The Density Profile of Dynamical Halos

Technical Summary: The Density Profile of Dynamical Halos

Problem Statement
The fundamental properties of dark matter halos, particularly their density profiles, are traditionally characterized by analytic fits (e.g., NFW, Einasto) to the total mass distribution within a reference radius like $R_{200m}$. However, this approach conflates two distinct dynamical populations: particles that are gravitationally bound and orbiting within the halo, and particles that are infalling but not yet bound. Recent proposals suggest defining a "dynamical halo" strictly as the collection of orbiting particles. While this definition avoids mixing internal structure with ongoing accretion, the structural properties (mean and scatter) of these orbiting profiles had not been fully characterized. Specifically, it was unclear which structural degrees of freedom remain at fixed orbiting mass, the magnitude of intrinsic scatter, and whether correlations with formation history are genuine properties of the orbiting component or artifacts of mixing populations.

Methodology
The authors utilized a Cold Dark Matter (CDM) simulation ($1024^3$ particles in a $1 \, h^{-1}$ Gpc box) run with GADGET-2. They generated a catalog of dynamical halos using the Rockstar halo finder as a starting point, then applied the algorithm of García et al. to tag particles as either "orbiting" or "infalling" based on their accretion time ($a_{acc}$) and time-averaged radial velocities.

• Orbiting Definition: Particles with early accretion times and near-zero time-averaged radial velocities are tagged as orbiting; those with late accretion times and negative radial velocities are infalling.
• Mass Binning: Halos were binned into nine narrow mass bins ($\log M_{orb} \in [13.40, 15.00]$).
• Parametric Model: The orbiting density profiles were fitted using the model from Salazar et al., defined as:
  $$\rho_{orb}(x|M_{orb}) = A \left(\frac{x}{\epsilon}\right)^{-\alpha(x|M_{orb})} \exp\left(-\frac{x^2}{2}\right)$$
  where $x \equiv r/r_h$, $r_h$ is the halo radius (exponential truncation scale), and the slope $\alpha(x)$ transitions from 0 to an asymptotic value $\alpha_\infty$. The parameter $\epsilon$ was fixed to 0.037.
• Fitting Strategy: Individual halos were fitted to determine the best-fit $r_h$ and $\alpha_\infty$. The authors then investigated the correlation between these parameters and the dependence on accretion history, characterized by a relative formation time variable $a_{RF}$.

Key Contributions and Results

1. Single Effective Degree of Freedom:
   The study demonstrates that at a fixed orbiting mass, the two free parameters of the profile ($r_h$ and $\alpha_\infty$) are tightly correlated. More extended halos (larger $r_h$) possess steeper inner slopes (larger $\alpha_\infty$). This correlation allows the profile of an individual dynamical halo to be characterized by a single free parameter: the halo radius $r_h$. The relationship is given by:
   $$\Delta\alpha_\infty = 0.05 + \ln R$$
   where $R = r_h / r_{h,st}(M_{orb})$ and $\Delta\alpha_\infty$ is the deviation from the stacked slope.

2. Scatter in Halo Radius:

   • At fixed orbiting mass, the distribution of halo radii is approximately log-normal. The intrinsic scatter in $\ln r_h$ is $\sigma_{\ln r_h | M_{orb}} \approx 0.16$ (16%).
   • The authors investigated the dependence of this scatter on accretion history. They defined a relative formation time $a_{RF}$ based on the 60th percentile of the particle accretion time distribution.
   • A correlation exists between $r_h$ and $a_{RF}$: late-forming halos are more compact (smaller $r_h$).
   • Accounting for this correlation reduces the scatter in halo radius at fixed mass and formation time to $\sigma_{\ln r_h | M_{orb}, a_{RF}} \approx 0.11$ (11%). The authors note that this reduction is marginal, suggesting that relative formation time alone is insufficient to fully explain the scatter.

3. Comparison with Accretion Rate:
   The authors compared the correlation between halo radius and relative formation time against the correlation between halo radius and the accretion rate per unit mass. They found the trends to be qualitatively similar, with the accretion rate correlation showing slightly tighter constraints but more pronounced non-Gaussian tails. They caution that they could not self-consistently estimate accretion rates for dynamical halos in this specific analysis.

4. Model Performance:
   The parametric fits to the orbiting profiles yield residuals comparable to NFW fits applied to total mass profiles (median residual $\approx -0.02$, scatter $\sim 0.15$). This validates the use of the Salazar et al. model for the orbiting component.

Significance and Claims
The paper claims to provide the first characterization of the accretion-history dependence of orbiting profiles for dynamically defined halos. By isolating the orbiting component, the authors argue that the measured correlations (e.g., between structure and formation time) are genuine properties of the bound system rather than definitional artifacts caused by mixing infalling material.

The primary significance lies in the ability to "read off" the mass and formation time of a halo from its structural properties ($r_h$ and $\alpha_\infty$). Unlike standard spherical overdensity halos, where the radius is definitionally tied to the enclosed mass, dynamical halos expose how a halo's extent and structure arise from orbital dynamics. This calibration serves as a necessary baseline for forward modeling, enabling more accurate dynamical decompositions of halo density profiles and improving the interpretation of mass distributions inferred from lensing and clustering data. The authors emphasize that while their model does not replace empirical fits for total matter distributions, it is essential for understanding the underlying dynamical structure.