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Scalar-Assisted Growth of Supermassive Black Holes: Boson-Mass Windows and Observational Constraints

This paper proposes a phenomenological model where supermassive black holes grow via simultaneous baryonic accretion and scalar cloud decay, revealing that while boson masses around $10^{-19}$ to $10^{-16}$ eV maximize early growth efficiency, observational constraints from quasar data favor lower masses near $10^{-21}$ to $10^{-20}$ eV, suggesting distinct bosonic scales may play complementary roles in black hole formation and dark matter phenomenology.

Original authors: Jorge Castelo Mourelle, Darío Núñez Zúñiga, Saeed Fakhry, Nicolas Sanchis-Gual

Published 2026-10-15

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CC BY 4.0

Original authors: Jorge Castelo Mourelle, Darío Núñez Zúñiga, Saeed Fakhry, Nicolas Sanchis-Gual

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

In the earliest moments of the universe, less than a billion years after the Big Bang, something impossible seemed to be happening. Astronomers have found massive black holes, weighing billions of times more than our sun, already fully formed and shining as bright quasars. According to standard theories, these objects should have taken much longer to grow. They are like giants that appeared fully grown in a nursery, defying the slow, steady pace of eating and growing that usually governs the cosmos. The puzzle is not just that they exist, but how they gathered so much mass so quickly. While some theories suggest these black holes started with unusually large seeds or ate gas at record-breaking speeds, there remains a nagging possibility that they were fed by something else entirely: a hidden form of matter that does not shine, does not push back with light, and can slip directly into the black hole's grasp.

This hidden matter is thought to be made of ultralight particles, a type of boson that behaves more like a wave than a solid object. In the vast space surrounding a black hole, these particles can form a dense, swirling cloud. The new research explores a specific scenario where this cloud acts as a second food source, feeding the black hole alongside the ordinary gas and dust that we can see. The scientists wanted to know if this invisible feeding channel could explain the rapid rise of these cosmic giants. They built a computer model to simulate the life of a black hole from its birth in the early universe to the present day. In this simulation, the black hole eats from two bowls at once: one bowl contains the normal gas of the universe, which it swallows at a rate limited by the pressure of its own light, and the other bowl contains the invisible cloud of bosonic particles, which it absorbs through a different, more direct mechanism.

The researchers ran thousands of simulations, changing the weight of the invisible particles and the size of the starting black hole to see what would happen. They found that the invisible cloud can indeed act as a powerful booster, but only under very specific conditions. The efficiency of this extra feeding depends on a precise match between the mass of the black hole and the mass of the invisible particles. If the particles are too heavy or too light for the size of the black hole, the cloud remains stable and barely feeds the hole at all. However, when the numbers align just right, the cloud collapses rapidly, dumping a huge amount of mass into the black hole in a short burst. This process can accelerate the growth of the black hole significantly, allowing it to reach the massive sizes we see in the early universe much sooner than it could by eating gas alone.

The study identified a specific range of particle masses where this rapid feeding works best. For the black holes the team modeled, the most effective invisible particles have a mass between 10^-19 and 10^-16 electron volts. In this range, the black hole can grow so fast that it reaches a billion times the mass of the sun at a much earlier time in the universe's history than standard models allow. This suggests that if such particles exist, they could solve the mystery of how these giants formed so quickly. However, the story becomes more complicated when the researchers compare their results with actual observations of the universe.

When the team checked their models against the real population of quasars and black holes we can see today, the picture changed. The models that produced the fastest-growing black holes did not match the observed distribution of these objects as well as models with much lighter particles did. The simulations that best fit the real data, matching both the mass and the brightness of the quasars we observe, pointed to a different range of particle masses, roughly between 10^-21 and 10^-20 electron volts. In this lighter range, the invisible cloud still feeds the black hole, but it does so more gently and over a longer period. This slower, steadier contribution helps the black hole grow in a way that looks more like the real universe we see, without the extreme, rapid bursts that would make the black holes look different from what astronomers actually measure.

