Primordial black holes as a natural consequence of scale-invariance

arXiv:2610.10677 · astro-ph.CO, hep-ph · Submitted 2026-10-07 · Read on arXiv

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Introduction to the show: ident: Astrophysics Radio. Generated commentary on the latest astrophysics papers.

Vera: Today's paper: "Primordial black holes as a natural consequence of scale-invariance".

Jocelyn: The gist: A scale-invariant U(1) gauge theory can naturally realize large curvature perturbations around the end of thermal inflation, seeding the formation of primordial black holes (PBHs).

Vera: First, who's behind it and why it matters.

Paper summary: Vera: So this paper "Primordial black holes as a natural consequence of scale-invariance" basically shows how these U(one) gauge theories can naturally produce those big density fluctuations we need for primordial black holes around the end of thermal inflation.

Jocelyn: It connects the abstract math about gauge theories to something that actually forms physical objects in our universe, which is what I always try to look for when I'm scanning those pulsar surveys.

Subrahmanyan: The authors are using a specific nonminimal gravitational interaction, labeled xi, of the Higgs field, and they find that if xi is positive and around one, it creates a negative mass-squared term at the Hubble scale.

Vera: That negative mass-squared term is what triggers this second order phase transition that stops thermal inflation, which is key because it's much more specific than just any random inflation model.

Jocelyn: So, they argue that this specific mechanism allows the gauge coupling to be much smaller than one, which changes how the fields behave during that transition.

Subrahmanyan: And because of this smaller coupling, the field fluctuations are dominated by thermal fluctuations right at the start of that phase transition, and these fluctuations get amplified enough to make a lot of primordial black holes with a wide range of masses.

Vera: A wide range is important because it suggests this isn't just one specific mass for these objects, but a whole distribution we can potentially look for in observations.

Jocelyn: I’m also interested in the gravitational waves they mentioned—the scalar-induced ones from the perturbations and those cosmic strings coming from the U(one) symmetry breaking.

Subrahmanyan: The scalar-induced gravitational wave spectrum relates directly to those PBH masses, with a characteristic frequency around five times ten-nine hertz times the peak black hole mass in solar masses squared.

Vera: So, this paper suggests that the way gravity couples to that field tells us a lot about both the structure of primordial fluctuations and what kind of gravitational signals we should expect to see.

Jocelyn: It really puts those theoretical calculations into a framework where I can start thinking about what kind of signals we might be hunting for with our instruments.

Subrahmanyan: The implication is that even in a relatively simple scale-invariant theory, you get a mechanism for generating PBHs spanning from light seeds to objects around fifty solar masses.

Vera: It’s an interesting piece because it shows how the details of the gravitational interaction can dictate whether we see these black holes and what kind of gravitational waves accompany them.

Jocelyn: This work opens up a new avenue for connecting inflation dynamics, phase transitions, and observable signals in a pretty concrete way.

Conclusion: Vera: So, to wrap up this whole discussion on "Primordial black holes as a natural consequence of scale-invariance," we're looking at how this simple U(one) gauge theory can actually generate density fluctuations big enough for primordial black holes around the end of thermal inflation.

Jocelyn: It really is interesting because it takes something that sounds purely theoretical, these gauge theories, and connects it to actual objects we might be able to detect out there in the sky.

Subrahmanyan: The authors are showing that by tweaking the nonminimal gravitational interaction xi of the Higgs field, they can get a negative mass-squared term at the Hubble scale when xi is positive and around one.

Vera: That specific way they end thermal inflation through a second order phase transition is what makes this model distinct from other inflation scenarios we've looked at.

Jocelyn: And because of that, the field fluctuations aren't just vacuum noise; they get amplified by thermal effects right around that moment the transition starts happening.

Subrahmanyan: This amplification leads to a broad range of possible black hole masses, not just one fixed mass for these objects.

Vera: That variability is what makes it compelling because it opens up many different possibilities for where we might find them in observational data.

Jocelyn: We've talked about how these perturbations generate both scalar and cosmic-string gravitational waves, and those signals are directly linked to the PBH mass range they predict.

Subrahmanyan: The main implication here is that this model offers a concrete pathway from basic symmetry breaking in a field theory to the formation of black holes across different mass scales.

Vera: It shows us how sensitive those final observable gravitational waves are to the details of the Higgs field coupling, xi.

Jocelyn: And that sensitivity means we need very precise measurements from instruments like pulsar timing arrays or future space-based interferometers to really nail down these parameters.

Subrahmanyan: So, this paper gives us a specific target: look for those GW signatures in the frequency range dictated by the peak black hole mass.

