STRAWBERRY: Finding haloes in the gravitational potential

TL;DR

This work addresses the ambiguous boundaries of dark matter haloes by introducing the strawberry algorithm, which uses a boosted gravitational potential to identify bound structures without ad-hoc density thresholds. By transforming to an accelerated reference frame and locating a saddle point in the boosted turn-around potential, particles are classified as bound if their energy $E$ is below the saddle energy $E_{ty}$; this yields a natural, parameter-free halo boundary that includes the influence of surrounding mass. The bound population is found to be virialised and to exhibit a well-defined edge, while an unbound, rapidly evolving exterior surrounds it, with binding typically completing within a dynamical time after infall. The approach provides insights into halo structure, evolution, and universal properties, and enables robust comparisons across redshift and cosmology, with practical applications for initial conditions and tidal evolution studies; the authors also provide public code for community use.

Abstract

Here, we present a novel algorithm that discriminates between bound and unbound particles by consideration of the gravitational potential from an accelerated reference frame -- also referred to as `the boosted potential'. Particles are considered bound if their energy does not exceed the escape energy of a potential well -- given by the closest saddle-point that connects to a deeper potential minimum. This approach has core benefits over previous approaches, since it does not require any ad-hoc thresholds (such as over-density criteria), it includes the gravitational effect of all particles in the binding criterion (improving over widely used self-potential binding checks) and it only operates with instantaneous information (making it simpler than approaches based on dynamical histories). We show that particles typically become bound between their first peri- and apo-centeric passage and that bound and unbound populations show very distinct characteristics through their distribution in phase space, their density profiles, their virial ratios, and their redshift evolution. Our findings suggest that it is possible to understand haloes as two-component systems, with one component being bound, virialized, of finite extent and evolving slowly in quasi-equilibrium and the other component being unbound, unvirialized and evolving rapidly.

Figures (16)

• Figure 1: Slice of the boosted gravitational potential field centred on a halo, the dark green contours mark the saddle point energy, $\phi_{\rm sad}$, the bound population of particles is shown as magenta points.
• Figure 2: Illustration of our binding notion for a fixed potential landscape. The region where particles may be bound to the minimum extends up to the first saddle-point that connects to a deeper minimum.
• Figure 3: Illustration of main particle assignment scheme implemented in strawberry. We start by defining the turn around boosted potential, $\phi_{\rm b,*}$, shown as a thick black line, and assigning the connectivity, thin grey lines, between the particles, drawn as circles. We then place ourselves at the bottom of the potential well and defining the group within the well, pink particle, and tracking particles adjacent to this group, in green, also known as surface particles. In the second panel, we grow the pink group by including surface particles if all of their neighbours with lower potentials are also part of the group. In the third panel we stop growing the group as we have found a surface particle, to the right, which is connected to particles, in blue, which lead to a deeper potential valley. This final particle sets the persistence of the group, $\Delta\phi_{\rm b, *}$.
• Figure 4: Median energy histories of three sets of particles from a sample of 100 haloes with masses $M_{\rm bound}\in[0.5\cdot10^{14}h^{-1}\mathrm{M}_\odot,1.5\cdot 10^{14}h^{-1}\mathrm{M}_\odot]$, selected according to their respective time of binding. The individual energy histories of particles from different haloes are normalised by the persistence $\Delta\phi_{\rm b}(a = 1)$ of the host halo before the median is computed. The median persistence history of the sample of haloes is shown in black.
• Figure 5: Fraction of bound particles as a function of the number of dynamical times since the first pericentric passage, shown as the median, (solid black line) and 1-$\sigma$ region (shaded area) as estimated over 100 haloes. The coloured points represent different events in a particle's history, green, the particle enters the potential well, orange, the particle passes the pericentre of its orbit for the first time, and red, the particle reaches its first apocentre.