Curvature Perturbations from First-Order Phase Transitions: Implications to Black Holes and Gravitational Waves

This work demonstrates that employing a fully covariant formalism to account for previously overlooked gauge dependencies reveals that the formation of primordial black holes and scalar-induced gravitational waves from first-order phase transitions is strongly suppressed, thereby calling into question their suitability as an explanation for the recent signals from pulsar timing arrays.

The Gist

The Big Picture: A Cosmic "Plop" That Didn't Scream

Imagine the early universe as a giant pot of cooling water. In our current universe, water freezes into ice smoothly. But in the very early universe, scientists think the "water" (the fundamental forces) might have frozen suddenly, like water turning to ice in a supercooled state. This is called a first-order phase transition (FOPT).

When this happens, bubbles of the new "ice" (the new vacuum state) begin to form within the old "water." These bubbles expand, collide with each other, and release a massive amount of energy.

For a long time, physicists believed these cosmic bubble collisions were so violent that they would produce two main things:

1. Primordial Black Holes (PBHs): Tiny black holes formed from the sheer weight of the collapsing bubbles.
2. Gravitational Waves (GWs): Ripples in spacetime, like the sound of a drumbeat, which we might hear today with special detectors (such as the Pulsar Timing Array).

The Problem: Previous studies used a "map" (a mathematical framework) that was slightly distorted. They viewed the universe from a specific, non-rotating perspective that made the bubbles appear much larger and more energetic than they actually were.

The New Discovery: This paper says, "Wait a minute, let's look at the map from every possible angle." When the authors applied a fully correct, "covariant" (angle-independent) method, they found that previous maps had drastically overestimated the power of these events.

The Analogy: The Foggy Window versus the Clear Lens

Imagine previous studies as looking at a storm through a foggy, distorted window. Through this window, the raindrops (bubbles) looked like giant hailstones, and the wind (energy) looked like a hurricane. Based on this view, they predicted the storm would smash houses (create black holes) and shake the ground (produce loud gravitational waves).

This paper is like wiping the window clean and using a high-resolution lens. When they looked through the clear lens, they realized:

• The hailstones were actually just small raindrops.
• The hurricane was merely a gentle breeze.

What They Found (The "So What?")

When they corrected the math, the results changed completely:

1. The Black Holes Disappeared

• Old View: The bubbles were so heavy they would easily collapse into black holes.
• New View: The bubbles are too light and too spread out. They simply don't have enough "momentum" to crush themselves into black holes.
• The Result: It is highly unlikely that these specific phase transitions produced the primordial black holes we are searching for. If we want to find evidence of these ancient bubble collisions, looking for black holes might be a dead end.

2. The Gravitational Waves Became Quieter

• Old View: The collisions produced a deafening roar of gravitational waves, loud enough to explain the signals we currently hear from the Pulsar Timing Array (a network of cosmic clocks).
• New View: The signal is much, much quieter. The authors calculated that previous estimates were off by a factor of 100,000 (or more).
• The Result: The "loud" signals we currently hear from the universe likely cannot be explained by these specific types of bubble collisions. The signal is too weak to be the main cause.

The "Gauge" Confusion (The Technical Glitch)

Why didn't the old math work? It comes down to something called "gauge dependence."

In physics, you can describe the universe using different coordinate systems (like stating a room's temperature in Celsius or Fahrenheit, or measuring a room's size from the corner versus from the center). Normally, the physical reality doesn't change, but the numbers you write down do.

• The Error: Previous researchers calculated the "density" (how much stuff is in a bubble) using a system called the "spatially-flat gauge." In this system, the numbers looked huge.
• The Reality: To know whether a bubble collapses into a black hole, one must use a different system called the "comoving gauge," which moves with the fluid.
• The Shock: When they translated the numbers from the "flat" system to the "comoving" system, the density dropped by a factor of 10. Since the formation of black holes depends on the density squared (or even higher powers), a density drop of 10 meant the probability of forming a black hole dropped by 100,000 or more.

The Bottom Line

This paper is a "reality check" for cosmology.

• Before: "Wow, these early bubble collisions in the universe were so violent that they created black holes and loud gravitational waves!"
• After: "Actually, if we do the math correctly, these collisions were much quieter. They probably didn't create any black holes and are not the source of the loud gravitational wave signals we detect today."

The authors have also released a new software tool (called deltaPT 2.0) so that other scientists can use this correct, "clear lens" method to study the early universe without making the same mistake.

In short: The early universe's "plop" was much quieter than we thought, and it likely didn't leave behind the heavy black holes or loud echoes we hoped to find.

Technical Summary

Technical Summary: Curvature Perturbations from First-Order Phase Transitions: Implications for Black Holes and Gravitational Waves

Problem Statement
The formation of primordial black holes (PBHs) and the generation of scalar-induced gravitational waves (SIGWs) during strong first-order phase transitions (FOPTs) in the early universe are critical topics in cosmology. These phenomena are particularly relevant for interpreting recent Pulsar Timing Array (PTA) signals. However, previous studies have relied on non-covariant treatments of cosmological perturbations, often implicitly choosing the spatially flat gauge. Since cosmological perturbations are inherently gauge-dependent, there is a risk that earlier predictions for PBH abundances and SIGW spectra are inaccurate. In particular, it remains an open question whether the energy stored in nucleating bubbles during a supercooled FOPT is sufficient to produce observable PBHs or significant SIGWs without violating existing constraints.

