Collisionless relaxation to equilibrium distributions in cold dark matter halos: origin of the Navarro-Frenk-White profile

Abstract

Collisionless self-gravitating systems such as cold dark matter halos are known to harbor universal density profiles despite the intricate non-linear physics of hierarchical structure formation in the $Λ$CDM paradigm. The origin of such states has been a persistent mystery, particularly because the physics of collisionless relaxation has remained poorly understood. To solve this long-standing problem, we develop a self-consistent quasilinear theory in action-angle space for the collisionless relaxation of inhomogeneous, self-gravitating systems by perturbing the governing Vlasov-Poisson equations. We obtain a quasilinear diffusion equation that describes the secular evolution of the mean coarse-grained distribution function $f_0$ of accreted matter in the fluctuating force field of a spherical isotropic halo. The diffusion coefficient not only depends on the fluctuation power spectrum but also on the evolving potential of the system, which reflects the self-consistency of the problem. Diffusive heating in the pre-assembled halo develops an $r^{-γ}$ inner density cusp, accretion and relaxation in which develops an $r^{-β}$ outer fall-off with $β\approx 5 - 2γ$ in the quasi-steady state. Spherical collapse theory dictates that a quasi-steady outer halo must settle to $β\approx 3$, for which the mass enclosed within a shell barely changes with time. This implies that $γ\approx 1$, which is possible in the quasilinear framework only if (i) the pre-assembled halo harbors an $r^{-γ_{\mathrm{P}}}$ profile with $γ_{\mathrm{P}} \gtrsim 0.5$, (ii) its fluctuations are correlated in time (red noise), and (iii) the initial value of $γ$ is smaller than $1$, implying that the $r^{-1}$ cusp is a neutral equilibrium. We demonstrate for the first time how the Navarro-Frenk-White (NFW) profile emerges as a quasi-steady state of collisionless relaxation.

Figures (3)

• Figure 1: Evolution of the inner log-slope $\gamma$ of a relaxing halo as a function of time $\tau=t/t_{\rm diff}$ ($t_{\rm diff} = I^2_0/D_0$) for different values of $\gamma_0=\gamma(\tau=0)$, $\gamma_{\rm P}$ and $n_\gamma$ as indicated, obtained by solving equation (\ref{['dgamma_dt']}) at ${\cal I}_r = I_r/I_0 = 1$. Dashed (dotted) line indicates the stable (unstable) fixed point. We adopt $\gamma_{\rm P}$ and $n_\gamma$ such that the stable fixed point is equal to $1$.
• Figure 2: Evolution of the outer log-slope $\beta$ as a function of $\tau=t/t_{\rm diff}$ ($t_{\rm diff} = I^2_0/D_0$), obtained by solving equations (\ref{['dgamma_dt']}) and (\ref{['dbeta_dt']}), for the combinations of $\gamma_0$, $\gamma_{\rm P}$ and $n_\gamma$ adopted in Fig. \ref{['fig:gamma_vs_t_diff_gamma0']} such that $\gamma_{\rm s} = 1$. We adopt $n_\beta = 0.1$, and restrict ourselves to $\gamma_0 < 1.5\left(1+n_\beta/2\right)$ so that the fixed point $\beta_{\rm s} = 5 - 2\gamma_{\rm s} + 3n_\beta/2$ is stable. As $\gamma$ approaches the stable fixed point $\gamma_{\rm s} = 1$ in Fig. \ref{['fig:gamma_vs_t_diff_gamma0']}, $\beta$ approaches $\beta_{\rm s} = 3.15$.
• Figure 3: Halo density $\rho_0$ (in units of $M_{\rm vir}/r^3_{{\rm s}}$) as a function of radius $r$ (in units of $r_{\rm vir}$). The solid blue line indicates the NFW profile, the constant flux quasi-steady state. The dot-dashed red and dashed green lines respectively indicate the central core and $r^{-1.5}$ profiles, which are zero flux steady states obtained by numerically integrating the Lane-Emden equation (\ref{['Lane_Emden_eq_1.5']}). The dashed black line indicates the isothermal sphere. The vertical dashed lines indicate the virial radius $r_{\rm vir}$ and the scale radius, $r_{\rm s}$, assumed to be $0.1 r_{\rm vir}$. The profiles are normalized such that the virial mass of the NFW and isothermal sphere profiles is the same as the total mass of the other two.