The Density Profile of Dynamical Halos

Among the most fundamental properties of a dark matter halo is its density profile. Motivated by the recent proposal by García et al. [R. García et. al., MNRAS 521, 2464 (2023)] to define a dynamical halo as the collection of orbiting particles in a gravitationally bound structure, we characterize the mean and scatter of the orbiting profile of dynamical halos as a function of their orbiting mass. We demonstrate that the orbiting profile of individual halos at fixed mass depends on a single dynamical variable – the halo radius $r_{\rm h}$ – which characterizes the spatial extent of the profile. The scatter in halo radius at fixed orbiting mass is $\approx 16%$. Only a small fraction of this scatter arises due to differences in halo formation time, with late-forming halos being more compact (smaller halo radii). Accounting for this additional correlation results in an $\approx 11%$ scatter in halo radius at fixed mass and halo formation time.

💡 Research Summary

The paper investigates the density structure of dark‑matter halos using a newly defined “dynamical halo” concept, which includes only particles that are gravitationally bound and orbiting within a self‑generated potential, as proposed by García et al. (2023). Starting from a large cosmological N‑body simulation (1 h⁻¹ Gpc box, 1024³ particles, run with GADGET‑2), the authors first generate a conventional halo catalog with the Rockstar finder. They then apply the García algorithm to each halo, tagging particles within 5 h⁻¹ Mpc as “orbiting” or “infalling” based on their first crossing of the one‑to‑two halo transition scale and their time‑averaged radial velocities. The set of orbiting particles defines a dynamical halo and its “orbiting mass” Mₒᵣb.

To describe the radial density profile of the orbiting component, the authors adopt the parametric form introduced by Salazar et al. (2022): ρₒᵣb(x) = A x^{ε‑α(x)} exp(‑x²/2), where x = r/r_h, ε = 0.037 (fixed), and α(x) = α_∞ x/(x+ε). The normalization A is set by the requirement that the profile integrates to Mₒᵣb, leaving r_h (the halo radius) and α_∞ as free parameters. They fit each individual halo using a χ² that combines Poisson uncertainties with a 5 % systematic error, excluding radii smaller than six times the force‑softening length.

The fits reveal that the two parameters are not independent. By defining Δα_∞ = α_∞ – α_∞,st(Mₒᵣb) and R = r_h/r_h,st(Mₒᵣb) (where the “st” quantities are the stacked‑halo best‑fit relations), the authors find a tight log‑linear correlation: Δα_∞ ≈ 0.05 + ln R. Consequently, the entire orbiting density profile of a halo of given mass can be described by a single effective degree of freedom, the halo radius r_h. The distribution of ln R at fixed Mₒᵣb is approximately Gaussian with mean ⟨ln R⟩ ≈ ‑0.064 (i.e., r_h ≈ 0.94 r_h,st) and intrinsic scatter σ_{ln r_h} ≈ 0.16. The scatter grows toward lower masses, which the authors attribute mainly to particle‑count noise.

To explore the physical origin of the radius scatter, the paper examines the accretion histories of the orbiting particles. For each particle they record the scale factor a_acc at which it first crossed the transition radius. They construct cumulative distribution functions (CDFs) of a_acc for halos split by R. The CDFs show that halos with larger r_h tend to have earlier accretion (older formation), while compact halos accreted later. To quantify this, the authors define a characteristic accretion time a_60 as the scale factor where the CDF reaches 0.6. Because a_60 correlates with halo mass, they introduce a relative formation time a_RF = a_60 / median(a_60|Mₒᵣb). Plotting a_RF versus R for massive halos (Mₒᵣb > 10¹⁴ h⁻¹ M_⊙) yields a linear relation: R = a₀ + s_a a_RF, with best‑fit a₀ ≈ 1.91 and s_a ≈ ‑0.98. Adopting s_a = ‑1 gives a simple model: r_h,mod(a_RF, Mₒᵣb) = (1.91 ‑ a_RF) r_h,st(Mₒᵣb). When this formation‑time correction is applied, the residual scatter in ln r_h at fixed mass and a_RF drops to σ_{ln r_h} ≈ 0.11, a modest reduction from the original 0.16. Thus, formation time explains only about a third of the radius variance; the remaining scatter likely reflects additional dynamical or environmental factors not captured by a single accretion‑time metric.

The authors compare their two‑parameter model (effectively one free parameter after the Δα_∞–R correlation) with the more complex Diemer (2021) description, which uses additional shape parameters. They argue that their approach, with fewer parameters and a clear physical interpretation of r_h as the spatial extent of the orbiting component, offers a more parsimonious yet accurate representation of halo structure, especially for large‑scale simulations where computational efficiency matters.

In summary, the paper demonstrates that:

1. The orbiting density profile of dynamical halos is well described by a truncated exponential form with a mass‑dependent scale radius r_h and asymptotic slope α_∞.
2. α_∞ and r_h are tightly correlated, reducing the effective degrees of freedom to a single radius parameter.
3. At fixed mass, the log‑normal scatter in r_h is ≈ 16 %; incorporating a relative formation time reduces this to ≈ 11 %.
4. The remaining unexplained scatter suggests that factors beyond simple formation time—such as tidal environment, merger history, or stochastic variations in orbital mixing—play a significant role.
5. The proposed r_h‑centric model provides a compact, physically motivated alternative to traditional halo‑boundary definitions (e.g., splashback radius) and may improve halo‑galaxy connection models and analytical predictions of halo statistics.