Thermodynamical uncertainties for primordial black holes from cosmological phase transitions

This paper employs high-precision thermodynamic analysis to establish a universal lower bound on the transition timescale for strongly supercooled phase transitions, revealing that the resulting primordial black hole abundance is severely constrained and unlikely to constitute viable dark matter within classically conformal gauge-Higgs theories.

The Gist

Imagine the early universe as a giant, super-hot pot of soup. As it cools down, it's supposed to change its state, much like water turning into ice. In physics, this is called a phase transition.

Usually, this happens smoothly, like water slowly freezing. But in some theories about the universe, this change happens violently and suddenly, like water that has been supercooled in a freezer and then suddenly explodes into ice all at once. This is a strongly supercooled first-order phase transition.

The paper you provided investigates a specific question: Could these violent "freezing" events in the early universe create "primordial black holes" (PBHs)? These are tiny black holes formed right after the Big Bang, which some scientists think could make up the mysterious "dark matter" that holds galaxies together.

Here is the breakdown of their findings using simple analogies:

1. The Problem with Previous Maps

Previous scientists tried to predict if these black holes would form, but they used "simplified maps." They assumed the physics was simple and ignored some of the messy, complex details of how the "soup" behaves when it's extremely hot. It's like trying to predict a hurricane by only looking at the wind speed, ignoring the humidity, pressure, and ocean temperature.

The authors of this paper say, "We need a better map." They used a highly advanced, state-of-the-art thermodynamic toolkit (called 3d Effective Field Theory) to calculate the physics with much higher precision. They looked at two specific theoretical models (the U(1)CW and SU(2)X models) which act like different recipes for this cosmic soup.

2. The "Bubble" Race

When the universe undergoes this violent phase transition, it doesn't freeze everywhere at once. Instead, pockets of the "new" state (true vacuum) form like bubbles in boiling water.

• The Race: These bubbles expand and crash into each other.
• The Danger Zone: If the bubbles form too slowly, the "old" state (false vacuum) gets trapped in big, isolated islands. If these islands are huge and dense enough, they can collapse under their own gravity to form black holes.

The key to this race is the timescale (how fast the transition happens). The authors calculated a specific number, $\beta/H^*$, which measures how fast the bubbles are forming relative to the expansion of the universe.

• Low number: Bubbles form slowly. Big islands of old vacuum get stuck. High chance of black holes.
• High number: Bubbles form fast. The transition finishes quickly. Low chance of black holes.

3. The "Speed Limit" Discovery

The authors ran their high-precision calculations and found a hard speed limit.

• No matter how they tweaked the parameters in their models, the transition could never be slow enough to create the massive islands needed for black holes.
• The slowest possible transition they found had a timescale of roughly 5 to 6.
• The Metaphor: Imagine trying to build a sandcastle before the tide comes in. The authors found that the tide (the universe expanding) always comes in too fast. Even in the "slowest" scenario, the sandcastle (the black hole) never has time to form because the water washes it away before it's built.

They call this a universal lower bound. It means that for these specific types of theories, the physics simply won't allow the transition to be slow enough to make black holes.

4. The "QCD" Accelerator

There is a twist. The universe also has a "QCD transition" (related to how quarks and gluons behave) that happens later.

• In some scenarios, the violent phase transition gets delayed so long that it waits for the QCD transition to happen first.
• When the QCD transition happens, it acts like a turbocharger. It breaks a symmetry and adds a "kick" that makes the phase transition happen even faster.
• Result: This turbocharger makes the transition speed up, which makes the formation of black holes even less likely.

5. The Final Verdict: No Dark Matter Candidates

The paper concludes that after using these precise, high-tech calculations:

• The "Sweet Spot" is Gone: Previous, less accurate studies suggested there might be a "sweet spot" where the transition is just slow enough to make black holes that could be dark matter.
• The Reality: With the new, precise math, that sweet spot disappears. The transition is always too fast.
• The Conclusion: In the specific models they studied, primordial black holes cannot be the dark matter. The universe simply doesn't stay in the "danger zone" long enough to create them.

Summary

Think of the early universe as a race between bubble formation and cosmic expansion.

• Old View: We thought the bubbles might form slowly enough to get stuck and collapse into black holes.
• New View (This Paper): We did the math with a much better calculator. We found that the bubbles always form too fast. The universe expands and finishes the transition before any big black holes can be born.

Therefore, for these specific theories, the search for dark matter among primordial black holes created by these phase transitions is likely a dead end.

Technical Summary

Technical Summary: Thermodynamical Uncertainties for Primordial Black Holes from Cosmological Phase Transitions

Problem Statement
Strongly supercooled first-order phase transitions (FOPTs) have been proposed as a mechanism for generating primordial black holes (PBHs) that could constitute dark matter. In these scenarios, the universe undergoes a period of supercooling where large regions of false vacuum persist within a background of true vacuum bubbles. Statistical fluctuations during nucleation can create large-scale inhomogeneities; if density perturbations are sufficiently large at horizon re-entry, these regions collapse into PBHs.

