⚛️ general relativity

Scalar Field Model for Dark Matter Spikes Surrounding Sgr A $^$ and M87 $^$

This paper proposes a real scalar field model with a non-standard potential to describe dark matter spikes around supermassive black holes like Sgr A $^*$ and M87 $^*$ , reverse-engineering the potential to match desired density profiles and enabling the study of testable geometric effects and gravitational perturbations including backreaction.

Original authors: Ramin G. Daghigh, Michael D. Green, Gabor Kunstatter

Published 2026-09-03

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ramin G. Daghigh, Michael D. Green, Gabor Kunstatter

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 most galaxies, including our own, lies a supermassive black hole, a region of space so dense that nothing, not even light, can escape its pull. For decades, astronomers have treated these cosmic giants as isolated objects, but recent thinking suggests they are rarely alone. They are typically surrounded by vast, invisible clouds of dark matter, a mysterious substance that makes up most of the universe's mass but does not emit or reflect light. When a black hole grows slowly within such a cloud, the gravitational tug of the black hole can pull the dark matter inward, compressing it into an incredibly dense, sharp peak right next to the event horizon. This dense region is known as a "spike." Understanding the shape and behavior of these spikes is crucial because they might subtly alter the signals we receive from black holes, such as the ripples in spacetime called gravitational waves or the dark silhouette, or shadow, that a black hole casts against the background of glowing gas. If we can detect the fingerprint of a dark matter spike, it would provide a rare, direct clue about the nature of this invisible substance.

In a new study, researchers set out to model these dark matter spikes around two of the most famous black holes in the sky: Sagittarius A* at the center of our Milky Way and M87* in a distant galaxy. Instead of treating the dark matter as a simple, invisible fluid, the team proposed a more complex description where the dark matter behaves like a real, physical field that permeates space, similar to how a magnetic field exists around a magnet. They started with a known idea: that the density of the dark matter spike should drop off in a specific, predictable way as you move away from the black hole. Working backward from this known density, they asked a reverse question: what kind of underlying force or potential energy would be required to hold this specific arrangement of matter in place? By solving the equations of gravity and matter together, they were able to construct a mathematical model that describes the dark matter spike as a stable structure held together by a specific type of self-interacting field.

The researchers found that to create a realistic spike, the dark matter cannot be completely passive; it must exert pressure. In many previous models, scientists assumed the dark matter had no pressure pushing outward, but this new work shows that such a scenario is impossible for a stable spike. The dark matter must push back against the crushing gravity of the black hole, and this pressure changes the geometry of the space around the hole. When the team calculated the effects of this pressure, they discovered that the dark matter spike acts like a lens, slightly stretching the space around the black hole. This stretching causes the black hole's shadow to appear larger than it would if the dark matter were absent. For the massive black hole in galaxy M87, the model predicts that the shadow could be enlarged by about 1.4 percent in the most optimistic scenario. This is a small but potentially measurable difference, one that current telescopes, like the Event Horizon Telescope which first imaged M87, might be able to detect.

Beyond the size of the shadow, the presence of this dense dark matter field also changes how the black hole rings like a bell after being disturbed. When a black hole is shaken, perhaps by a collision with another object, it emits gravitational waves that fade away in a specific pattern known as a ringdown. The study shows that the dark matter spike alters the frequency of these waves, shifting them slightly from what we would expect for a black hole in empty space. The researchers calculated that these shifts are large enough to be noticeable, changing the second decimal place of the wave's frequency. This suggests that if we listen closely enough to the gravitational waves from black holes, we might hear the echo of the dark matter surrounding them.

The team also explored the mathematical nature of the force holding this dark matter together. They found that the potential energy required to sustain the spike follows a very simple, bell-shaped curve, a form often seen in basic physics problems. This simplicity is surprising given the complex environment near a black hole. However, the researchers are careful to note that their model is specific to the conditions around these two black holes and does not yet claim to be a universal theory of dark matter for the entire universe. The parameters that define the dark matter field around Sagittarius A* are different from those around M87, suggesting that the dark matter might behave differently depending on its local environment. While the study does not prove that dark matter is a scalar field, it demonstrates that this idea is a viable way to model the spikes and that such a model produces observable effects. By providing a complete mathematical framework that includes the back-reaction of the dark matter on the black hole's gravity, the work opens a new door for testing these ideas against real astronomical data, turning the invisible dark matter spike from a theoretical curiosity into a potential target for observation.

Technical Summary: Scalar Field Model for Dark Matter Spikes Surrounding Sgr A and M87**

Problem Statement
Theoretical models suggest that the adiabatic growth of a supermassive black hole (SMBH) immersed in a dark matter (DM) halo leads to the formation of high-density regions, known as "spikes," near the event horizon. While previous studies have analytically modeled these spikes using cold dark matter (CDM) treated as a perfect fluid with negligible radial pressure, such approaches fail to account for the backreaction of the DM spike on the background geometry. This limitation is critical because DM spikes are substantially denser near the horizon than galactic halos, where non-linear gravitational effects may be significant. Furthermore, standard phenomenological stress-energy tensors are insufficient for studying coupled dynamics, such as gravitational perturbations and the backreaction of the spike itself. The authors posit that a field-theoretic description, specifically using a real scalar field, offers a more manageable framework to derive a covariant action, thereby enabling the study of linearized perturbations and backreaction effects.

