⚛️ phenomenology

Dark Matter phase space distribution under a fully relativistic treatment. Eddington inversion according to general relativity

This paper presents a fully relativistic extension of the Eddington inversion method to reconstruct the dark matter phase-space distribution in static, spherically symmetric spacetimes, demonstrating that relativistic effects significantly alter velocity distributions and capture rates near supermassive black holes compared to classical Newtonian predictions.

Original authors: Jaime Hoefken Zink, Pellegrino Piantadosi

Published 2026-10-27

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Jaime Hoefken Zink, Pellegrino Piantadosi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Deep in the heart of our galaxy, where the gravity is so intense that it warps the very fabric of space and time, a hidden population of matter is gathering. This invisible substance, known as dark matter, makes up most of the mass in the universe, yet it refuses to emit or reflect light, revealing itself only through its gravitational pull. For decades, scientists have tried to map how this mysterious material moves and spreads, because knowing its speed and direction is essential for detecting it in laboratories on Earth or understanding how it interacts with stars. However, the tools used to create these maps were built for a universe where gravity is weak and gentle. They break down when applied to the extreme environment surrounding a supermassive black hole, the massive object anchoring the center of the Milky Way. In this region, the old rules of physics no longer apply, and a new, more rigorous approach is needed to see what is really happening.

Two researchers, Jaime Hoefken Zink and Pellegrino Piantadosi, have developed a new method to solve this problem by applying the full power of Einstein's theory of general relativity. Instead of treating gravity as a simple force that pulls objects together, their work treats it as a curvature of space and time that changes how particles move and how we measure their energy. They started with a known map of how dense the dark matter is at different distances from the galactic center. Using complex equations that describe how matter and gravity balance each other in a static, spherical system, they worked backward to reconstruct the speed distribution of the dark matter particles. This process, known as Eddington inversion, had previously been limited to simpler, non-relativistic calculations. By upgrading this technique to handle the extreme gravity near the black hole, the team created a fully relativistic version that reveals how dark matter behaves when it is pushed to its limits.

When the researchers applied their new method to a model of the Milky Way, which includes a standard halo of dark matter and a dense spike of material swirling around the central black hole, the results were striking. Far away from the galactic center, where gravity is weaker, their new relativistic calculations matched the old, classical ones perfectly. The speed of the dark matter particles looked the same whether calculated with the new, heavy-duty math or the traditional, lighter approach. However, as they moved closer to the black hole, the two methods began to diverge. At distances of about one ten-thousandth of a parsec from the center, the difference became visible. Closer still, within a few millionths of a parsec, the discrepancy grew dramatic. The new relativistic model showed that the dark matter particles were moving significantly faster than the old models predicted, with peak speeds reaching nearly twice the values estimated by classical physics.

The study also examined how these different speed distributions affect the likelihood of dark matter being captured by other objects, specifically neutron stars, which are the incredibly dense remnants of exploded stars. The researchers calculated a factor that determines how much dark matter a neutron star can grab as it moves through the galaxy. They found that using the standard, classical approach or the common assumption that dark matter moves like a gas of particles in a simple, random pattern leads to significant errors near the galactic center. The classical method overestimates the capture rate by a factor of about two, while the simple gas assumption overestimates it by nearly ten times. In contrast, the new relativistic calculation shows that the capture rate is much lower than previously thought in these extreme regions. This suggests that previous predictions about how much dark matter accumulates in stars near the black hole were too high.

The researchers also addressed a technical issue that often plagues these calculations: a mathematical glitch that can cause the results to blow up into infinity at the edges of the model. They developed a procedure to smooth out the density profile at the boundaries, ensuring that the final results remain physically realistic and free of these unphysical spikes. By doing so, they provided a stable and reliable way to calculate the speed of dark matter anywhere in the galaxy, from the quiet outskirts to the violent heart. Their work demonstrates that to truly understand dark matter in the most extreme environments, scientists must abandon the simplified rules of the past and embrace the full complexity of Einstein's gravity. This new perspective is crucial for the next generation of experiments, ensuring that when we finally detect dark matter, we know exactly what we are looking for and where to look.

Technical Summary: Dark Matter Phase Space Distribution under a Fully Relativistic Treatment

Problem Statement
The Standard Model of particle physics cannot account for dark matter (DM), a dominant component of the universe. Accurate knowledge of the local and global DM velocity distribution is critical for predicting recoil rates in direct detection experiments, calculating annual modulation signatures, evaluating capture rates in stellar bodies, and determining flux predictions for indirect searches. Historically, these studies have relied on the Standard Halo Model (SHM), which assumes an isotropic Maxwell-Boltzmann speed distribution truncated at an ad hoc escape velocity. However, the SHM lacks dynamical self-consistency in realistic, non-isothermal galactic potentials.

