Relativistic accretion and burdened primordial black holes

This paper investigates how the combined effects of relativistic accretion and memory-burdened evaporation alter the evolution of primordial black holes, potentially allowing low-mass black holes to survive to the present day as dark matter while also analyzing their impact on dark radiation and dark matter emission during earlier evaporation phases.

The Gist

Imagine the early universe as a bustling, chaotic kitchen just after the Big Bang. In this kitchen, tiny, invisible "kitchen sinks" called Primordial Black Holes (PBHs) were formed. These aren't the massive black holes at the centers of galaxies; they are microscopic, some weighing as little as a single grain of sand or even a speck of dust.

For decades, scientists thought these tiny black holes had a very short lifespan. According to standard physics (Hawking radiation), they were supposed to act like ice cubes in a hot oven: they would slowly melt away, evaporating completely into particles of light and energy until they vanished. By today, scientists believed any black hole smaller than a mountain would have already melted away.

However, this paper introduces two new ingredients that change the recipe: Relativistic Accretion and Memory Burden.

1. The "Memory Burden": A Heavy Backpack

Imagine a black hole is a person trying to empty a backpack full of information (its "memory") by throwing things out the back door.

• The Old View: The person throws things out at a steady, fast pace until the bag is empty.
• The New View (Memory Burden): As the person throws out more items, the bag gets heavier with the "burden" of the information they are trying to keep track of. This weight slows them down. They start throwing things out much more slowly, or maybe even stop entirely.

In the paper's language, this "memory burden" creates a backreaction that slows down the evaporation. This means tiny black holes that should have vanished billions of years ago might actually be stuck in a slow-motion fade, surviving all the way to the present day.

2. Relativistic Accretion: The Snowball Effect

Now, imagine our tiny black hole isn't just sitting in an empty oven; it's in a blizzard.

• Accretion is the process of the black hole swallowing matter from its surroundings.
• Relativistic means the matter it's eating is moving at speeds close to the speed of light.

Think of the black hole as a snowball rolling down a steep, snowy hill. As it rolls, it picks up more snow. The faster it goes (relativistic speeds), the more snow it grabs, and the bigger it gets. The paper shows that in the early, dense universe, these black holes could have "ate" enough fast-moving matter to grow significantly larger than they started.

The Big Picture: Two Scenarios

The authors combined these two effects (the heavy backpack slowing the melt, and the snowball growing bigger) to see what happens to these black holes. They looked at two main stories:

Story A: The Black Holes That Disappeared (Before Nucleosynthesis)
Some black holes were so small that even with the "memory burden" slowing them down and the "snowball" growth, they still evaporated completely before the universe formed its first atoms (a time called Big Bang Nucleosynthesis).

• What they found: Even though they disappeared, they left a trace. As they evaporated, they shot out particles. The paper calculates how much Dark Matter (the invisible stuff holding galaxies together) and Dark Radiation (invisible energy) these dying black holes created.
• The Twist: Because the "snowball" effect made them bigger before they died, and the "memory burden" made them live longer, the amount of Dark Matter they produced is different than scientists previously thought. It actually restricts the types of particles that could have been created.

Story B: The Black Holes That Survived (Until Today)
Because of the "memory burden," some black holes that should have been too small to survive actually made it to the present day.

• What they found: These surviving black holes could be the Dark Matter we are looking for.
• The Twist: The "snowball" effect (accretion) means that a black hole that started out tiny could have grown just enough to survive. This opens up a "new window" of possibilities. It suggests that tiny black holes, which we thought were impossible candidates for Dark Matter, might actually be hiding in plain sight, provided they grew fast enough and slowed their evaporation enough.

The Constraints: The Rules of the Game

The paper doesn't just say "anything is possible." It checks these ideas against the rules of the universe we observe:

• The Gamma-Ray Rule: If these black holes are still evaporating today, they should be shooting out gamma rays. We look for these rays in the sky. If we don't see them, the black holes can't be too common or too heavy.
• The Cosmic Microwave Background (CMB) Rule: If black holes evaporated too late, they would have messed up the "baby picture" of the universe (the CMB). The paper checks if their new model fits these ancient photos.
• The "Effective Number" Rule: The evaporation creates extra invisible energy (Dark Radiation). This changes a specific number called $N_{eff}$ that cosmologists measure. The paper shows how the new "snowball" and "backpack" effects change this number, potentially making it detectable by future telescopes.

Summary

In simple terms, this paper argues that tiny black holes are tougher and more adaptable than we thought.

1. They have a "memory" that slows down their death.
2. They can grow up by eating fast-moving matter in the early universe.

Because of this, some might have survived to become the Dark Matter of today, while others might have died earlier but created a specific signature of particles that we can now calculate. The authors provide a new map for scientists to look for these elusive objects, showing that the "safe zone" where they could exist is different than previously believed.

Technical Summary

Technical Summary: Relativistic Accretion and Burdened Primordial Black Holes

Problem Statement
Primordial Black Holes (PBHs) are candidates for Dark Matter (DM) and sources of various cosmological phenomena. However, standard semiclassical Hawking evaporation imposes strict constraints: PBHs with initial masses $M \lesssim 10^{15}$ g are expected to have evaporated completely by the present epoch, while those with $M \lesssim 10^9$ g would have evaporated before Big Bang Nucleosynthesis (BBN), potentially disrupting light element abundances. Recent theoretical developments suggest two mechanisms that could alter this picture: the "memory burden" effect, which slows evaporation due to quantum backreaction, and relativistic accretion, which increases PBH mass. This paper investigates the joint effects of these two processes on PBH evolution, specifically examining how they modify the survival probability of low-mass PBHs, the production of Dark Matter and Dark Radiation (DR) during evaporation, and the resulting constraints on PBH abundance.

