}\rangle=(2n+1)\langle(\Delta u_L)^{2n}\rangle\langle\Delta u_L\rangle\propto{r}$ and satisfy a generalized stable clustering hypothesis (GSCH); iii) Scale dependence is studied for longitudinal velocity $u_L$ or $u_L^{'}$, pairwise velocity (velocity difference) $\Delta u_L$=$u_L^{'}$-$u_L$ and velocity sum $\Sigma u_L$=$u^{'}_L$+$u_L$. Fully developed velocity fields are never Gaussian on any scale; iv) On small scale, both $u_L$ and $\Sigma u_L$ can be modelled by a $X$ distribution to maximize system entropy. Distributions of $\Delta u_L$ is different with its moments analytically derived; v) On large scale, both $\Delta u_L$ and $\Sigma u_L$ can be modelled by a logistic function; vi) Redshift evolution of velocity distributions follows prediction of $X$ distribution with a decreasing shape parameter $\alpha(z)$ to continuously maximize system entropy.
分类： 天文学 >> 天文学 提交时间： 2023-02-19
摘要： N-body simulation is an invaluable tool to understand cosmic velocity field. However, simulation samples velocity only at particle locations that leads to information loss when projecting particle fields onto structured grid. Here we extract two-point statistics without field projection. These statistics, i.e. correlation, structure, dispersion functions in real space and spectrum functions in Fourier space, are modeled on both small and large scales. Kinematic relations between statistical measures are developed for incompressible, constant divergence, and irrotational flow. The nature of dark matter flow is identified by these relations. Much more complex than incompressible flow, peculiar velocity of dark matter flow is of constant divergence on small scale and irrotational on large scale. Incompressible and constant divergence flow share same kinematic relations for even order correlations. The limiting correlation of velocity $\rho_L=1/2$ on the smallest scale ($r=0$) is a unique feature of collisionless flow ($\rho_L=1$ for incompressible flow). Assuming gravity is the only interaction and no radiation produced, this leads to an increase in particle mass converted from kinetic energy upon "annihilation". On large scale, transverse velocity correlation has an exponential form $T_2\propto e^{-r/r_2}$ with a comoving scale $r_2$=21.3Mpc/h that maybe related to the size of sound horizon. All other correlation, structure, dispersion and spectrum functions for velocity, density, and potential are derived analytically from kinematic relations for irrotational flow. On small scale, structure function follows one-fourth law of $S^l_2\propto r^{1/4}$. All other statistical measures can be derived for constant divergence flow. Vorticity is negatively correlated for scale $r$ between 1 and 7Mpc/h. Divergence is negatively correlated for $r$>30Mpc/h that leads to negative density correlation.
分类： 天文学 >> 天文学 提交时间： 2023-02-19
摘要： 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 $m_hm_{hc}$, 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-$\lambda$ 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.
分类： 天文学 >> 天文学 提交时间： 2023-02-19
摘要： Halo abundance and structures are critical to understand the small scale challenges for $\Lambda$CDM cosmology. We present a unified theory and analytical models for both halo mass function and halo density profile based on a random walk of halos and dark matter particles. The position dependent waiting time in random walk leads to a stretched Gaussian for mass function and density with a power-law on small scale and an exponential decay on large scale. Both Press-Schechter mass function and Einasto density profile can be easily recovered. This new perspective provides a simple theory for universal halo mass function and density profile.
分类： 天文学 >> 天文学 提交时间： 2023-02-19
摘要： Smalls scale challenges suggest some missing pieces in our current understandings of dark matter. A cascade theory for dark matter flow is proposed to provide extra insights, similar to the cascade in hydrodynamic turbulence. The energy cascade from small to large scales with a constant rate $\varepsilon_u$ ($\approx -4.6\times 10^{-7}m^2/s^3$) is a fundamental feature of dark matter flow. Energy cascade leads to a two-thirds law for kinetic energy $v_r^2$ on scale $r$ such that $v_r^2 \propto (\varepsilon_u r)^{2/3}$, as confirmed by N-body simulations. This is equivalent to a four-thirds law for mean halo density $\rho_s$ enclosed in the scale radius $r_s$ such that $\rho_s \propto \varepsilon_u^{2/3}G^{-1}r_s^{-4/3}$, as confirmed by data from galaxy rotation curves. By identifying relevant key constants, critical scales of dark matter might be obtained. The largest halo scale $r_l$ can be determined by $-u_0^3/\varepsilon_u$, where $u_0$ is the velocity dispersion. The smallest scale $r_{\eta}$ is dependent on the nature of dark matter. For collisionless dark matter, $r_{\eta} \propto (-{G\hbar/\varepsilon_{u}}) ^{1/3}\approx 10^{-13}m$, where $\hbar$ is the Planck constant. A uncertainty principle for momentum and acceleration fluctuations is also postulated. For self-interacting dark matter, $r_{\eta} \propto \varepsilon_{u}^2 G^{-3}(\sigma/m)^3$, where $\sigma/m$ is the cross-section. On halo scale, the energy cascade leads to an asymptotic slope $\gamma=-4/3$ for fully virialized halos with a vanishing radial flow, which might explain the nearly universal halo density. Based on the continuity equation, halo density is analytically shown to be closely dependent on the radial flow and mass accretion such that simulated halos can have different limiting slopes. A modified Einasto density profile is proposed accordingly.