While the classical Eddington inversion method allows for the reconstruction of the phase-space distribution function (PSDF) from a density profile, it is based on a weak-field, Newtonian gravitational framework. This approximation ceases to be valid in the strong gravitational fields near the Galactic Center, specifically in the vicinity of the supermassive black hole Sagittarius A* (Sgr A*), where a dense "dark matter spike" is hypothesized to exist. There is a need for a self-consistent method to reconstruct the DM PSDF under a fully relativistic (Einsteinian) description of gravity to accurately model interactions in these extreme environments.

Methodology
The authors develop a fully relativistic extension of the Eddington inversion method. The core methodology involves the following steps:

Relativistic Framework: Dark matter is treated as an effective perfect fluid within a static, spherically symmetric spacetime. The gravitational background is determined using the Tolman-Oppenheimer-Volkoff (TOV) equations, which relate the DM density profile $\rho(r)$ to the metric components, specifically the time component $g_{tt}$ and pressure $P(r)$ .
Relativistic Inversion: The authors derive an Abel-type integral equation relating the density to the distribution function $f(E)$ , where $E$ is the conserved particle energy measured at infinity. By transforming variables to isolate the distribution function, they derive an expression for $f(E)$ in terms of the density profile and metric derivatives.
Boundary Term Regularization: The authors identify that the relativistic inversion formula contains a boundary term that can introduce unphysical divergences when the integration is performed over a finite radius (finite escape velocity). They propose a regularization procedure that modifies the density profile $\rho \to \tilde{\rho}$ to ensure the boundary term vanishes identically, thereby removing the divergence while maintaining consistency.
Velocity Distribution (VDF) Transformation: The authors derive the transformation required to convert the energy-based distribution $f(E)$ into a velocity distribution function $f_v(v)$ in the relativistic regime, noting the presence of an additional $\gamma^6$ factor compared to the classical case.
Application to the Milky Way: The method is applied to a Milky Way model consisting of a generalized Navarro-Frenk-White (gNFW) halo and a Gondolo-Silk dark matter spike around Sgr A*. Baryonic contributions are neglected, as the focus is on the relativistic effects near the black hole.

Key Contributions

Derivation of Relativistic Eddington Inversion: The paper presents a closed-form expression for the DM phase-space distribution function derived directly from the density profile using General Relativity, recovering the classical Eddington result in the weak-field limit.
Regularization of Boundary Divergences: A specific procedure is introduced to regularize the density profile to eliminate unphysical divergences arising from finite boundary terms in the inversion integral.
General Capture Rate Formula: A general expression for the dark matter capture rate by neutron stars is derived, accommodating arbitrary ambient velocity distributions under both classical and relativistic approaches.
Comparative Analysis: The study provides a direct comparison between the fully relativistic (REL), non-relativistic (NR/Eddington), and Maxwell-Boltzmann (MB) approaches across different galactic radii.

Results

Velocity Distributions: Far from the Galactic Center (radii $> 10^{-5}$ pc), the relativistic and Newtonian velocity distributions agree. However, at distances of order $10^{-5}$ pc and closer to Sgr A*, the distributions diverge. In the strong-field region, the relativistic peak speed can reach approximately twice the value predicted by the Newtonian Eddington inversion. The relativistic distribution also exhibits a more concentrated profile with less spread compared to the Newtonian and MB models.
Maxwell-Boltzmann Discrepancy: The standard Maxwell-Boltzmann approximation predicts significantly lower velocities near the Galactic Center compared to both inversion methods and fails to capture the relativistic peak.
Accretion Rates: The differences in velocity distributions significantly impact the calculated accretion (capture) rates of DM by neutron stars. Near Sgr A*, the relativistic capture factor is nearly an order of magnitude smaller than the Maxwell-Boltzmann prediction. Conversely, using the classical Newtonian Eddington result can overestimate the capture rate by a factor of approximately two compared to the fully relativistic treatment.
Stability: The relativistic method remains numerically stable at all positions in the galaxy, whereas the classical method requires analytical expressions for the potential and density derivatives to avoid numerical instability near the center.

Significance
The paper demonstrates that the reconstruction of the dark matter phase-space distribution in strong gravitational fields requires a fully relativistic treatment to avoid significant systematic errors. The authors conclude that for phenomenology involving dark matter near supermassive black holes, such as capture by neutron stars or indirect detection signals, the use of Newtonian approximations or Maxwell-Boltzmann distributions can lead to overestimations or incorrect predictions of interaction rates. The developed method provides a necessary tool for precision particle astrophysics in the era of high-precision observations near the Galactic Center.

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