The many boundaries of the stratified dark matter halo

This paper reviews the physics of dark matter halo collapse and the resulting stratified structure defined by multiple boundaries—such as the splashback, depletion, and turnaround radii—that characterize ongoing halo growth, while also introducing a Python package called SpheriC to implement the underlying spherical collapse models.

The Gist

Imagine the universe isn't just a random scattering of stars and galaxies, but a giant, organized city. In this city, Dark Matter Halos are the invisible neighborhoods that hold everything together. For decades, astronomers thought these neighborhoods were simple, solid balls of invisible matter with a clear, hard edge, like a tennis ball. They called the edge the "Virial Radius."

But this paper argues that the tennis ball analogy is wrong. Real dark matter halos are more like giant, swirling whirlpools or onion-like structures with many distinct layers and fuzzy boundaries. The author, Jiaxin Han, explains that because these halos are constantly eating new material from the surrounding universe, they have complex "skin" layers that we need to understand to see how the universe grows.

Here is a breakdown of the paper's key ideas using everyday analogies:

1. The Old View: The Static Tennis Ball

For a long time, scientists defined a halo by its Virial Radius.

• The Analogy: Imagine a spinning top that has settled down. It's spinning steadily, and everything inside is in a stable dance. The "Virial Radius" is the edge of that stable dance floor.
• The Problem: The universe is expanding, and halos are constantly growing. If you define a halo only by this stable dance floor, you miss all the new dancers running in from the outside. Also, because the "dance floor" definition relies on the background of the universe changing, the size of the ball seems to change even when the halo isn't actually growing. This is called "Pseudo-evolution" (fake growth).

2. The New View: The Stratified Onion

The paper proposes that a halo is a stratified structure, meaning it has different layers, each with its own boundary. Think of it like an onion or a set of Russian nesting dolls, but the layers are made of moving particles.

Here are the four main "zones" or boundaries, moving from the center outwards:

A. The Virial Radius (The Inner City)

• What it is: The core.
• The Analogy: This is the downtown district. The traffic (particles) is chaotic but balanced; cars are going in and out at the same rate, so the population stays stable. This is the "classic" halo we used to study.

B. The Splashback Radius (The Commuter Belt)

• What it is: The edge of the first layer of new material falling in.
• The Analogy: Imagine a crowd of people running toward a stadium (the halo). They run in, dive into the center, and then bounce back out because they can't stop immediately. The Splashback Radius is the point where these "bouncers" reach their furthest point before falling back in again.
• Why it matters: It's like a "splash" of water hitting a wall and splashing back. This creates a sharp drop in density. It tells us how fast the halo is currently eating new matter. If the halo is hungry (growing fast), this splash happens closer in. If it's full, the splash happens further out.

C. The Depletion Radius (The Empty Zone)

• What it is: A region where the density actually drops because the halo is sucking everything in.
• The Analogy: Imagine a vacuum cleaner running in a room. The dust (matter) near the nozzle is sucked in so fast that the area around the nozzle becomes empty before the dust even reaches the center. The Depletion Radius is the edge of this "empty zone."
• Why it matters: It marks the boundary where the halo is actively draining its neighborhood. It's a better indicator of the halo's "growth rate" than the old definitions.

D. The Turnaround Radius (The Farmland)

• What it is: The very outer edge of the halo's influence.
• The Analogy: Imagine a giant magnet in the middle of a field. Far away, the wind (the expansion of the universe) blows everything outward. But as you get closer to the magnet, the wind slows down, stops, and then reverses, pulling things toward the magnet. The Turnaround Radius is the exact spot where the wind stops and the pull begins.
• Why it matters: This is the true limit of the halo's territory. Beyond this, the universe's expansion wins; inside this, the halo's gravity wins.

3. Why Does This Matter? (The "So What?")

Fixing the "Halo Exclusion" Problem:
If you try to map the universe by tiling it with these "tennis balls" (Virial halos), you end up with gaps. There is matter in the gaps that doesn't belong to any halo.

