Primordial Black Hole Hotspots Beyond Flat Spacetime

This paper formulates the diffusion equation for primordial black hole hotspots in an expanding universe, revealing that while cosmological expansion does not alter the robust formation or spatial temperature profile, it significantly accelerates the cooling process and ensures all hotspots eventually disappear within a finite time, contrary to flat-spacetime predictions.

The Gist

The Big Picture: Cosmic Campfires in an Expanding Room

Imagine the early universe as a giant, expanding room filled with a thick, hot fog (plasma). Inside this room, tiny, invisible "ghosts" called Primordial Black Holes (PBHs) are evaporating. As they vanish, they spit out high-energy particles, acting like tiny, super-hot campfires.

These campfires heat up the fog immediately around them, creating localized "hotspots" that are much hotter than the rest of the room.

For a long time, scientists studied these hotspots as if the room were a static, non-moving box. They calculated how the heat spreads out (diffusion) and how long the hotspots last. However, this paper argues that the room isn't static; it's expanding (like the actual universe). The authors wanted to know: Does the expansion of the room change how these campfires behave?

The Setup: How the Hotspots Form

When a PBH evaporates, it shoots out energy. This energy doesn't instantly turn into heat; it has to bounce around and mix with the fog first.

• The Old View: Scientists thought the heat spreads out in a specific pattern, creating a warm center (a plateau) that fades out into the distance.
• The New Finding (Formation): The authors did the math with the expanding room included. They found that the formation of the hotspot is surprisingly robust. Even though the room is stretching, the heat manages to build up the same way it did in the static box. The "shape" of the hotspot remains the same.

The Twist: The Cooling Phase

The real surprise happens after the PBH completely evaporates. The campfire is gone, and the hotspot is just a pocket of hot fog trying to cool down.

In the old "static room" models, this cooling happened slowly. The heat would spread out, and the temperature would drop, but in some scenarios, the hotspot would theoretically stay warm forever, never quite cooling down to the temperature of the surrounding room.

The authors found that in an expanding universe, this changes dramatically:

1. The Rapid Drop: When the PBH disappears, the hotspot doesn't just slowly fade. It takes a hard, rapid hit to its temperature immediately.
2. The Steeper Slide: After that initial drop, the temperature continues to fall, but it falls much faster than the old models predicted.
   • Analogy: Imagine a hot cup of coffee in a still room. It cools down slowly. Now, imagine that same cup of coffee is being pulled apart by an invisible force that stretches the air around it. The heat doesn't just spread; the very act of stretching the air sucks the heat away faster.
3. The "Finite" Lifespan: Because the universe is expanding, it acts like a double-whammy:
   • It stretches the heat out (redshifting).
   • It makes it harder for the heat to move around (suppressing diffusion).
   • Result: In the old models, some hotspots were "immortal." In this new model, every single hotspot dies out within a finite amount of time. They all eventually cool down to match the background temperature of the universe.

Why the Old Math Didn't Work

You might think, "If the universe expands, we can just take the old answer and shrink it a bit (redshift it) to get the new answer."

The authors say no. That's like trying to fix a leaky boat by just painting over the hole.

• The Problem: The expansion of the universe doesn't just cool the heat; it also slows down the process of heat spreading.
• The Analogy: Imagine trying to run through a hallway to deliver a message.
  • Static Hallway: You run at a normal speed.
  • Expanding Hallway: As you run, the hallway stretches behind you. Not only does the destination get further away, but the floor itself is moving against you, making it harder to run.
  • The paper shows that you can't just calculate your speed and then "stretch" the result. You have to recalculate the whole run because the floor is moving.

The Bottom Line

• Formation: The universe's expansion doesn't stop the hotspots from forming. They look the same as before.
• Cooling: The expansion changes the cooling process completely. The hotspots cool down much faster and die out sooner than previously thought.
• The "Immortal" Myth: The idea that some of these hotspots could last forever is incorrect when you account for the expanding universe. They all eventually disappear.

This research refines our understanding of how energy behaves in the early universe, ensuring that our models of these "cosmic campfires" match the reality of a stretching, expanding cosmos.

