Dark matter mounds from the collapse of supermassive stars: a general-relativistic analysis

This paper develops a fully general-relativistic formalism to model the formation of shallower dark matter "mounds" around supermassive black holes resulting from the non-adiabatic collapse of supermassive stellar progenitors, revealing significant reshaping of the dark matter phase-space distribution that is crucial for interpreting future extreme mass-ratio inspiral observations.

The Gist

The Big Picture: The Cosmic "Haircut"

Imagine the center of a galaxy as a giant, invisible dance floor. On this floor, Dark Matter particles are dancing in a slow, organized circle around a central point. Usually, scientists thought that if a massive black hole formed right in the middle of this dance floor, it would act like a vacuum cleaner, sucking the dancers closer and closer until they formed a super-dense, sharp peak right at the center. They called this a "Spike."

However, this new paper argues that the reality is messier and more interesting. Instead of a sharp spike, the black hole creates a "Mound"—a hill that is still high, but much flatter and less crowded at the very top.

The Story: How the Black Hole Was Born

To understand why the shape changes, we have to look at how the black hole was born. The paper compares two different ways a black hole can grow:

1. The Slow Growth (The "Spike" Scenario)

Imagine a black hole seed that is tiny, like a grain of sand. Over millions of years, it slowly eats more and more matter, growing into a giant.

• The Analogy: Think of a slow-growing tree in a forest. The birds (dark matter) have plenty of time to adjust their nests as the tree grows. They slowly shuffle inward, getting closer and closer to the trunk. Because the change is so slow, the birds can perfectly rearrange themselves into a tight, dense cluster.
• The Result: A sharp, steep Spike of dark matter.

2. The Fast Collapse (The "Mound" Scenario)

This is what the paper focuses on. Imagine a "Supermassive Star" (a star 100,000 times heavier than our Sun) that forms and then suddenly collapses into a black hole. This happens very fast—like a building imploding in seconds.

• The Analogy: Imagine the same dance floor, but instead of a tree growing slowly, the floor suddenly drops out from under the dancers, or a giant trapdoor opens instantly. The dancers (dark matter) are caught off guard. They are still moving at their old speeds, but the gravity has suddenly changed.
• The Result: Many dancers get thrown off balance. Some are flung outward, and some fall straight into the trap (the black hole). The ones that stay don't form a tight cluster; they form a loose, bumpy Mound.

The "Relativistic" Twist

The authors didn't just guess this; they used Einstein's theory of General Relativity (the rulebook for how gravity works when things are huge and moving fast) to simulate it.

They found that when the star collapses, the "dance floor" changes so quickly that the dark matter particles don't have time to adjust.

• The "Depletion": The particles that were closest to the center (the ones with the lowest energy) get swallowed by the new black hole immediately.
• The "Flattening": The particles that survive are pushed into wider orbits. This leaves a "hole" or a depletion right at the very center of the distribution. Instead of a sharp needle (spike), you get a rounded hill (mound).

Why Does This Matter? (The "Ear" Analogy)

You might ask, "Who cares if it's a spike or a mound?"

The answer lies in Gravitational Waves. These are ripples in space-time, like sound waves, created when two heavy objects orbit each other. Future space telescopes (like LISA) will listen for these waves.

• The Analogy: Imagine you are trying to hear a specific violin note (the gravitational wave) in a crowded room.
  • If the room is filled with a Spike of people (dense dark matter), the air is thick. The violin sound gets muffled and distorted in a specific way.
  • If the room has a Mound (less dense center), the air is thinner in the middle. The sound travels differently.

By listening to the "dephasing" (the slight change in the rhythm of the wave), scientists can tell if the black hole grew slowly (Spike) or formed from a sudden collapse (Mound).

The Conclusion: A Reset Button

The paper also notes that if this new black hole keeps eating matter for a long time after the collapse, it might eventually smooth out the "Mound" and turn it back into a "Spike." It's like if you keep adding sand to the hill; eventually, the shape might look like the slow-growth version again.

In summary:
This paper tells us that the history of a black hole leaves a fingerprint on the dark matter around it. If the black hole was born from a sudden, violent collapse of a giant star, the dark matter around it won't be a sharp spike, but a flatter, "depleted" mound. Detecting this difference will help us understand how the giants of the universe were born.

Technical Summary

1. Problem Statement

The formation and growth of Supermassive Black Holes (SMBHs) significantly alter the distribution of Dark Matter (DM) in their vicinity. While the "adiabatic growth" scenario (where a black hole grows slowly from a seed) is well-understood and predicts a steep density enhancement known as a "spike," the formation of SMBHs via direct collapse of Supermassive Stars (SMS) remains less explored.

In direct-collapse scenarios, the gravitational potential changes on a timescale comparable to or shorter than the dynamical timescale of inner DM orbits. This breaks the adiabatic approximation, potentially leading to a shallower DM overdensity, termed a "mound." Previous studies (e.g., Ref. [47]) addressed this using semi-relativistic Monte Carlo simulations with an instantaneous collapse approximation. However, a fully general-relativistic (GR) treatment of the collisionless DM phase-space distribution during the non-adiabatic collapse phase was lacking. This gap is critical because the precise DM density profile affects the orbital evolution of Extreme Mass-Ratio Inspirals (EMRIs), which are prime targets for future space-based gravitational wave (GW) detectors like LISA.