Methodology
The authors employ a "reverse engineering" approach to determine the scalar field potential $V(\phi)$ required to model the DM spikes around M87* and Sgr A*. The methodology proceeds as follows:

Model Setup: The system is modeled using a real Klein-Gordon scalar field minimally coupled to gravity. The action includes the Einstein-Hilbert term and the scalar field Lagrangian with an unknown potential $V(\phi)$ .
Constraint by Density Profile: The authors fix the DM spike density profile, $\rho(r) \propto r^{-\gamma_{sp}}$ , based on established theoretical predictions for the adiabatic growth of black holes. They utilize specific parameters for M87* and Sgr A* found in the literature.
Solving for Pressure and Metric: Using the Tolman-Oppenheimer-Volkoff (TOV) equations combined with the scalar field stress-energy tensor, the authors solve for the radial pressure $p_r(r)$ and the metric functions numerically. A key finding in this step is that configurations with zero radial pressure are not possible for this scalar field model; the pressure must be non-zero to satisfy the equations of motion and energy conditions.
Deriving the Potential: Once the density, pressure, and metric components are determined as functions of the radial coordinate $r$ , the authors numerically integrate to find the scalar field profile $\phi(r)$ . By inverting this relationship, they determine the scalar potential $V(\phi)$ that yields the desired density profile as a static solution.
Analytic Fitting: The numerically derived potential is fitted to a simple analytic form (a Gaussian function) to facilitate further analysis.
Observational Consequences: With the full metric and potential established, the authors calculate the effective potential for scalar perturbations to determine the ringdown waveform (Quasi-Normal Modes or QNMs) and the enlargement of the black hole shadow radius.

Key Contributions and Results

Reverse-Engineered Potential: The study successfully derives a scalar field potential that supports a DM spike density profile. The resulting potential is found to be well-approximated by a simple Gaussian function: $\hbar V(\phi) = -A e^{-(\sqrt{\hbar}\phi - C)^2 / B^2}$ .
Necessity of Radial Pressure: The authors analytically and numerically demonstrate that a DM spike modeled by this scalar field cannot have zero radial pressure. Analytic exploration of the asymptotic behavior shows that pressure dropping faster than density is inconsistent, while dropping at the same rate is mathematically consistent. However, numerical calculations indicate that the radial pressure actually drops more slowly than the density (specifically, $p_r \propto 1/(r^2 \ln r)$ ), a behavior supported by the numerical results.
Effective Mass Discrepancy: The effective mass of the scalar field, derived from the curvature of the potential at $\phi=0$ , is calculated for both M87* and Sgr A*. The results show a significant dependence on the black hole mass, with the scalar field mass around Sgr A* being three orders of magnitude larger than that around M87*. The authors note this dependence as a limitation, indicating the model is not a fundamental theory of dark matter applicable universally.
Ringdown Waveform Modifications: The presence of the scalar field spike alters the Regge-Wheeler effective potential. Consequently, the ringdown waveform and the first two QNM frequencies differ from the standard Schwarzschild case. The authors report that these differences are visible up to the second decimal place in the frequency values.
Shadow Radius Enlargement: The presence of the DM spike introduces a redshift factor $C$ , which enlarges the apparent black hole shadow radius by a factor of $\sqrt{C}$ . For the most optimistic parameters for M87* ( $\alpha_\gamma = 1.94$ ), the shadow radius enlargement is calculated to be approximately 1.4%, which the authors state is potentially detectable with current Event Horizon Telescope (EHT) data. For other parameter sets and for Sgr A*, the enlargement is significantly smaller (below 0.1%), falling below current EHT sensitivity.

Significance and Claims
The paper claims that its primary significance lies in providing a covariant action-based framework for modeling DM spikes, which is a prerequisite for studying the backreaction of the spike on gravitational perturbations—a feature ignored in earlier literature. By establishing a specific scalar potential, the authors enable the calculation of the coupled dynamics of the metric and the DM field.

The authors are modest regarding the broader implications of their work. They explicitly state that the goal is not to construct a fundamental theory of dark matter, evidenced by the fact that the scalar potential parameters differ between M87* and Sgr A*. Instead, the work serves as a proof-of-concept to model the spike using a covariant action to study the QNM spectrum and ringdown of the coupled gravity-DM system. The authors note that a full study of the coupled scalar and gravitational perturbations (analogous to Reissner-Nordström black holes) will be the subject of a future publication. The current results regarding ringdown and shadow radius are presented as qualitative analyses based on perturbations in a fixed background, with the caveat that the scalar field mass dependence on the black hole mass limits the model's universality.

More like this