Methodology
The authors apply a fully covariant formalism to investigate cosmological perturbations generated during a strongly supercooled FOPT. The methodology includes:

1. Covariant Framework: The authors model the background evolution using a relativistic bag equation of state and decompose the energy density into radiation ($\rho_R$) and vacuum ($\rho_V$) components. They treat the nucleation rate $\Gamma$ stochastically.
2. Gauge Analysis: The study explicitly compares perturbations in the spatially flat gauge (F) and the comoving gauge (C). It derives transformation rules between these gauges, specifically relating the density contrast $\delta$ and the comoving curvature perturbation $\mathcal{R}$.
3. Numerical Implementation (deltaPT 2.0): The authors utilize and extend their public code deltaPT 2.0 to calculate the false vacuum fraction $F$ and the resulting non-adiabatic pressure perturbation $\delta p_{\text{nad}}$. They perform $N_{\text{sim}} = 10^6$ realizations of random bubble nucleation histories within a comoving volume to generate the probability density functions (PDFs) of the perturbations.
4. Statistical Evaluation:
   • PBHs: The abundance is estimated using the density contrast in the comoving gauge, $\delta^{(C)}$, at the time of Hubble crossing ($k \simeq H$). This choice aligns with numerical relativity simulations that define the collapse threshold based on the compaction function in the comoving gauge.
   • SIGWs: The SIGW spectrum is calculated using the gauge-invariant curvature perturbation $\mathcal{R}$, treating the source as a second-order effect. The authors also investigate the influence of non-Gaussian distributions (skewness) on the SIGW signal.

Main Contributions

1. Identification of Gauge Dependence: The work demonstrates that earlier non-covariant analyses (which often employed the spatially flat gauge) significantly overestimate the amplitude of perturbations. The authors find that at the time of Hubble crossing, the density contrast in the spatially flat gauge is linked to the comoving gauge by $\delta^{(F)} \simeq 10 \delta^{(C)}$.
2. Revised Collapse Threshold: By correctly identifying the comoving gauge as the appropriate framework for PBH collapse criteria (where $\delta^{(C)}_c \in [0.40, 0.67]$), the authors show that using the spatially flat gauge leads to an underestimation of the collapse threshold by a factor of $\sim 10$.
3. Correction of SIGW Amplitude: The study corrects the calculation of the curvature power spectrum. It identifies a factor of $2/3\pi$ between the variance of the volume-averaged curvature perturbation and its power spectrum, which was previously overlooked.
4. Analysis of Non-Gaussian Distribution: The authors quantify the non-Gaussian distribution of curvature perturbations and show that non-Gaussian corrections to the SIGW spectrum are negligible compared to the Gaussian contribution.

Results

• PBH Abundance: The PDF of the density contrast in the comoving gauge, $\delta^{(C)}$, exhibits negative skewness, associated with an exponential suppression of large overdensities. Even for optimistic completion rates ($\beta/H = 4$), the probability of exceeding the collapse threshold $\delta^{(C)}_c \gtrsim 0.4$ is extremely low. The authors conclude that slow FOPTs do not produce PBHs in observable quantities.
• SIGW Spectrum: The SIGW contribution is found to be strongly suppressed. Due to the gauge correction and the factor between variance and power spectrum, the SIGW amplitude is lower than previous estimates (particularly Ref. [20]) by a factor of approximately $2 \times 10^5$. The SIGW signal never forms a secondary maximum in the full gravitational wave spectrum and affects only the far-infrared tail.
• Gauge Comparison: The analysis confirms that a misidentification of the gauge leads to a dramatic overestimation of the PBH abundance (by many orders of magnitude) and the SIGW amplitude (by a factor of $10^4$).

Significance and Claims
The work claims that the previously proposed interpretation of the PTA signal via strong FOPTs, based on the production of PBHs or large SIGWs, is significantly challenged by these results.

• PBH Constraints: The results imply that using PBHs as a probe for strong FOPTs in the primordial universe is much more difficult than previously assumed, as the production rate is negligible.
• PTA Interpretation: The findings suggest that the "loudest" region of the posterior distribution for FOPT interpretations of the PTA signal is not necessarily excluded by PBH overproduction constraints (e.g., from LIGO-Virgo-KAGRA). However, the suppression of SIGWs implies that FOPTs with reheating temperatures above a GeV may not explain the PTA signal via the scalar-induced mechanism, contrary to some earlier claims.
• Methodological Rigor: The work emphasizes the necessity of covariant, gauge-invariant treatments when linking phase transitions in the early universe to late-time observations, correcting a systematic error in the literature regarding gauge choice.

The authors point out limitations, including the neglect of gravitational effects on nucleation rates and bubble expansion, as well as gradient energy in bubble walls, and propose these as directions for future refinements.