The researchers conclude that there is a tension between the mechanism that creates the fastest growth and the mechanism that best fits the observations. The most efficient way to build a giant black hole quickly requires a heavier particle mass, but the way the universe actually looks suggests a lighter particle mass is more likely. This does not mean the idea is wrong; rather, it suggests that the role of these particles might be more nuanced. It is possible that different masses play different roles, or that the most efficient growth phase happens early and briefly, while the lighter particles provide a steady background contribution that shapes the final population. The work highlights that while invisible matter could be a key ingredient in the recipe for supermassive black holes, the exact type of ingredient matters. The study provides a clear map of where these particles could be hiding and how they might be shaping the history of the cosmos, leaving the door open for future observations to pinpoint exactly which mass, if any, is responsible for feeding the giants of the early universe.

Technical Summary: Scalar-Assisted Growth of Supermassive Black Holes

Problem Statement
The existence of supermassive black holes (SMBHs) with masses $M \sim 10^9\text{--}10^{10} M_\odot$ at high redshifts ( $z \gtrsim 6$ ), and potentially up to $z \sim 10\text{--}11$ as revealed by JWST, poses a significant challenge to standard astrophysical models. Conventional growth via Eddington-limited baryonic accretion requires massive seeds, high duty cycles, and prolonged gas supply, often struggling to assemble such masses within the available cosmic time. While mechanisms like super-Eddington accretion or massive direct-collapse seeds offer solutions, they rely on specific, often fine-tuned astrophysical conditions. This paper investigates whether an alternative or complementary channel—scalar-assisted growth via the absorption of an ultralight bosonic scalar field (a candidate for dark matter)—can accelerate SMBH formation. Specifically, the authors address whether this mechanism remains astrophysically relevant when ordinary baryonic growth is included simultaneously, and how the efficiency of this process depends on the boson mass and initial conditions.

Methodology
The authors develop a phenomenological cosmological model evolving a single black hole coupled to two finite matter reservoirs: a baryonic reservoir and a bosonic scalar cloud. The system is evolved directly in redshift ( $z$ ) within a flat $\Lambda$ CDM background.

Dynamical Equations: The evolution is governed by a set of coupled nonlinear ordinary differential equations (ODEs) for the black hole mass ( $M_{BH}$ $M_{B H}$ ), scalar cloud mass ( $M_c$ $M_{c}$ ), and baryonic reservoir mass ( $M_b$ $M_{b}$ ).
- Baryonic Channel: Modeled via a standard Eddington-like accretion prescription with a radiative efficiency $\epsilon = 0.1$ , limited by the finite mass of the baryonic reservoir.
- Scalar Channel: Modeled via an effective quasibound-state decay rate, $\Gamma \propto \mu^6 M_{BH}^5$ , where $\mu$ is the boson mass. This rate captures the strong parametric dependence on the dimensionless coupling $\alpha = \mu M_{BH}$ . The scalar transfer is capped by a phenomenological upper bound to prevent numerical stiffness in extreme regimes.
Numerical Implementation: The system is integrated using the implicit Radau method (a fifth-order Runge-Kutta scheme) from an initial redshift $z_{ini} = 120$ down to $z=0$ .
Parameter Scan: A comprehensive scan is performed over a four-dimensional parameter space:
- Boson mass: $\mu \in [10^{-22}, 10^{-16}]$ eV.
- Initial seed mass: $M_0 \in \{10^2, 10^3, 10^4, 10^5\} M_\odot$ .
- Initial scalar cloud mass: $M_{c,0} \in \{10^7, 10^8, 10^9, 10^{10}\} M_\odot$ .
- Initial baryonic reservoir mass: $M_{b,0} \in \{10^8, 10^9, 10^{10}\} M_\odot$ .
Observational Comparison: The resulting trajectories are compared against a catalog of over 70,000 high-redshift quasars and AGN. Two primary diagnostics are used:
1. Mass Agreement: The redshift at which the model reaches $10^9 M_\odot$ ( $z_{reach}$ ) and the root-mean-square error (RMSE) in $\log M_{BH}$ compared to observed masses.
2. Luminosity Agreement: A comparison of the reconstructed baryonic bolometric luminosity (derived from the baryonic depletion history) against observed quasar luminosities, using a weighted RMSE metric.