• The Fix: By using the Depletion Radius or Turnaround Radius, we can define halos that actually touch each other and fill the entire universe, leaving no gaps. It's like realizing the neighborhoods don't have hard fences, but rather overlapping suburbs that blend into one another.

A New Tool for Cosmology:
These new boundaries act like a speedometer for the universe.

• Because the size of the "Splashback" and "Depletion" zones changes depending on how fast the halo is growing, measuring them tells us about the history of the universe's expansion and the nature of Dark Energy. It's like looking at the ripples in a pond to figure out how hard the rain is falling.

4. The "SPHERIC" Package

The author didn't just write a theory; they built a tool. They created a free software package called SPHERIC (which stands for Spherical Collapse).

• The Analogy: It's like a physics simulator app. Instead of needing a supercomputer to figure out how these layers form, astronomers can use this app to calculate the exact size of the Splashback or Depletion radius for any given halo, helping them interpret real telescope data.

Summary

This paper tells us to stop looking at Dark Matter Halos as simple, static balls. Instead, we should see them as dynamic, layered structures with a stable core, a "splashy" outer layer of falling matter, an empty "drainage" zone, and a massive outer limit where the universe's expansion gives way to gravity.

By understanding these layers, we can:

1. Stop making "fake" measurements of halo growth.
2. Fill in the gaps in our map of the universe.
3. Use the shape of these layers to measure the secrets of Dark Energy and the expansion of the cosmos.

It's a shift from seeing the universe as a collection of static islands to seeing it as a flowing, breathing ecosystem.

Technical Summary

1. Problem Statement

The conventional definition of a dark matter halo relies on the virial radius ($R_{vir}$), which assumes the halo is a monolithic, virialized object in equilibrium. However, this definition suffers from three fundamental limitations in the context of modern cosmology and structure formation:

1. Pseudo-evolution: For halos that have stopped accreting mass, the virial radius evolves over time solely due to the changing background density of the universe, rather than reflecting physical structural changes.
2. Halo Exclusion: Virial radii do not account for the non-virialized material surrounding halos. This creates gaps in the cosmic density field when modeling large-scale structure, as the space between virial radii is left unaccounted for.
3. Assembly Bias: The spatial clustering of halos (bias) depends not just on mass, but also on the environment outside the virial radius. The virial definition is incomplete for describing halo properties on the largest scales.

The paper argues that a realistic dark matter halo is a stratified structure undergoing continuous growth, requiring multiple physical boundaries to characterize its inner equilibrium, growth layer, and infall region.

2. Methodology

The paper employs a combination of analytical derivations, theoretical modeling, and numerical simulation analysis:

• Analytical Framework: The author derives and reviews solutions for the Spherical Collapse (SC) model under $\Lambda$CDM cosmology. This includes:
  • Monolithic SC: Deriving the turnaround and virial radii for a uniform spherical perturbation.
  • Self-Similar Secondary Infall: Extending the SC model to account for continuous mass accretion (secondary infall) where shells cross. The author derives equations for self-similar collapse with power-law initial perturbations ($\delta \propto r^{-\epsilon}$) and generalizes them to realistic NFW-like profiles.
  • New Derivations: The paper provides the first analytical derivation of the depletion radius within the self-similar collapse framework.
• Numerical Tools: The author introduces SPHERIC, a Python package implementing these monolithic and self-similar collapse models to calculate key boundary radii.
• Simulation Analysis: The theoretical predictions are compared against cosmological N-body and hydrodynamical simulations (e.g., IllustrisTNG, FLAMINGO) to validate boundary locations and their dependence on mass accretion rates ($\Gamma$) and cosmology.
• Tracer Comparison: The study analyzes boundaries across different tracers: Dark Matter (DM), gas (via shock radii), galaxies, subhalos, and halo stars.

