Accretion Effects on Primordial Black Hole Reheating… — Plain-Language Explanation

The Big Picture: A Universe That Forgot to Cook

Imagine the early Universe as a giant kitchen. According to standard physics, after the "Big Bang" (the explosion that started everything), the Universe was too hot and chaotic to form stars or atoms. It needed to "reheat" and cool down to a specific temperature to start the process of creating life and galaxies. Usually, we think a mysterious field called the "inflaton" did the cooking.

But this paper explores a different chef: Primordial Black Holes (PBHs).

These are tiny black holes that formed right at the beginning of time from random bumps in the fabric of space. The theory suggests that if enough of these tiny black holes formed, they would take over the Universe, acting like a heavy, cold soup that eventually evaporated (via Hawking radiation) to heat the Universe back up, allowing the "Big Bang" to truly begin.

The Problem: The "Accretion" Surprise

The author of this paper, Chenhuan Wang, asks a simple question: What if these black holes ate while they were waiting to evaporate?

In physics, black holes don't just sit there; they can suck in surrounding gas and energy. This process is called accretion.

The Old View: Scientists previously thought, "Okay, if a black hole eats a little bit, it just gets slightly heavier. We can just adjust our math to say the black hole started heavier, and we're good."
The New View (This Paper): The author found that it's not that simple. Eating changes the timing of everything.

The Analogy:
Imagine a group of runners (the black holes) in a race against time.

Without Eating: They run at a steady pace. They finish the race (evaporate) at a specific time, heating up the track (the Universe) just right.
With Eating: As they run, they stop to grab snacks (accretion). This makes them heavier.
- Result A: Because they are heavier, they run slower.
- Result B: Because they are heavier, they take longer to finish the race.
- Result C: Because they are running slower, they dominate the track for a longer time, pushing back the moment the race ends.

The paper shows that this "snacking" doesn't just change the weight of the runners; it fundamentally changes the duration of the race. The black holes take over the Universe for longer than we thought, and they evaporate later.

The Two Big Constraints (The Rules of the Game)

To see if this "Black Hole Chef" theory works, the author checks it against two strict rules (constraints) that the Universe must follow.

1. The "Gravitational Wave" Noise (The Isocurvature Constraint)

When the black holes finally evaporate, they suddenly turn into radiation. Imagine a drumbeat that stops abruptly. The sudden change causes the fabric of space-time to "ring" like a bell, creating Gravitational Waves (ripples in space).

The Rule: Big Bang Nucleosynthesis (BBN) is the process where the first atoms formed. It's a very sensitive recipe. If there is too much "noise" (gravitational waves) in the kitchen, the recipe fails, and the Universe looks nothing like what we see today.
The Finding: Because the black holes ate (accretion), they dominated the Universe for longer. This made the "ringing" of space-time much louder and more dangerous.
The Consequence: To keep the noise low enough to pass the test, the black holes must be smaller and less abundant than we previously thought. The "safe zone" for this theory has shrunk significantly.

2. The "Black Hole Mergers" (The Clumping Constraint)

While the black holes are dominating the Universe, they aren't just floating alone. Because they are heavy, they start to clump together, like magnets. When they get close, they might crash into each other and merge, creating huge black holes.

The Rule: If these merged black holes get too big (heavier than a certain limit), they would still be around today or would have evaporated in a way that we would have already detected.
The Finding: Accretion makes the black holes heavier to begin with. This means they merge into even bigger monsters faster.
The Consequence: This puts a limit on how many black holes can exist. However, the author found that this rule is less strict than the "Gravitational Wave" rule. The "noise" from the evaporation is the bigger problem than the "clumping" of the black holes.

The Bottom Line

The paper concludes that accretion is a game-changer.

You cannot simply say, "Oh, the black holes ate a little, so let's just pretend they started heavier." That doesn't work. The act of eating changes the timeline of the Universe's history.

Before this paper: We thought the "safe" range for these black holes was quite wide.
After this paper: The "safe" range has shifted. To make this theory work, the black holes must have formed with less mass and fewer numbers than previously allowed.

In short, if the Universe was cooked by tiny black holes, they must have been very small and very scarce, and they must have been very careful not to eat too much, or they would have ruined the recipe for our existence.

Technical Summary: Accretion Effects on Primordial Black Hole Reheating Constraints

Problem Statement
Primordial Black Holes (PBHs) formed from enhanced primordial fluctuations can dominate the energy density of the early Universe, leading to an epoch of early matter domination (eMD). If these PBHs are sufficiently light ( $m \lesssim 10^9$ g), they can evaporate via Hawking radiation before Big Bang Nucleosynthesis (BBN), thereby reheating the Universe. This scenario is constrained by two primary mechanisms:

Isocurvature-induced Gravitational Waves (GWs): The sudden transition from PBH domination to radiation domination causes the gravitational potential to oscillate, sourcing second-order GWs. The energy density of these GWs is constrained by BBN limits on the effective number of relativistic species ( $\Delta N_{\text{eff}}$ ).
Merger-induced Extended Mass Functions: During eMD, density contrasts grow linearly, leading to PBH clustering and mergers. This can produce heavier PBHs that may survive until BBN or later, violating observational constraints on PBH abundance in the $10^{10}$ – $10^{17}$ g mass range.