Key Contributions and Results

Selective Efficiency of Scalar Transfer:
The study finds that scalar-assisted growth is highly selective in parameter space. The efficiency is not determined by the boson mass $\mu$ alone but by the dimensionless coupling $\alpha = \mu M_{BH}$ .
- For the mass scales relevant to early SMBH formation ( $M_{BH} \sim 10^9 M_\odot$ ), the most efficient scalar absorption occurs in the window $\mu \sim 10^{-19}\text{--}10^{-16}$ eV.
- In this regime, the scalar cloud undergoes rapid depletion, triggering a positive feedback loop where the growing black hole mass increases the decay rate, leading to catastrophic absorption. This can advance the formation of $10^9 M_\odot$ black holes to significantly higher redshifts compared to baryonic-only growth.
- Outside this window (specifically for $\mu \lesssim 10^{-19}$ eV), the scalar transfer remains suppressed, and evolution is predominantly baryonic.
Divergence Between Dynamical Efficiency and Observational Compatibility:
A central finding is the distinction between the boson mass that maximizes growth efficiency and the mass that best fits observational data.
- Dynamically Efficient Regime: $\mu \sim 10^{-19}\text{--}10^{-16}$ eV.
- Observationally Preferred Regime: When comparing model trajectories to the observed quasar population (both mass and luminosity), the best agreement is found for much lighter boson masses, broadly around $\mu \sim 10^{-21}\text{--}10^{-20}$ eV.
- Specifically, when requiring substantial observational coverage (comparing against many quasars rather than just a few), the preferred mass shifts to $\mu \approx 3.16 \times 10^{-21}$ eV. In this lower-mass regime, the scalar contribution is weaker, and baryonic accretion plays a more dominant role in shaping the evolutionary tracks, which happens to align better with the bulk of the observed quasar population.
Luminosity Imprints:
The scalar field does not radiate directly in this model but affects the baryonic luminosity indirectly. By accelerating the growth of $M_{BH}$ , the scalar channel modifies the subsequent Eddington-limited baryonic accretion rate. The analysis shows that while scalar-assisted growth alters the accretion history, the configurations that best match observed luminosity trends are those with moderate scalar contributions (lower $\mu$ ), rather than the most extreme scalar-dominated scenarios.

Significance and Claims
The paper claims to demonstrate that scalar-assisted growth is a viable complementary channel for SMBH assembly but highlights a tension between maximizing early growth speed and matching the observed quasar population.

Non-Contradictory Windows: The authors argue that the separation between the "efficient growth window" ( $\mu \sim 10^{-19}\text{--}10^{-16}$ eV) and the "observationally compatible window" ( $\mu \sim 10^{-21}\text{--}10^{-20}$ eV) is not necessarily contradictory. It suggests that different bosonic mass scales may play complementary roles: heavier scalars could drive rapid early assembly, while lighter, fuzzy-dark-matter-like scalars could remain compatible with the observed quasar population at later epochs.
Phenomenological Nature: The authors explicitly state that their results are derived from a phenomenological framework. The scalar transfer rate is an effective prescription based on small-coupling scaling, and the model does not include full relativistic backreaction, BH spin evolution, or detailed hydrodynamics. Therefore, the identified mass ranges are presented as results of this specific framework rather than formal constraints on ultralight boson dark matter.
Future Directions: The paper concludes that a more complete treatment involving scalar-cloud formation, self-gravity, and realistic accretion physics is required to determine if a single intermediate mass scale can simultaneously satisfy both dynamical efficiency and observational constraints.

Technical Summary: Scalar-Assisted Growth of Supermassive Black Holes

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