2. Methodology

The authors developed a fully self-consistent general-relativistic framework to track the evolution of the DM distribution function ($f$) through three distinct phases:

• Initial State (NFW Halo): The simulation begins with a DM halo described by a Navarro-Frenk-White (NFW) profile. The initial distribution function is derived using the Eddington inversion procedure in the Newtonian limit, assuming isotropy.
• Adiabatic SMS Growth: The supermassive star grows via accretion ($\sim 0.1 M_\odot \text{yr}^{-1}$) until it reaches $\sim 10^5 M_\odot$. During this phase, the growth is slow compared to DM orbital timescales. The authors use relativistic adiabatic invariants (conservation of angular momentum $L$ and radial action $I_r$) to map the initial DM distribution to the pre-collapse state around the SMS. The SMS interior is modeled as a uniform density perfect fluid satisfying the Tolman-Oppenheimer-Volkoff (TOV) equations.
• Non-Adiabatic Collapse (Oppenheimer-Snyder): The SMS collapses into a Schwarzschild black hole. The authors model this using the Oppenheimer-Snyder (OS) solution for a pressureless dust sphere.
  • Matching: They ensure continuity of the metric and extrinsic curvature across the collapse hypersurface ($\Sigma_0$) using Israel's junction conditions.
  • Geodesic Tracking: They numerically integrate the geodesic equations for DM particles backward in time from a final static state ($t_f$, where the star radius $r_* \leq 4M$) to the pre-collapse state ($t=0$). This accounts for particles that remain outside the star, those that cross the surface twice, and those that start inside.
  • Phase Mixing: Recognizing that the collapse induces violent relaxation and phase mixing, the authors compute the coarse-grained post-collapse distribution function ($f_c$) by averaging the distribution over the orbital period in the final static metric, rather than assuming a time-dependent fine-structure.
• Regrowth: The framework also models a subsequent phase of adiabatic accretion to see if the "mound" features are erased as the black hole grows further.

3. Key Contributions

• First Fully GR Treatment of Direct Collapse: This is the first study to provide a self-consistent general-relativistic mapping between the pre- and post-collapse DM phase-space distributions for a direct-collapse scenario, moving beyond semi-relativistic or instantaneous approximations.
• Refined "Mound" Profile: The authors derive the specific density profile of the resulting DM "mound," showing it is significantly shallower than the adiabatic "spike" but less suppressed than predicted by instantaneous collapse models.
• Phase-Space Depletion Analysis: They explicitly demonstrate that the non-adiabatic collapse leads to a depletion of low-binding-energy orbits (specifically circular and near-circular orbits) in the phase space $(L, 1-E)$. This is a direct consequence of the rapid change in the potential well, which ejects or captures particles that would have remained bound in an adiabatic scenario.
• Quantification of Regrowth Effects: The paper establishes a critical threshold for black hole regrowth. They find that if the black hole mass increases by a factor of $\approx 5.5$ (specifically $M_f \approx M \sqrt{R_*/8M}$) after the collapse, the memory of the non-adiabatic formation is erased, and the profile converges to the standard adiabatic spike.

4. Key Results

• Density Profiles: The resulting DM rest-mass density profile (the "Relativistic Mound") shows a sharp enhancement near the black hole compared to the initial NFW halo, but the inner density is approximately 5 times lower than that of an adiabatically formed spike at the innermost stable circular orbit ($r=6M$).
• Slope Differences: The logarithmic slope ($\gamma = -d \log \rho / dr$) of the mound is significantly flatter than the spike in the inner regions, which directly impacts the dynamical friction and accretion rates for EMRIs.
• Phase Space Structure:
  • Adiabatic Spike: Retains a population of particles on circular orbits with low binding energy.
  • Direct Collapse Mound: Exhibits a clear depletion in the region of phase space corresponding to low binding energy and high angular momentum (circular orbits). These particles were either captured by the black hole or scattered to unbound states during the rapid potential change.
• Impact of Approximations: Comparing the full OS collapse to an "instantaneous" collapse limit (Appendix B), the realistic OS model yields a less suppressed central density. The instantaneous approximation overestimates the depletion of the spike because it neglects the finite time evolution of the metric and the specific geodesic dynamics during the collapse.

5. Significance

• Gravitational Wave Astronomy: The density and velocity distribution of DM near SMBHs directly influence the dephasing of gravitational wave signals from EMRIs. The distinct "mound" profile implies a different cumulative phase shift compared to the "spike" scenario. Future observations by LISA could potentially distinguish between these formation histories.
• SMBH Formation History: The results provide a theoretical tool to infer the formation mechanism of SMBHs (direct collapse vs. light-seed adiabatic growth) based on the observed DM environment.
• Dark Matter Nature: By constraining the DM distribution, these findings help refine indirect detection limits and the interpretation of GW data regarding the nature of dark matter (e.g., self-interacting or annihilating properties).
• Methodological Advancement: The developed formalism for evolving relativistic distribution functions across non-adiabatic metric transitions is applicable to other astrophysical scenarios involving rapid gravitational changes.

In conclusion, the paper establishes that while direct collapse creates a significant DM overdensity, the non-adiabatic nature of the event fundamentally reshapes the phase space, creating a "mound" with a depleted core of circular orbits. This signature is robust enough to potentially be detected by future gravitational wave observatories, offering a new window into the earliest stages of black hole formation.