Technical Summary

Technical Summary: Primordial Black Hole Hotspots Beyond Flat Spacetime

Problem Statement
Primordial Black Holes (PBHs) with masses below $10^9$ g evaporate via Hawking radiation before Big Bang Nucleosynthesis, injecting high-energy particles into the surrounding plasma. Previous analyses of the resulting "hotspots"—localized regions of elevated temperature—have treated the subsequent energy transport and cooling within a flat spacetime framework. This approach neglects the expansion of the Universe, specifically the Hubble dilution term and the scale-factor suppression of spatial gradients. Given that the lifetime of light PBHs ($\tau_{PBH}$) is comparable to the Hubble time ($H^{-1}$) during the radiation-dominated era, the omission of cosmological expansion may significantly alter the evolution of these hotspots, particularly during the cooling phase.

Methodology
The authors formulate a diffusion equation for energy density in a Friedmann-Robertson-Walker (FRW) background to incorporate Hubble expansion effects.

1. Formulation: Starting from the covariant conservation of the stress-energy tensor and the relativistic energy flux, they derive the Hubble-corrected diffusion equation (Eq. 10). This equation includes a Hubble dilution term ($4H\rho$) and a $1/a^2$ factor suppressing spatial gradients in comoving coordinates.
2. Validity Conditions: The study establishes the conditions under which the diffusion approximation remains valid, requiring that the mean free path is well-defined over a scattering event and that the scattering rate exceeds the Hubble expansion rate ($v/\lambda \gg H$).
3. Critical Scale Identification: The authors identify a critical length scale, $R_H = \sqrt{\lambda v / H}$, where Hubble expansion begins to dominate over diffusion. They demonstrate that this scale coincides with the decoupling radius ($r_{dec}$) defined in previous flat-spacetime studies.
4. Numerical Solution: To analyze the cooling phase, the authors introduce dimensionless variables (redshift-corrected temperature $u$, dimensionless time $\tau$, and dimensionless comoving radius $x$) to transform the partial differential equation into a universal form (Eq. 28) independent of specific physical parameters. They solve this equation numerically to track the evolution of the temperature profile after the PBH has fully evaporated.

Key Contributions and Results

• Robustness of Formation: The study confirms that the formation of hotspots and the resulting temperature profile at the moment of PBH evaporation ($t = t_{ev}$) are robust against cosmological expansion. The envelope profile $T \propto r^{-7/11}$ and the central plateau temperature remain effectively unchanged because the evaporation process accelerates near the end of the PBH's life, minimizing the time available for redshift to distort the inner shells.
• Modified Cooling Dynamics: The cooling stage is substantially modified by Hubble expansion.
  • Initial Drop: The plateau temperature undergoes a rapid initial drop before settling into a power-law decline.
  • Scaling Exponent: In the late-time power-law regime, the plateau temperature scales as $T_{plt} \propto t^{-11/15}$. This is steeper than the flat-spacetime prediction of $T_{plt} \propto t^{-7/15}$.
  • Mechanism: This change in scaling cannot be reproduced by simply redshifting the flat-spacetime solution. The steeper decline arises because Hubble expansion simultaneously redshifts the temperature and suppresses the efficiency of diffusive transport via the $1/a^2$ factor.
• Finite Disappearance Time: A critical consequence of the modified scaling is that the hotspot temperature eventually falls below the background temperature of the Universe. Unlike the flat-spacetime prediction, where hotspots in certain parameter spaces could persist indefinitely (everlasting hotspots), the inclusion of Hubble expansion ensures that all hotspots disappear within a finite time.

Significance and Claims
The paper claims that incorporating Hubble expansion is essential for a proper treatment of PBH hotspot dynamics, particularly for heavy PBHs with large couplings where the cooling phase extends into the power-law regime. The authors assert that their findings correct the flat-spacetime scaling exponent and resolve the unphysical prediction of everlasting hotspots.

The significance of these results lies in their potential impact on phenomenological applications sensitive to the local temperature and lifetime of hotspots, such as sphaleron-mediated baryogenesis, topological defect production, dark matter production, and catalyzed vacuum decay. The authors note that while the diffusion-based treatment breaks down for very light PBHs where the maximum temperature is below the background or scattering is too infrequent, the corrected cooling dynamics apply to the valid parameter space. They also suggest that analogous "dark hotspots" could arise in dark sectors with gauge interactions, though a detailed study of such scenarios is reserved for future work.