分类： 天文学 >> 天文学 提交时间： 2023-02-19
摘要： Inverse mass cascade is a key feature of statistically steady state for self-gravitating collisionless dark matter flow (SG-CFD). Continuous mass transfer from small to large mass scales (inverse) is formulated. Direct effect of mass cascade on halo mass function is presented. Mass cascade is local, two-way, and asymmetric in mass space. Halos inherit/pass their mass from/to halos of similar size. Two regimes are identified: a propagation range with 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 scale factor. Particle mass $m_p$ can be determined if that frequency is known. Rate of mass transfer ${\epsilon}_m{\sim}a^{-1}$ is independent of halo mass, a key feature of propagation range. Typical halos grow as $m_h{\sim}a^{3/2}$ and halo lifespan scales as ${\sim}m_h^{-\lambda}$. Chain reaction of mass cascade provides non-equilibrium dark matter flow a mechanism to continuously release energy and maximize entropy. Continuous injection of mass ("free radicals") at the smallest scale is required to sustain the everlasting inverse mass cascade such that 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. 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 Press-Schechter mass function can be fully derived without relying on any specific collapse models. A double-$\lambda$ mass function is proposed with different $\lambda$ for two ranges and agrees with N-body simulations.
分类： 天文学 >> 天文学 提交时间： 2023-02-19
摘要： MOND is an empirically motivated theory using modified gravity to reproduce many astronomical observations without invoking the dark matter hypothesis. Instead of falsifying the existence of dark matter, we propose MOND is an effective theory naturally emerging from the long-range interaction and collisionless nature of dark matter flow. It describes the dynamics of baryonic mass suspended in fluctuating dark matter fluid. The long-range interaction requires a broad size of halos to be formed that facilitate inverse mass and energy cascade with a constant rate of cascade $\varepsilon _{u} \approx -4.6\times 10^{-7}{m^{2}/s^{3}}$. In addition to the velocity fluctuation with a typical scale $u$, the long-range interaction leads to fluctuation in acceleration with a typical scale $a_{0}$. The velocity and acceleration fluctuations satisfy $\varepsilon_{u}=-{a_{0}u/(3\pi)^{2}}$ due to energy cascade, where factor $3\pi$ is from the angle of incidence. With $u_{0} \equiv u(z=0)\approx 354.61{km/s}$, $a_{0}(z=0)\approx 1.2\times 10^{-10}{m/s^{2}}$ can be easily obtained. While Planck constant $\hbar$, gravitational constant $G$, and $\varepsilon_{u}$ are proposed to find the dark matter particle properties on the smallest scale, $u$, $G$, and $\varepsilon_{u}$ determine halo properties on the largest scale. For given particle velocity $v_{p}$, maximum entropy distributions of dark matter flow lead to kinetic energy $\varepsilon_{k} \propto v_{p}$ for $aa_{0}$. Combining this with constant $\varepsilon_{u}$, Newtonian dynamics and "deep-MOND" behavior can be fully recovered. A notable coincidence of cosmological constant $\Lambda \propto ({a_{0}/c})^{2}$ or a linear relation for light speed $c=(3\pi)^3u_0$ might point to an entropic origin of dark energy from acceleration fluctuation with its density $\rho_{vac} \propto {a_{0}^{2}/G}$.
分类： 天文学 >> 天文学 提交时间： 2023-02-19
摘要： Dark matter can be characterized by the mass and size of its smallest constituents, which are challenging to directly probe and detect. After years of null results in the search for thermal WIMPs, a different prospective might be required beyond the standard WIMP paradigm. We present a new approach to estimate the dark matter particle mass, size, density, and many other relevant properties based on the nature of flow of dark matter. A comparison with hydrodynamic turbulence is presented to reveal the unique features of self-gravitating collisionless dark matter flow, i.e. an inverse mass and energy cascade from small to large scales with a scale-independent rate of energy cascade $\varepsilon_u\approx -4.6\times 10^{-7}m^2/s^3$. For the simplest case with only gravitational interaction involved and in the absence of viscosity in flow, the energy cascade leads to a two-thirds law for pairwise velocity that can be extended down to the smallest scale, where quantum effects become important. Combining the rate of energy cascade $\varepsilon_u$, the Planck constant $\hbar$, and the gravitational constant $G$ on the smallest scale, the mass of dark matter particles is found to be around $0.9\times10^{12}GeV$ with a size around $3\times10^{-13}m$. Since the mass scale $m_X$ is only weakly dependent on $\varepsilon_u$ as $m_X \propto (-\varepsilon_u\hbar^5/G^4)^{1/9}$, the estimation of $m_X$ should be robust for a wide range of possible values of $\varepsilon_u$. If gravity is the only interaction and dark matter is fully collisionless, mass of around $10^{12}GeV$ is required to produce the given rate of energy cascade $\varepsilon_u$. In other words, if mass has a different value, there must be new interaction beyond gravity. This work suggests a heavy dark matter scenario produced in the early universe ($\sim 10^{-14}s$). Potential extension to self-interacting dark matter is also presented.
点击量
待评议
点击量
下载量
评论
