Universal numerical convergence criteria for subhalo tidal evolution

TL;DR

This work tackles artificial disruption and biased subhalo demographics in cosmological N-body simulations by performing an extensive AMR convergence study for anisotropic subhalos on diverse orbits. It derives a universal force-resolution criterion requiring the instantaneous tidal radius to be resolved by at least 20 cells, applicable across mass resolutions, AMR strategies, and subhalo properties, and an independent, universal expression for the discreteness-noise-driven scatter in the bound-mass fraction that depends only on $N_ ext{par}$ and $f_{ m bound}$. The authors show that up to about 50% of subhalos in state-of-the-art simulations may be force- or mass-unresolved, implying substantial systematic uncertainties in subhalo statistics. They then discuss the implications for AMR- and tree-based codes, and advocate tidal-radius-based adaptive refinement as a robust path forward, including vetting strategies for cosmological subhalo catalogs. The results reconcile numerical convergence issues with real astrophysical implications for subhalo abundances, radial distributions, and satellite populations across cosmic time.

Abstract

Dark matter subhalos and satellite galaxies in state-of-the-art cosmological simulations still suffer from the ``overmerging'' problem, where inadequate force and/or mass resolution cause artificially enhanced tidal mass loss and premature disruption. Previous idealized simulations addressing this issue have been restricted to a small subset of the subhalo orbital parameter space, and all assumed subhalos to be isotropic. Here, we present the first extensive simulation suite that quantifies numerical convergence in the tidal evolution of anisotropic subhalos under varying numerical resolutions and orbits. We report a universal force resolution criterion: the subhalo's instantaneous tidal radius must always be resolved by at least 20 cells in adaptive mesh refinement (AMR)-based simulations, or by 20 softening lengths (Plummer equivalent) in tree-based simulations, regardless of refinement details or subhalo physical properties such as concentration or velocity anisotropy. We also report a universal expression for the discreteness-noise-driven scatter in the bound-mass fraction of subhalos that depends only on the subhalo mass resolution at infall and the instantaneous bound mass fraction, agnostic of any further subhalo properties. Such stochastic discreteness noise causes both premature disruption and, notably, spurious survival of poorly mass-resolved subhalos. We demonstrate that as many as 50 percent of all subhalos in state-of-the-art cosmological simulations are likely to be either force and/or mass unresolved. Our findings advocate for adaptive softening or grid refinement based on the instantaneous tidal radius of the subhalo.

Figures (13)