While accretion of surrounding fluid onto PBHs is known to increase their mass, its specific impact on these two constraints within the PBH reheating scenario has not been fully quantified. This work investigates how accretion modifies the background evolution and the resulting observational bounds.

Methodology
The authors model the cosmological evolution using the Friedmann-Lemaître-Robertson-Walker (FLRW) metric with a general equation of state parameter $\omega \in (0, 1]$ for the background fluid prior to PBH domination. The system is governed by coupled differential equations for the energy densities of PBHs ( $\rho_{\text{PBH}}$ ), radiation ( $\rho_{\text{rad}}$ ), and the background fluid ( $\rho_{\omega}$ ), alongside an equation for the PBH mass evolution ( $m$ ).

Accretion Model: The mass evolution includes a Hawking evaporation term ( $\Gamma_{\text{evap}}$ ) and an accretion term ( $\Gamma_{\text{accret}}$ ) derived from spherical symmetric accretion in a general relativistic context. The accretion rate is proportional to the background density and the PBH mass.
Background Evolution: The authors solve the background equations numerically and derive analytical fits for the duration of the $\omega$ -domination era ( $N_{\text{eq}}$ ) and the PBH domination era ( $N_{\text{eva}} - N_{\text{eq}}$ ). They introduce a fitting function $f(\omega)$ to account for the shift in the equality epoch caused by accretion.
Isocurvature GWs: Using the Press-Schechter formalism and linear perturbation theory, the authors calculate the power spectrum of isocurvature perturbations. They derive the resulting second-order GW spectrum generated by the oscillation of the gravitational potential after PBH evaporation. The total GW energy density is integrated and compared against the BBN bound ( $\Omega_{\text{GW}} \lesssim 0.068$ ).
Merger Dynamics: The authors model PBH clustering using the Press-Schechter formalism. They treat clusters as isolated systems where two competing processes occur: binary mergers (driven by gravitational wave emission) and cluster evaporation (driven by ejection of high-velocity PBHs). They solve differential equations for the cluster mass and number of PBHs to determine the final relic mass spectrum.

Key Contributions and Results

Impact of Accretion on Background Evolution:
- Accretion increases the PBH mass, but its effect is not merely a rescaling of the initial mass ( $m_f \to m_{\text{accret}}$ ).
- Accretion causes the PBHs to dominate the Universe earlier (advancing the equality epoch) and prolongs the duration of the PBH domination phase.
- To mimic the effects of accretion in a non-accreting model, one must not only increase the mass but also increase the initial abundance parameter $\beta$ by an order of magnitude ( $\beta \to 10\beta$ ).
Constraints from Isocurvature-Induced GWs:
- The inclusion of accretion shifts the BBN constraints significantly. The allowed parameter space for the formation mass ( $m_f$ ) and initial abundance ( $\beta$ ) moves toward smaller values for both parameters.
- The upper bound on the formation mass is reduced because accretion increases the final PBH mass, which lowers the reheating temperature ( $T_{\text{rh}} \propto m^{-3/2}$ ). To satisfy $T_{\text{rh}} \gtrsim 4$ MeV, the initial mass must be smaller.
- The constraints are more pronounced for larger values of the background equation of state parameter $\omega$ , where accretion effects are stronger.
- Future detectors like the Big Bang Observer (BBO) and Cosmic Explorer (CE) are projected to probe only the upper-right corner (large $m_f$ , large $\beta$ ) of the allowed parameter space, regardless of accretion.
Constraints from PBH Mergers:
- Accretion increases the initial mass of PBHs entering the merger phase, which shifts the resulting extended mass function toward higher masses.
- However, accretion also reduces the fraction of dark matter in PBHs ( $f_{\text{BH}}$ ) due to the lower reheating temperature, which lowers the overall amplitude of the mass function.
- For certain parameters (e.g., $\omega = 2/3$ ), the reduction in the mass function amplitude can allow scenarios with formation masses around $10^7$ g to "escape" constraints that would otherwise rule them out.
- The GWs produced directly by the mergers themselves are found to be sub-dominant, contributing an energy density several orders of magnitude below the BBN $\Delta N_{\text{eff}}$ bound.

Significance and Claims
The paper claims that accretion is a critical factor in evaluating the viability of the PBH reheating scenario. The authors demonstrate that the effects of accretion cannot be trivially absorbed by simply adjusting the initial mass parameter; it fundamentally alters the cosmological timeline and the resulting observational constraints.

Primary Finding: The constraints derived from isocurvature-induced gravitational waves are typically stronger than those derived from PBH mergers.
Shift in Bounds: Including accretion shifts the allowed parameter space for PBH reheating toward smaller formation masses and smaller initial abundances.
Modesty: The authors note that their merger analysis relies on simplified assumptions (e.g., isolated clusters, mean mass tracking) and that a full treatment of the mass distribution (e.g., via coagulation equations) is beyond the scope of this work. They also acknowledge that non-monochromatic initial mass functions could further alter the GW constraints.

In conclusion, the work provides a refined set of constraints on PBH reheating scenarios, highlighting that accretion tightens the bounds on the initial PBH abundance and mass, particularly for stiff equations of state ( $\omega$ ).

Accretion Effects on Primordial Black Hole Reheating Constraints