Angular momentum of vacuum bubbles in a first-order phase transition

This paper calculates the angular momentum of spherical false vacuum bubbles induced by cosmological perturbations during a first-order phase transition in a dark sector, revealing that the resulting dimensionless spin parameter of potential primordial black holes spans a wide range from $10^{-5}$ to $10$ and follows a specific scaling relation with the transition time scale, bubble wall velocity, and temperature ratio between the dark and visible sectors.

The Gist

Imagine the early universe not as a smooth, calm ocean, but as a pot of water just starting to boil. As it heats up (or cools down, depending on how you look at it), it undergoes a phase transition—like water turning into ice or steam. In the world of particle physics, this is called a First-Order Phase Transition (FOPT).

This paper is about what happens when this "cosmic boiling" occurs in a hidden, invisible part of the universe (the "dark sector") and how it might create tiny, spinning black holes.

Here is the story of the paper, broken down into simple concepts:

1. The Setting: A Hidden Boiling Pot

Usually, we think of the universe as filled with normal matter (stars, gas, us). But physicists suspect there's a "dark sector" we can't see.

• The Scenario: Imagine this dark sector is a fluid that suddenly changes its state. It goes from a "False Vacuum" (a high-energy, unstable state) to a "True Vacuum" (a lower-energy, stable state).
• The Bubbles: As this change happens, bubbles of the new "True Vacuum" start popping into existence, like bubbles forming in boiling water. These bubbles expand and eventually collide, filling the universe with the new state.

2. The Mystery: Why Do Black Holes Spin?

When these bubbles crash together, they can sometimes collapse into Primordial Black Holes (PBHs)—black holes formed right at the beginning of time, not from dying stars.

• The Question: We know black holes spin (like a spinning top). But why? If a bubble is perfectly round and the universe is perfectly smooth, the bubble should just collapse straight in, with no spin.
• The Answer: The universe isn't perfectly smooth. It has tiny ripples and bumps called cosmological perturbations. Think of these as tiny wind gusts or uneven spots on the surface of the boiling water.

3. The Mechanism: The "Twist" from the Wind

The authors calculated how these tiny ripples affect the spinning bubbles.

• The Analogy: Imagine a perfectly round balloon (the bubble) floating in a room. If the air in the room is perfectly still, the balloon stays still. But if there are tiny, random breezes (perturbations) blowing from different directions, they push on the balloon unevenly.
• The Result: Even though the balloon is round, these uneven pushes create a torque (a twisting force). The bubble starts to spin as it collapses. The paper calculates exactly how fast these bubbles would spin based on the strength of these "breezes."

4. The Calculation: Tracking the Chaos

The team had to do some heavy math to figure this out.

• The Sound Speed Dip: During the phase transition, the "dark fluid" behaves strangely. Its ability to transmit sound (sound speed) drops dramatically, even becoming negative for a split second.
• The Amplifier: This strange behavior acts like a microphone amplifier. It takes the tiny, weak "breezes" (perturbations) and boosts them significantly. This means the bubbles can spin much faster than we might have guessed.
• The Spin Factor: They calculated a "spin parameter" (a number that tells us how fast it spins relative to its size). They found that depending on the specific conditions of the phase transition, these bubbles could have a wide range of spins:
  • Some spin very slowly (like a lazy turntable).
  • Some spin incredibly fast (like a figure skater pulling in their arms).

5. The Big Picture: What Does This Mean?

• No Magic Needed: Unlike other theories that require the early universe to have huge, unnatural spikes in energy to make black holes, this theory says black holes can form just from the natural "bubbling" of a phase transition.
• A New Fingerprint: If we ever detect these primordial black holes (perhaps through gravitational waves), their spin could tell us if they were born from this "boiling" process. If they are spinning fast, it might be a signature of this specific dark sector event.
• The Range: The paper shows that these black holes could have spins ranging from almost zero to very high values. Interestingly, a spin value greater than 1 isn't a problem here because the object isn't a black hole yet—it's just a bubble of energy that might become one.

Summary Analogy

Think of the early universe as a giant, invisible lava lamp.