Methodology
The authors model the mass evolution of PBHs by combining the rate of mass gain from accretion with the rate of mass loss from evaporation.

1. Mass Evolution: The total mass change is defined as $\dot{M} = \dot{M}_{\text{acc}} + \dot{M}_{\text{evap}}$.
   • Accretion: The authors utilize a relativistic accretion formalism (extending Bondi-Hoyle) where the accretion rate depends on the PBH mass, the background energy density, and the equation of state (EoS) parameter $w$. They consider two background scenarios: radiation domination ($w=1/3$) and a stiffer fluid ($w=2/3$).
   • Evaporation: The evaporation rate incorporates the "memory burden" effect. Initially, PBHs evaporate via standard Hawking radiation. Once the mass drops to a fraction $q$ of its initial mass ($M < q M_{\text{in}}$), the evaporation rate is suppressed by a factor $S^{-\kappa}$, where $S$ is the entropy and $\kappa$ is a phenomenological parameter. This suppression delays or stabilizes the remnant.
2. Scenarios: The analysis covers two primary regimes:
   • Complete Evaporation: PBHs evaporate entirely before BBN. The authors calculate the resulting relic density of emitted Dark Matter particles and the contribution to Dark Radiation ($\Delta N_{\text{eff}}$).
   • Survival: PBHs survive until the present epoch due to the combined effects of mass gain (accretion) and delayed evaporation (memory burden). The authors calculate the fraction of DM contributed by these stable remnants ($f_{\text{PBH}}$).
3. Constraints: The study applies observational constraints from the Cosmic Microwave Background (CMB), extragalactic $\gamma$-rays, BBN, and warm dark matter limits to the derived parameter spaces.

Key Contributions and Results

• Extended Survival Window: The combination of relativistic accretion and memory burden significantly extends the lifetime of low-mass PBHs.

  • Without accretion, the memory burden effect alone allows PBHs with $M \lesssim 10^{15}$ g to survive.
  • With relativistic accretion, the minimum initial mass required for a PBH to survive to the present day is drastically reduced. For example, in a radiation-dominated background ($w=1/3$) with $\kappa=2$ and $q=1/2$, the minimum surviving mass drops from $\sim 5600$ g (no accretion) to $\sim 1400$ g (relativistic accretion). In stiffer backgrounds ($w=2/3$), this limit can reach $\sim 640$ g.
  • Conversely, for PBHs to evaporate completely before BBN, the maximum allowed initial mass is significantly lowered by accretion. For $\kappa=2, q=1/2$, the limit drops from $\sim 21$ g (no accretion) to $\sim 5$ g (relativistic accretion).

• Dark Matter Production: The paper analyzes the production of DM particles from evaporating PBHs in two regimes:

  • PBH Domination ($\beta > \beta_{\text{cr}}$): When PBHs dominate the energy density before evaporation, the relic abundance of emitted DM is derived. The authors find that increasing the memory burden parameter $\kappa$ shrinks the allowed parameter space in the $M_{\text{in}} - m_j$ plane. Relativistic accretion further reduces this viable region.
  • Subdominant PBHs ($\beta < \beta_{\text{cr}}$): When PBHs do not dominate, the relic abundance depends on the initial abundance $\beta$ and the background EoS. Accretion significantly impacts the parameter space, particularly for relativistic cases, by altering the effective initial mass and evaporation timescale.
  • Constraints: The study incorporates constraints from warm dark matter (Ly-$\alpha$ forest) and gravitationally produced DM, showing that accretion tightens the bounds on the mass of emitted DM particles.

• Stable PBHs as Dark Matter: For PBHs that survive to the present, the authors calculate the fraction of DM they constitute ($f_{\text{PBH}}$).

  • Accretion shifts the constraints on $f_{\text{PBH}}$ toward higher initial masses. A PBH that would have evaporated in the standard scenario may survive and contribute to DM if it accretes sufficient mass.
  • The analysis shows that the "new window" opened by the memory burden effect for low-mass PBHs is further modified by accretion, shifting the allowed mass range and abundance constraints.

• Dark Radiation ($\Delta N_{\text{eff}}$): The paper investigates the contribution of light particles (DR) emitted during evaporation to the effective number of relativistic degrees of freedom.

  • The authors find that the contribution to $\Delta N_{\text{eff}}$ is sensitive to the memory burden parameters and the accretion regime.
  • They note that upcoming CMB missions (CMB-S4, CMB-HD) could potentially rule out all DR contributors except the massless graviton, depending on the specific values of $\kappa$ and the accretion efficiency.

Significance and Claims
The paper claims that the interplay between relativistic accretion and memory-burdened evaporation creates a "new window" in the parameter space for Primordial Black Holes. Specifically:

1. It challenges the standard exclusion of low-mass PBHs ($M < 10^{15}$ g) as Dark Matter candidates, demonstrating that accretion can stabilize remnants that would otherwise evaporate.
2. It provides updated constraints on the initial abundance ($\beta$) and mass ($M_{\text{in}}$) of PBHs required to produce specific Dark Matter abundances or to survive until today, showing that neglecting accretion leads to inaccurate bounds.
3. It highlights that the memory burden effect, combined with accretion, can significantly alter the production rates of Dark Matter and Dark Radiation, thereby influencing cosmological observables like $\Delta N_{\text{eff}}$ and the relic density of DM.

The authors conclude that while their work focuses on specific toy models (sudden transitions, specific EoS values), the results suggest that a comprehensive understanding of PBH cosmology must account for both relativistic accretion and quantum memory effects simultaneously. They suggest future extensions could include rotating (Kerr) black holes and gradual transitions to the memory-burdened regime.