Vera: Next time we talk about this, we'll dig into the actual constraints they found on xi and how that parameter space shifts when you try to find heavier or lighter black holes.

TaeHun Kim, *Wan-Il Park

Korea Institute for Advanced Study · International Center for Quantum-field Measurement Systems for Studies of the Universe and Particles · The Pennsylvania State University · Division of Science Education and Institute of Fusion Science, Jeonbuk National University

astro-ph.CO, hep-ph

Submitted: 2026-10-07

Updated: 2026-10-07

Comments: 14 pages, 8 figures

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 78/100

The gist: The gist: A scale-invariant U(1) gauge theory can naturally realize large curvature perturbations around the end of thermal inflation, seeding the formation of primordial black holes (PBHs).

Key concepts

Scale-invariant U(1) gauge theory
This is a specific type of particle physics model used in cosmology where the fundamental forces are described by a $U(1)$ symmetry that does not change its strength across different energy scales. It involves a complex scalar field whose spontaneous symmetry breaking sets the Planck scale, providing the framework for generating early universe dynamics.
Nonminimal gravitational interaction ($\xi$)
This term describes how the Higgs field couples to gravity in a non-standard way. When $\xi$ is positive and of order one, it introduces a negative mass-squared term at the Hubble scale. This specific effect triggers a second-order phase transition, which terminates thermal inflation in a controlled manner.
Spinodal amplification
This process describes how fluctuations in the early universe are exponentially amplified during an unstable phase of the field evolution, specifically around the onset of a phase transition. These amplified fluctuations are what convert into the large-scale curvature perturbations necessary to form primordial black holes.
Primordial Black Holes (PBHs)
These are hypothetical black holes formed from extremely large density fluctuations in the very early universe. In this model, they are seeded by the amplified curvature perturbations generated during thermal inflation's end. The mechanism allows for a broad range of possible PBH masses, including seeds for dark matter and supermassive black holes.

Terminology

Summary

The gist: A scale-invariant U(1) gauge theory can naturally realize large curvature perturbations around the end of thermal inflation, seeding the formation of primordial black holes (PBHs).

How it works

  1. The model starts with a scale-invariant U(1) gauge theory with a complex scalar field where spontaneous symmetry breaking generates the Planck scale pg2.

  2. A nonminimal gravitational interaction ξ of the Higgs field induces a Hubble-scale negative mass-squared term for positive ξ ∼ O(1) which triggers a second order phase transition that ends thermal inflation while keeping the extra e-folds small pg3.

  3. This mechanism allows the gauge coupling to be much smaller than unity, and field fluctuations are dominated by thermal fluctuations around the onset of the phase transition, which are sufficiently amplified to produce an appreciable amount of PBHs with a wide range of possible masses pg3.

Thermal Inflation and Phase Transition

In the early universe with finite temperature, the temperature-dependent contributions are added to Eq. (11)

**"The effective potential in the region of ϕ ≲ T /g can be approximated as V (ϕ, T) ≈ V0 − π/2 **

T 4 − ξV0/M 2Pϕ squared + g 2/8T 2ϕ 2−g3/4πT ϕ3 + 3g464π 16π 1/2ϕ4ln ϕ2v2ϕ!−1/2! (A13)"

If the reheating temperature after the primordial inflation was sufficiently high, the thermal corrections could trap the Higgs field ϕ at the origin

The end of thermal inflation is marked by a phase transition where the effective mass squared at the origin m 2 eff vanishes pg3. The nature of this transition depends on ξ, g, and vϕ, as shown in Figure 2 pg3.

Cosmological Perturbations and PBH Formation

In our scenario, when the second-order phase transition starts, thermal fluctuations dominate over the vacuum fluctuations

Spinodal amplification of these fluctuations generates the curvature perturbations

The power spectrum of field fluctuations can be converted to the power spectrum of curvature perturbation ζ

The resulting dimensionless curvature power spectra Pζ,k can reach O(10-1–10-2) pg7. The PBH mass follows the mass-horizon relation MPBH ≈ 32π/2√2γM 3P g 2v 4ϕ e 2∆NR pg8, and the formation probability fPBH is highly sensitive to ξ pg9.

Gravitational Waves and Observational Signatures

Curvature perturbations that are sufficient to produce PBHs also induce significant secondary scalar-induced GWs

Another source of GWs in our scenario is cosmic strings from local U(1) breaking

The present-day spectrum for scalar-induced GWs is given by omegaSI GW(f)h squared ≈ 4 × 10-4 × Z 1-1 dx Z ∞ 1 dy Pζ,(x+y)πfPζ,(y−x)πfJ 2(x, y)pg8. The characteristic frequency of the scalar-induced GW is related to the PBH mass as fpeak ≈ 5 × 10-9 Hz × MPBH peak M⊙-1/2 pg8.