3. Key Contributions and Definitions

The paper defines and characterizes a hierarchy of boundaries that replace the single virial radius:

1. Turnaround Radius ($R_{ta}$): The largest scale where particles decouple from the Hubble flow ($v_r = 0$) and begin falling inward. It encloses the entire infall region.
2. Splashback Radius ($R_{sp}$): Defined as the location of the steepest slope in the density profile. Physically, it corresponds to the first apocenter of particles falling into the halo. It marks the boundary where infalling material "splashes back" due to the growing gravitational potential.
   • Key Finding: $R_{sp}$ is sensitive to the mass accretion rate; faster accretion leads to a smaller $R_{sp}$.
3. Depletion Radius ($R_{id}$): Defined as the radius where the net mass inflow rate is maximized (or where the density profile transitions from growing to decreasing). It marks the outer edge of the "splashback" region where the density begins to drop due to the draining of the surrounding environment.
   • Key Finding: In simulations, $R_{id}$ is typically $\sim 2 \times R_{vir}$, significantly larger than $R_{sp}$ ($\sim 1.25 \times R_{vir}$).
4. Edge Radius ($R_{edge}$): The radius where the fraction of orbiting particles drops to a negligible level (e.g., 1%). It is found to be close to the depletion radius.
5. Shock Radius ($R_{shock}$): For gas tracers, this is the location of the accretion shock. It is generally larger than $R_{sp}$ and depends on the adiabatic index and merger history.

4. Key Results

• Stratified Structure: The halo is confirmed to be a multi-layered object:
  • Inner: Virialized region ($< R_{vir}$).
  • Middle: Growing/Splashback region ($R_{vir} < r < R_{sp}$).
  • Outer: Depleting region ($R_{sp} < r < R_{id}$).
  • Outermost: Expanding/Infall region ($R_{id} < r < R_{ta}$).
• Accretion Rate Dependence: Both $R_{sp}$ and $R_{id}$ are strongly dependent on the halo's mass accretion rate ($\Gamma$).
  • $R_{sp}/R_{vir} \approx 0.75 + 1.04 e^{-\Gamma/3}$.
  • $R_{id}/R_{vir} \approx 1.35 + 0.42 \ln \Omega_m + [1.12 - 0.51 \ln \Omega_m]e^{-\Gamma/3}$.
• Tracer Differences:
  • Dark Matter: Shows clear caustics and splashback features.
  • Gas: The shock radius is often larger than the DM splashback radius and is more sensitive to mergers (runaway shocks).
  • Galaxies/Stars: Follow DM dynamics but with slight offsets due to dynamical friction (massive satellites splash back closer) and substructure.
• Halo Model Improvement: The paper demonstrates that using the Depletion Radius as the halo boundary resolves the "halo exclusion" problem.
  • A new Depletion Halo Model (DHM) is proposed, which treats the matter field as a convolution of the halo field and halo profiles.
  • The DHM achieves $\sim 5\%$ accuracy in predicting halo-matter and matter-matter clustering across linear and nonlinear scales without fine-tuning, outperforming classical virial-based models.
• Bias Minimum: The minimum in the nonlinear bias profile around a halo corresponds physically to the depletion radius, providing an observational proxy for it.

5. Significance and Future Outlook

• Physical Halo Definition: The paper shifts the paradigm from a static, virialized view to a dynamic, stratified view of halos, providing a more physically consistent definition for cosmological modeling.
• Cosmological Probes: The new boundaries offer independent probes for cosmology.
  • $R_{sp}$ is sensitive to the dark energy equation of state and modified gravity screening mechanisms.
  • $R_{ta}$ (and its upper limit) can constrain dark energy and the expansion history.
  • The infall region contains complementary information to cluster abundance, potentially breaking the $\sigma_8$-$\Omega_m$ degeneracy.
• Galaxy Formation: The boundaries provide a natural framework for classifying central vs. satellite galaxies and understanding the quenching of star formation (e.g., the transition in quenched fraction aligns with the depletion radius).
• Theoretical Tools: The release of the SPHERIC package and the analytical derivations for the depletion radius provide essential tools for the community to test these models against upcoming high-precision data from surveys like Euclid, Rubin Observatory, and Roman Space Telescope.

In conclusion, this review synthesizes the physics of halo boundaries, validates them through simulations, and demonstrates their critical role in solving long-standing issues in the halo model and large-scale structure formation.