Accretion history dependence of the halo depletion radius

1. Problem Statement

Dark matter halos are the fundamental building blocks of cosmic structure, yet their outer boundaries remain physically ambiguous. While the virial radius ($R_{vir}$) is commonly used, it relies on arbitrary overdensity thresholds and suffers from pseudo-evolution. The splashback radius ($R_{sp}$), defined by the steepest slope in the density profile (first apocenter of accreted matter), offers a dynamical boundary but is primarily sensitive to the recent mass accretion rate (MAR).

A complementary boundary, the depletion radius ($R_{id}$), was recently proposed (Fong & Han 2021). It is defined as the radius where the local density evolution changes sign ($\partial\rho/\partial t = 0$), marking the transition from an inner growth region to an outer depletion region. While $R_{id}$ has been observed to be approximately $2R_{vir}$ across various conditions, its quantitative dependence on accretion history, redshift, and secondary halo properties remains uncharacterized. This paper aims to systematically investigate these dependencies to determine if $R_{id}$ serves as a unique probe of halo growth modes distinct from $R_{sp}$.

2. Methodology

• Simulation Data: The authors utilized a large cosmological $N$-body simulation from the CosmicGrowth suite (Jing 2019).
  • Resolution: $3072^3$ dark matter particles in a $600 \, h^{-1}\text{Mpc}$ box ($m_p \approx 5.54 \times 10^8 \, h^{-1}M_\odot$).
  • Cosmology: Flat $\Lambda$CDM ($\Omega_m = 0.268, \Omega_\Lambda = 0.732$).
  • Sample: $\sim 5.6 \times 10^6$ halos at $z=0$, selected with $M_{vir} \ge 10^{12} h^{-1}M_\odot$, central subhalo stability, and isolation from massive neighbors.
• Boundary Definitions:
  • Mass Flow Rate (MFR): Calculated as $MFR(r) = 4\pi r^2 \rho(r) v_r(r)$.
  • Depletion Radius ($R_{id}$): Defined as the location of the local minimum in the MFR profile (where density evolution $\partial\rho/\partial t = 0$).
  • Splashback Radius ($R_{sp}$): Defined as the minimum of the logarithmic density slope ($d\log\rho/d\log r$).
  • Smoothing: Profiles were smoothed using a Savitzky-Golay filter to locate minima accurately.
• Accretion Rate: The mass accretion rate ($\Gamma_{dyn}$) was defined as the logarithmic growth rate of virial mass over one dynamical timescale ($t_{dyn} = 2R_{vir}/V_{vir}$).
• Analysis Strategy:
  • Halos were binned by mass and $\Gamma_{dyn}$ across redshifts $0 \le z \le 3$.
  • Merger Analysis: A subset of halos excluding recent major mergers (mass ratio $>1:9$ within $t_{dyn}$) was analyzed to isolate smooth accretion effects.
  • Secondary Properties: Dependence on shape ($e$), spin ($j$), concentration ($V_{max}/V_{vir}$), and formation time ($a_{1/2}$) was tested.

3. Key Contributions & Results

A. Dependence on Mass Accretion Rate ($\Gamma_{dyn}$)

• Strong Correlation: $R_{id}/R_{vir}$ exhibits a strong, monotonic dependence on $\Gamma_{dyn}$. Higher accretion rates lead to smaller $R_{id}/R_{vir}$ (more compact growth layers).
• Relation to Splashback: $R_{id}$ and $R_{sp}$ are tightly correlated. The relationship is approximately linear:
  $$R_{id}/R_{vir} \approx -0.50 + 1.72 (R_{sp}/R_{vir})$$
  However, $R_{id}$ is consistently larger than $R_{sp}$ (roughly $1.5 \times R_{sp}$ at high accretion rates).

B. Redshift Evolution and the "Universal" Ratio

• Redshift Dependence: Unlike $R_{sp}/R_{vir}$, which shows weak redshift evolution, $R_{id}/R_{vir}$ exhibits a systematic redshift evolution. At fixed $\Gamma_{dyn}$, $R_{id}/R_{vir}$ increases with redshift.
• Physical Origin: This evolution is driven by major mergers. When halos with recent major mergers are excluded, the redshift dependence of $R_{id}$ largely vanishes, and the relation converges to a single curve consistent with $\Omega_m \approx 1$ (high redshift limit).
• Resolution of the "Universal" $2R_{vir}$: The previously reported constant ratio $R_{id}/R_{vir} \simeq 2$ is not a fundamental physical constant. It arises from a statistical compensation:
  1. Higher redshift implies higher typical $\Gamma_{dyn}$ (which lowers $R_{id}/R_{vir}$).
  2. Higher redshift implies a steeper $R_{id}/R_{vir}$ vs. $\Gamma_{dyn}$ relation (which raises $R_{id}/R_{vir}$).
     These opposing trends cancel out for the median halo population, creating an apparent universality.

C. Dichotomy in Drivers (Smooth vs. Perturbed Accretion)

The paper identifies a critical dichotomy in what drives $R_{id}$:

• Slow Accretion ($\Gamma_{dyn} \lesssim 1$): $R_{id}$ is primarily determined by $\Gamma_{dyn}$. Secondary properties (shape, spin, formation time) have negligible impact. The system behaves like a smooth spherical collapse.
• Fast Accretion ($\Gamma_{dyn} \gtrsim 1$): $R_{id}$ becomes sensitive to secondary halo properties, particularly the formation time ($a_{1/2}$) of the central subhalo.
  • Halos with early formation ($low \, a_{1/2}$) but high current $\Gamma_{dyn}$ show significantly smaller $R_{id}$.
  • Interpretation: This indicates merger-dominated growth. The central subhalo grew early and is now quiet, while the total halo mass grows rapidly via massive satellite infall. This anisotropic, merger-driven accretion enhances coherent inflow and suppresses the outflow component, shifting $R_{id}$ inward.

D. Mass Dependence

• $R_{id}/R_{vir}$ shows weak dependence on halo mass when binned by $\Gamma_{dyn}$.
• More massive halos exhibit slightly deeper MFR troughs (stronger net mass transport), but the location of the characteristic radii remains self-similar.

4. Significance and Implications

1. New Physical Probe: The depletion radius is established as a sensitive probe of the detailed accretion history of dark matter halos, complementary to the splashback radius. While $R_{sp}$ traces the orbital dynamics of recently accreted particles, $R_{id}$ traces the time-evolution of the density field, making it uniquely sensitive to the mode of accretion (smooth vs. merger-driven).
2. Merger Diagnostics: The sensitivity of $R_{id}$ to secondary properties (like formation time) in the fast-accretion regime provides a new method to identify halos undergoing major mergers or anisotropic accretion, even when the merger is not immediately obvious in the density profile.
3. Theoretical Refinement: The results challenge simple self-similar spherical collapse models, which fail to predict the redshift dependence of $R_{id}$ in the presence of mergers. The study highlights that the "virial radius" itself is a dynamic quantity affected by accretion modes, complicating the interpretation of $R_{id}/R_{vir}$.
4. Observational Utility: Since $R_{id}$ has been detected via weak lensing and satellite kinematics, these findings suggest that measuring $R_{id}$ can constrain not just the current growth rate of a halo, but also its recent merger history and the nature of its surrounding environment.

In summary, Zhou & Han demonstrate that the depletion radius is a robust, physically motivated boundary that encodes rich information about the accretion mode of dark matter halos, distinguishing between smooth growth and merger-driven assembly.