1. The Phase Transition: The wax inside is heating up and starting to form blobs.
2. The Bubbles: These blobs are the "False Vacuum" bubbles.
3. The Perturbations: The lamp isn't sitting on a perfectly flat table; it's shaking slightly, and the wax has tiny imperfections.
4. The Spin: As the blobs rise and merge, those tiny shakes and imperfections cause them to twirl.
5. The Discovery: This paper is the first to measure exactly how fast those blobs would twirl before they potentially collapse into black holes.

The Takeaway: The universe is full of hidden dynamics. Even in the "dark" parts we can't see, the chaotic process of phase transitions can spin up tiny black holes, leaving a unique fingerprint of rotation that future telescopes might one day detect.

Technical Summary

1. Problem Statement

The formation of Primordial Black Holes (PBHs) during a first-order phase transition (FOPT) in a dark sector is a subject of growing interest. While the mass distribution of such PBHs has been studied, their spin (angular momentum) remains largely unexplored in this specific context.

• The Gap: Existing literature on PBH spin often relies on enhancements in the primordial curvature power spectrum (rare peaks) or assumes ellipsoidal collapse geometries. This paper addresses the scenario where PBHs form from the direct collapse of false vacuum (FV) bubbles or via intermediate Fermi balls (FBs) without requiring a large enhancement in the primordial power spectrum.
• The Core Question: How much angular momentum is induced in these spherical FV bubbles by standard cosmological perturbations (Gaussian random variables) during the phase transition, and what is the resulting dimensionless spin parameter $s = J/(G_N M^2)$?

2. Methodology

The authors employ a multi-step theoretical and numerical framework to calculate the spin of FV bubbles at the time of percolation.

A. Model Framework

• Dark Sector Dynamics: They introduce a dark sector containing a real scalar field $\phi$ (triggering the FOPT) and potentially dark Dirac fermions $\chi$ (for Fermi ball formation).
• Effective Potential: The dynamics are governed by a finite-temperature effective potential $V_{\text{eff}}(\phi, T)$ with a quartic form. The transition involves the nucleation of true vacuum (TV) bubbles within the false vacuum (FV).
• Thermodynamics: The dark plasma is treated as a single fluid with two phases (FV and TV). The authors track the evolution of the equation of state ($w$) and sound speed squared ($c_s^2$), noting that $c_s^2$ can become negative during the rapid drop of the FV fraction, leading to instability.

B. Angular Momentum Formalism

• Definition in GR: The authors define angular momentum in a perturbed Friedmann-Robertson-Walker (FRW) spacetime using the uniform Hubble gauge (UHG).
• Second-Order Perturbation: They demonstrate that for a spherical volume, the first-order contribution to angular momentum vanishes. The leading non-zero contribution arises at second order in perturbations, specifically as a product of density ($\delta$) and velocity ($v$) perturbations.
• Statistical Approach: Since initial curvature perturbations are Gaussian random variables, the angular momentum $\vec{J}$ and spin $s$ are also Gaussian random variables with zero mean. The goal is to calculate their Root-Mean-Square (RMS) values.

C. Evolution of Perturbations

• Gauge Choice: Calculations are performed in the Uniform Hubble Gauge (UHG), which simplifies the tracking of perturbations from superhorizon scales to the percolation time.
• Background Evolution: They solve a system of coupled ordinary differential equations (ODEs) to track the background evolution (temperature, scale factor, FV fraction $F$) from the critical point ($\eta_c$) to percolation ($\eta_*$).
• Perturbation Evolution:
  • Superhorizon: Modes evolve according to standard radiation-dominated solutions.
  • Subhorizon: The authors derive analytical approximations using Airy functions to describe the behavior of perturbations when the sound speed $c_s^2$ drops significantly (becoming negative).
  • Enhancement: They find that modes with wavelengths comparable to the duration of the sound speed dip are exponentially enhanced. This leads to a divergence in the mean square density contrast, establishing a natural cutoff scale ($k_{\text{cut}}$) related to the bubble size.

