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PBH runaway during reheating

This paper extends the analysis of primordial black hole runaway growth to the reheating era, deriving analytical solutions that reveal how reheating dynamics and formation time modify critical conditions for runaway absorption and establish a composition law for combining mass-growth mechanisms across cosmological eras.

Original authors: Md Riajul Haque, Mathieu Gross, Mathieu Houlier, Yann Mambrini

Published 2026-09-11

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CC BY 4.0

Original authors: Md Riajul Haque, Mathieu Gross, Mathieu Houlier, Yann Mambrini

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, a fraction of a second after the Big Bang, space was filled with a seething, ultra-hot soup of particles and energy. Within this chaotic environment, regions of extreme density could have collapsed under their own gravity to form primordial black holes. Unlike the black holes we observe today, which are born from the death of massive stars, these ancient objects would have appeared instantly, potentially as small as a grain of sand or as massive as a mountain. Once formed, their fate depended on a delicate tug-of-war. On one side was Hawking evaporation, a process where black holes slowly lose mass by emitting radiation, eventually disappearing entirely. On the other side was accretion, the act of swallowing the surrounding hot plasma. If a black hole swallowed matter fast enough, it could grow uncontrollably, becoming a cosmic giant. For decades, scientists believed that in the standard era of radiation domination that followed the Big Bang, there was a universal tipping point. If a black hole formed with a certain efficiency, it would inevitably cross this threshold and begin an endless, runaway growth spurt, regardless of when or how it was born.

However, the universe did not jump immediately into that standard era. Before the hot, radiation-filled phase began, there was a distinct period known as reheating. During this time, the universe was dominated by a different energy source, a field called the inflaton, which was oscillating and slowly decaying into the particles that would eventually make up the hot soup. This new paper investigates what happens to primordial black holes if they form during this specific, transitional window. The researchers found that the old rules of universal tipping points do not apply here. Because the environment during reheating is fundamentally different from the later radiation era, the conditions required for a black hole to start its runaway growth change. The critical threshold is no longer a single, unchanging number. Instead, it depends on exactly when the black hole formed and how the reheating process unfolded.

The team, led by researchers from Shanghai Jiao Tong University and the University of Paris-Saclay, developed a new mathematical framework to track these black holes as they grew. They discovered that the mass a black hole accumulates during reheating acts as a head start. Even if a black hole does not grow fast enough to trigger runaway absorption while reheating is still happening, the extra mass it gathers can push it over the edge the moment the universe transitions into the standard radiation era. This means that a black hole that seemed safe during the chaotic reheating phase could suddenly become a runaway monster once the universe cooled and settled. The study identifies two distinct scenarios for this growth: one where the black hole explodes in size while reheating is still ongoing, and another where it quietly accumulates mass during reheating and only begins its rapid, unbounded expansion once the radiation era begins.

To understand how these different growth mechanisms work together, the authors derived a simple rule for combining them. They showed that the effects of swallowing radiation and swallowing the inflaton field can be added together to determine the total growth potential. This composition law allows scientists to calculate the exact conditions needed for runaway growth in complex, multi-stage cosmic histories. By applying this rule to the specific case of reheating, they found that the combined effect of these two growth channels lowers the threshold for runaway growth. In other words, it is easier for a black hole to become a runaway giant during this early epoch than previously thought, provided it forms at the right time. The researchers also refined their calculations to account for the messy reality of the transition between the inflaton-dominated era and the radiation era, where both energy sources exist simultaneously. This refinement removed an artificial glitch in their earlier models and confirmed that their analytical predictions match perfectly with full numerical simulations.

The findings suggest that the history of the universe is more nuanced than a simple switch from one state to another. The specific dynamics of the reheating phase leave a permanent mark on the fate of primordial black holes. The critical efficiency required for a black hole to enter a runaway growth phase is not a fixed constant but a value that shifts based on the formation time and the specific details of the reheating process. For black holes formed very early in the reheating period, the threshold is much higher, making runaway growth difficult. For those formed later, closer to the end of reheating, the threshold drops, making it easier for them to grow. The study concludes that while a universal threshold exists for the later radiation era, it is fundamentally broken during the reheating epoch. This work provides a clear, analytical tool for predicting how these ancient black holes evolve, offering a more complete picture of the universe's earliest moments and the potential seeds of the massive black holes we see today.