• Figure 1: Evolution of the bound mass fraction $f_{\rm bound}$ (top row) and mean velocity anisotropy $\beta_{50\%}$ (bottom row) of subhalos with an initial velocity anisotropy $\beta = 0.5$ along circular orbits with $\mathcal{R}_E = 0.4$ (orbital period indicated by vertical dotted lines). Different columns correspond to different mass resolutions (i.e., different $N_\text{par}$), as indicated at the top. Colored lines in each panel show the results for ten independent random realizations and are compared against the benchmark result (thick black curve) obtained using a simulation with $N_\text{par} = 5 \times 10^7$. The ten-realization-averaged disruption time $t_\text{dis}$ (colored vertical dashed lines) and the associated one-sigma scatter are denoted in the top-right corner of each column. Note that the realization-to-realization variance becomes appreciable for $N_\text{par} \lesssim 10^5$.
• Figure 1: Numerical scatter $\sigma_{\log f}(t=10~\text{Gyr})$ of $\beta = 0.25$ subhalos of $N_\text{par} = 10^3, 10^4, 10^5, 10^6$ (left to right columns), evolved along orbits of different $\mathcal{R}_E$ and eccentricities $e$. Top row: Each pixel is measured from ten independent simulations and annotated with the number of disrupted subhalos, if non-zero; pixels with all ten realizations disrupted are in black. See text for the annotation color and treatment of pixels with partial disruption. Bottom row: Model predictions Eq. (\ref{['eqn:sigma_fbound_fit']}) from the respective $N_\text{par}$ and ten-realization-averaged $f_{\rm bound}$ for each corresponding pixel in the top row. We observe remarkable agreement between simulation measurements and model predictions. Empirically, we find that the threshold $\sigma_{\log f} \geq 0.3$ is indicative of partial disruption.
• Figure 2: Numerical convergence of the tidal disruption time $t_\text{dis}$ of $\beta = 0.5$ subhalos evolved on $\mathcal{R}_E = 0.4$ circular orbits with varying mass resolutions $N_\text{par} = 10^3\text{\textendash}5\times10^7$. Colored dots are simulation results from Fig. \ref{['fig:Mass_Resolution']}, while the light-gray dots correspond to results based on additional simulations. Upward arrows indicate that the subhalos survive for longer than 45 Gyr. All simulations are based on a refinement scheme that assures adequate force resolution (based on the criteria discussed in §\ref{['ssec:Case_Study_Force']}). The black curve indicates the $N_\text{par}$-dependent disruption time defined as the time when a subhalo of $N_\text{par}$ particles that follows $f^\text{truth}_\text{bound}(t)$ drops below 10 particles. Hence, in the absence of discreteness-noise-induced scatter, all data points should fall along this line. As is evident, discreteness noise becomes appreciable for $N_\text{par} \lesssim 10^5$ and can cause both premature disruption (subhalo disrupts prior to the expected disruption time) as well as spurious survival (subhalo disrupts after the expected disruption time).
• Figure 3: Evolution of $f_{\rm bound}$ (top) and $\beta_{50\%}$ (bottom) for $\beta = 0.5$ subhalos evolved along circular orbits with $\mathcal{R}_E = 0.4$, using different maximum-allowed refinement levels (color-coded as indicated). For each choice of $\Delta x_\text{min}$, we show the results for ten random realizations with $N_\text{par} = 10^7$. The black solid lines indicates the "truth" obtained using a simulation with $N_\text{par} = 5\times 10^7$ (black) and the fiducial $\Delta x_\text{min} = 0.0012r_{\rm vir}$. Note that results are converged for $\Delta x_\text{min} \lesssim 0.01r_{\rm vir}$. For $\Delta x_\text{min} = 0.02r_{\rm vir}$, the subhalos are force-unresolved, giving rise to bound mass fractions that are too small which ultimately causes premature disruption. The vertical dotted line indicates the orbital period, while the vertical dashed lines mark the average disruption times.
• Figure 4: Evolution of the bound mass fractions, $f_{\rm bound}$, normalized by $f^\text{truth}_\text{bound}$, of subhalos with $\beta=0.5$ evolved along circular orbits with $\mathcal{R}_E=0.4$. Different rows correspond to simulations with different mass resolution, ranging from $N_\text{par} = 10^7$ to $10^3$, as indicate at the right-hand side. Different columns correspond to different maximum-allowed refinement levels of, from left to right, 9, 7, 6 and 5, resulting in the minimum cell sizes indicated at the top. Each panel shows the results for ten independent realizations that are individually color-coded based on the highest refinement level that is instantaneously achieved; 9 (orange), 8–7 (dark blue), 6 (light blue), 5 (dark red), or 4 and below (pink). With $N_\text{par} = 10^7$ the target refinement level is always achieved (i.e. $\Delta x^\text{ins}_\text{min} = \Delta x_\text{min}$). However, at lower mass resolution the instantaneous refinement level is lower than the target at late times due to the decreasing central particle number density. Note how force-unresolved subhalos (those with $\Delta x^\text{ins}_\text{min} > 0.01r_{\rm vir}$, indicated as dark red or pink) typically experience artificially enhanced tidal mass loss, ultimately resulting in premature numerical disruption. However, if $N_\text{par}$ is sufficiently small, the discreteness noise becomes sizable enough to cause some of the force-unresolved systems to have bound mass fractions that are artificially high.