The cosmic-string GW spectrum depends on the UV embedding of the U(1) symmetry and on the cosmological history pg8. The SMBH and "DM" benchmarks provide interesting observational targets, with the “SMBH” signal peaking at frequencies of order 10-9 Hz, overlapping the projected SKA sensitivity range pg8.

Summary and Discussion

The model demonstrates that a nonminimal coupling ξ ∼ O(1) triggers an instability earlier than in the minimally coupled case, reaching m 2 eff = 0 by the ξ contribution pg11. This results in a broad producible PBH mass range in our scenario, including SMBH seeds and PBH DM pg9. The observational status of PBHs strongly depends on mass, with constraints excluding the light gray shaded regions in Figure 6 pg9.

This work shows that small g values do not necessarily lead to an excessively long thermal inflation, leaving room for primordial inflation to generate the observed large-scale perturbation pg9. The model permits light evaporating PBHs, under possible constraint by cosmic-string GWs pg9.

The mass function fPBH can be above or below unity, and is highly sensitive to ξ, due to the exponential sensitivity of βR to the curvature power spectrum through Eqs. (44) and (45) pg9. The observational status of PBHs strongly depends on mass pg9.

Viable PBH populations include PBH DM and SMBH seeds, where the latter can be in the mass range of MPBH ≈ 50M⊙ with 10-12MPBH/M⊙ ≤ fPBH ≤ 10-8MPBH/M⊙ pg9. The constraints on PBHs are sensitive to ξ, and widen with decreasing ξ pg9.

The model also permits light evaporating PBHs, under possible constraint by cosmic-string GWs pg9. The constraints on PBHs are sensitive to ξ, and widen with decreasing ξ pg9.

The model also permits light evaporating PBHs, under possible constraint by cosmic-string GWs pg9.

Improvements for AI systems

  1. Incorporation of Nonminimal Coupling Effects in Perturbation Theory: Improve AI systems to predict how a nonminimal gravitational interaction ξ of the U(1) Higgs field enhances curvature perturbations, as this term is shown to make the induced mass term triggers a second order phase transition when thermal fluctuations dominate over vacuum fluctuations.

  2. PBH Mass and Abundance Prediction via Parameter Space Mapping: Enhance the AI's ability to map parameter spaces (g, vϕ/GeV, ξ) to specific PBH populations by using the derived relations in Eq. (46) and Eq. (48), allowing it to predict the broad producible PBH mass range in our scenario, 10−30M⊙ ≲ MPBH ≲ 105M⊙.

  3. GW Signal Spectrum Modeling for Observational Constraints: Develop the system to calculate the scalar-induced GW spectrum using Eq. (50) and Eq. (51), enabling it to distinguish between benchmark points by predicting frequencies relevant to detectors like NANOGrav or LISA, and assessing constraints from cosmic-string GWs.

  4. Distinguishing Thermal vs. Vacuum Fluctuations in Phase Transitions: Improve the AI's diagnostic capability to determine the nature of the phase transition by comparing conditions related to Eq. (29) and Eq. (30), specifically identifying when thermal fluctuations dominate over vacuum fluctuations to predict whether a first or second order phase transition can take place.

  5. Modeling Stochastic Field Evolution with Time-Dependent Noise: Implement the Langevin equation (B1) and its resulting power spectrum evolution in Eq. (B7) to simulate the dynamics of field modes during the tachyonic stage, allowing for predictions of how the numerical solution agrees with the thermal equilibrium solution Pk = (2nk + 1)χk2/a2 at T ≫ Tc and evolves smoothly into the tachyonic regime.

Abstract

We show that a scale-invariant U(1) gauge theory can naturally realize large curvature perturbations around the end of thermal inflation, seeding the formation of primordial black holes (PBHs). The allowed nonminimal gravitational interaction ξ of the Higgs field induces a Hubble-scale negative mass-squared term for positive ξ about O(1), and triggers a second order phase transition that ends thermal inflation while keeping the extra e-folds small. This sharp contrast to the minimally coupled case allows the gauge coupling to be much smaller than unity. It turns out that field fluctuations are dominated by thermal fluctuations around the onset of the phase transition, and sufficiently amplified to produce an appreciable amount of PBHs with a wide range of possible masses. The associated scalar-induced gravitational waves (GWs) are accompanied by cosmic-string induced GWs coming from the breaking of the U(1) symmetry. As a result, the generated PBH population can have a correlated GW signature relevant to pulsar timing arrays, space-based interferometers, or ground-based detectors.

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