Technical Summary: PBH Runaway During Reheating

Problem Statement
The evolution of Primordial Black Holes (PBHs) is governed by the competition between Hawking evaporation and the absorption of the surrounding plasma. In a radiation-dominated universe, it has been established that PBHs formed with a collapse efficiency $\gamma$ exceeding a universal critical threshold ( $\gamma_c \simeq 0.395$ ) enter a "runaway" regime where their mass grows without bound. However, this universality relies on the scale-free nature of radiation domination. In realistic cosmological scenarios, the universe undergoes a reheating phase between inflation and the standard hot Big Bang, during which the expansion is driven by an oscillating inflaton field rather than radiation. The central problem addressed is whether the universal critical collapse efficiency survives in this more general background, or if the reheating dynamics and the specific time of PBH formation modify the conditions for runaway growth.

Methodology
The authors extend the analytical framework for PBH mass evolution to the reheating era. The methodology involves:

Analytical Derivation: Solving the evolution equation for PBH mass, which includes both Hawking evaporation and absorption terms, under the assumption of an inflaton-dominated background with a specific equation of state ( $w_\phi$ ) and a reheating temperature evolution characterized by a parameter $\zeta$ .
Regime Classification: Distinguishing between two distinct physical scenarios: (a) runaway growth occurring during reheating itself, and (b) runaway growth triggered after the onset of radiation domination due to mass accumulated during reheating.
Composition Law Formulation: Developing a general "addition theorem" to describe how independent mass-growth mechanisms (specifically radiation absorption and inflaton accretion) combine within a single cosmological era and how these effects propagate across successive eras.
Refinement of Background Evolution: Moving beyond the piecewise approximation (where only the dominant energy component determines the Hubble rate) to a refined treatment that retains both inflaton and radiation energy densities near the transition to radiation domination.
Numerical Validation: Comparing the derived analytical solutions against full numerical integrations of the mass-evolution equations to verify accuracy, particularly near the reheating-radiation transition.

Key Contributions and Results

Breakdown of Universality: The paper demonstrates that the universal critical collapse efficiency $\gamma_c \simeq 0.395$ is not preserved during reheating. The critical condition for runaway growth becomes dependent on the PBH formation time (relative to the end of reheating) and the specific reheating dynamics (equation of state $w_\phi$ and temperature evolution $\zeta$ ).
Two Runaway Regimes: The authors identify two distinct criteria for runaway growth:
1. During Reheating: The perturbative solution diverges before the end of reheating. This requires a higher collapse efficiency, particularly for PBHs formed very early in the reheating epoch.
2. At Onset of Radiation Domination: Even if the PBH remains in the perturbative regime throughout reheating, the mass accumulated during this phase increases the effective collapse efficiency ( $\gamma_{\text{eff}}$ ) at the start of radiation domination. If $\gamma_{\text{eff}}$ exceeds the universal $\gamma_c$ , runaway growth is triggered immediately after reheating. The authors argue this second criterion is the physically relevant one for determining subsequent evolution.
Composition Law (Addition Theorem): A general law is derived showing that the inverse critical collapse efficiencies of independent mass-growth processes add linearly within a given era: $1/\gamma_{c, \text{era}} = \sum 1/\gamma_{c,i}$ . This allows for the combination of radiation absorption and inflaton accretion effects.
Analytical vs. Numerical Agreement: The derived analytical expressions for the critical collapse efficiency, including the combined effects of inflaton accretion and radiation absorption, show excellent agreement with full numerical simulations.
Refined Transition Treatment: By including both inflaton and radiation contributions to the Hubble rate near the transition, the authors eliminate a spurious shallow minimum in the critical efficiency curve found in the piecewise approximation. The refined analytical result confirms that the critical efficiency remains bounded from below by the universal radiation-dominated value ( $\gamma_c$ ) and interpolates smoothly between reheating-era thresholds and the standard radiation-era limit.

Significance
The paper claims to provide a unified analytical framework for describing PBH evolution across successive cosmological histories, specifically bridging the reheating and radiation-dominated eras. By deriving analytical solutions for general reheating histories and establishing a composition law for multiple growth mechanisms, the work offers a self-consistent description of how the critical collapse threshold is modified by pre-radiation dynamics. The results indicate that the "universal" nature of the runaway threshold is an artifact of assuming immediate radiation domination; in realistic scenarios involving reheating, the threshold is a function of formation time and reheating parameters. The refined treatment of the transition ensures that analytical predictions remain accurate without relying on artificial approximations near the end of reheating.

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