D. Numerical Implementation

• Transfer Functions: They numerically compute transfer functions ($U_\delta, U_{\hat{\psi}}$) relating initial curvature perturbations to density and velocity perturbations at percolation.
• Integration: The RMS spin is calculated by integrating the product of transfer functions over Fourier modes, weighted by a geometric form factor determined by the bubble size.
• Scans: They perform extensive scans over the physical parameter space, varying the latent heat ($L_c/T_c^4$), potential parameters ($A, \lambda$), critical temperature ($T_c$), and the dark-to-visible sector temperature ratio ($r_T$).

3. Key Contributions

1. First Calculation of FOPT-Induced Spin: This is the first study to calculate the spin of PBHs formed specifically from the collapse of false vacuum bubbles induced by standard cosmological perturbations, without relying on enhanced power spectra.
2. Second-Order Mechanism: The paper rigorously establishes that for spherical bubbles, the spin is a second-order quantity ($J \propto \delta \times v$), contrasting with ellipsoidal collapse models where spin can be first-order.
3. Sound Speed Instability: The authors identify a critical physical mechanism where the sound speed squared in the dark plasma becomes negative during the phase transition. This leads to an exponential enhancement of subhorizon density and velocity perturbations, which is crucial for generating significant angular momentum.
4. Scaling Relations: They derive a scaling relation for the RMS spin ($s_{\text{rms}}$) linking it to:
   • The inverse time scale of the FOPT ($\beta_*/H_*$).
   • The bubble wall velocity ($v_w$).
   • The dark-to-visible temperature ratio ($r_T$).
   • The effective degrees of freedom ($g_{\rho}$).

4. Results

• Spin Magnitude: The dimensionless spin parameter $s$ for FV bubbles spans a vast range, from $O(10^{-5})$ to $O(10)$.
  • Values $s > 1$ are physically permissible in this context because the objects are pre-collapse bubbles, not yet black holes.
  • The spin is highly sensitive to the FOPT parameters.
• Scaling Dependencies:
  • $s_{\text{rms}} \propto (\beta_*/H_*)^2 / v_w^2$: Faster transitions (larger $\beta$) and slower walls (smaller $v_w$) generally increase spin.
  • $s_{\text{rms}} \propto r_T^{-4}$: A colder dark sector relative to the visible sector significantly enhances the spin.
  • $s_{\text{rms}} \propto g_{\rho, \text{SM}}$: Higher degrees of freedom in the visible sector increase the spin.
• Benchmark Points: The authors provide specific benchmark scenarios (BP-1 to BP-6) covering critical temperatures from 10 keV to 100 GeV.
  • For $T_c \sim 10$ keV and $r_T \sim 0.4$, spins are generally small ($\sim 10^{-5}$).
  • For higher $T_c$ or lower $r_T$ (e.g., $r_T = 0.1$), spins can reach $O(1)$ or higher.
• Cutoff Scale: The integration for spin calculation is naturally cut off at the scale of the bubble radius ($k_{\text{cut}} \approx \pi/R_*$), as modes smaller than the bubble resolve the internal structure of the vacuum, invalidating the single-fluid approximation.

5. Significance

• PBH Characterization: This work provides a crucial missing piece in the characterization of PBHs formed via FOPTs. Knowing the spin distribution is essential for predicting gravitational wave signatures from PBH mergers and understanding their accretion history.
• Dark Sector Probes: The results suggest that the spin of PBHs could serve as a probe for the thermodynamics of the dark sector, particularly the temperature ratio $r_T$ and the dynamics of the phase transition ($\beta/H$).
• Theoretical Robustness: By demonstrating that significant spin can be generated even without a large enhancement in the primordial power spectrum, the paper broadens the viable parameter space for PBH formation mechanisms.
• Future Observables: The wide range of predicted spins implies that if such PBHs exist, their merger rates and gravitational wave waveforms could exhibit distinct features compared to standard astrophysical black holes or PBHs formed via other mechanisms.

In summary, the paper establishes that cosmological perturbations, amplified by the unique thermodynamic instabilities of a first-order phase transition, can generate a wide spectrum of angular momentum in primordial vacuum bubbles, potentially leading to rapidly spinning primordial black holes.