However, previous studies estimating PBH abundance from such transitions have relied on simplified models with limited thermodynamic precision. These simplifications often fail to capture the full complexity of nucleation dynamics within realistic extensions of the Standard Model (SM), leading to potentially unreliable predictions regarding the transition timescale ($\beta/H_*$) and the resulting PBH abundance.

Methodology
This work provides a state-of-the-art thermodynamic analysis of classically conformal extensions of the SM, specifically the $U(1)_{CW}$ (Coleman-Weinberg) and $SU(2)_X$ models. The authors employ high-temperature dimensional reduction to construct a three-dimensional effective field theory (3d EFT). This framework allows for the computation of the nucleation rate with next-to-leading order (NLO) precision, including one-loop fluctuation determinants for both scalar and gauge modes.

The analysis addresses four specific sources of theoretical uncertainty:

1. Effective Action at NLO: The authors utilize the derivative expansion to derive the effective action for arbitrary background fields. They compute the soft, static, infrared modes described by the 3d EFT, including non-analytic temporal-scalar contributions to the potential barrier. The action is evaluated numerically to account for the full soft potential.
2. Bubble Nucleation Rate: The rate $\Gamma(T)$ is factorized into statistical ($A_{stat}$) and dynamical ($A_{dyn}$) parts. The statistical part is computed up to NLO in the soft expansion, incorporating fluctuation determinants ($\det S$ and $\det V$) around the critical bubble solution. The dynamical part is approximated without damping, providing a lower bound on the transition timescale.
3. Percolation Criterion: The transition temperature ($T_p$) is determined by the percolation condition, ensuring a sufficient fraction of the universe converts to the true vacuum. The authors examine two criteria: $I(T_p) = 1$ and $I(T_p) = 0.34$. They also verify that the false vacuum volume decreases around $T_p$ to avoid eternal inflation, identifying regions where the transition may terminate at a lower temperature ($T_{shrink}$) if standard percolation fails.
4. QCD-Sourced Transitions: The authors account for the possibility that the universe supercools below the QCD scale ($T_{QCD} \approx 100$ MeV). In this regime, chiral symmetry breaking induces a vacuum expectation value for the SM Higgs, breaking classical scale invariance (CSI). This generates a negative mass term in the effective potential, which counteracts the thermal barrier and accelerates the phase transition.

Key Contributions and Results

• Universal Lower Bound on Timescale: By computing the nucleation dynamics with high precision, the authors determine the slowest possible transition timescales for the models considered. They find a universal lower bound on the inverse timescale $\beta/H_*$:
  • $\beta/H_* \simeq 5.99$ for the $U(1)_{CW}$ model.
  • $\beta/H_* \simeq 5.50$ for the $SU(2)_X$ model.
    These values are derived using the standard percolation criterion $I(T_p) = 1$. When using the stricter criterion $I(T_p) = 0.34$, the transition often fails to percolate at the estimated $T_p$, pushing the termination to lower temperatures where QCD effects further accelerate the transition, maintaining the $\beta/H_* \gtrsim 5$ bound.
• Impact of QCD Effects: The study demonstrates that QCD-sourced transitions, triggered by chiral symmetry breaking at low temperatures, significantly enhance $\beta/H_*$ by reducing the bounce action. This effect prevents the transition from becoming slow enough to produce significant PBHs in the deep supercooling regime.
• PBH Abundance Estimation: Using numerical simulations (deltaPT) and updated fitting formulae, the authors estimate the PBH abundance. They find that achieving a PBH abundance sufficient to explain dark matter requires $\beta/H_*$ values significantly lower than those found in their analysis (specifically $\beta/H_* \ll 4$ based on recent gauge-covariant studies, or $\beta/H_* \lesssim 7.5$ based on older fits).
• Exclusion of Viable Parameter Space: The combination of the precise thermodynamic lower bound ($\beta/H_* \gtrsim 5$) and the updated constraints on PBH formation leads to the conclusion that no region in the parameter space of the $U(1)_{CW}$ or $SU(2)_X$ models allows for a sizable PBH abundance. The parameter space where PBHs could be viable dark matter candidates is severely limited or entirely excluded.

Significance and Claims
The paper claims that reliable theoretical predictions for PBH production from supercooled FOPTs require precise nucleation dynamics within realistic BSM extensions, rather than simplified models. The authors assert that their state-of-the-art thermodynamic analysis reveals a universal thermodynamic lower limit on the transition timescale ($\beta/H_* \simeq 5$).

Consequently, the paper concludes that for the classically conformal gauge-Higgs theories examined, PBHs cannot constitute a significant fraction of dark matter. The authors emphasize that while future work may refine the required threshold for PBH formation (e.g., by including spacetime curvature effects or energy transfer from bubble walls), any new threshold must remain above the universal lower limit of $\beta/H_* \simeq 5$ allowed by thermodynamics. Thus, the viability of PBHs as dark matter in these specific models is ruled out under current theoretical understanding.