QCD and electroweak phase transitions with hidden scale invariance: implications for primordial black holes, quark-lepton nuggets and gravitational waves

Technical Deep Dive: Cosmological Implications of Scale-Invariant Standard Model

Overview

This work investigates the cosmological implications of a framework where the Standard Model features spontaneously broken scale invariance. The key findings are:

• The electroweak phase transition is delayed until after the QCD chiral phase transition, which occurs at around 28 MeV. This triggers the electroweak symmetry breaking almost instantaneously.
• Primordial black holes could form during the QCD hadronization or chiral-electroweak phase transitions, with typical masses of ~3 M☉ or ~40 M☉, respectively. Their abundance is highly sensitive to the details of the underlying inflationary model.
• Gravitational waves from the chiral-electroweak phase transition have a peak frequency of ~2×10^-5 Hz but negligible amplitude, well below current and planned detector sensitivities.
• Metastable quark-lepton nuggets with masses around 10^9 kg could form during the hadronization transition, but they contribute less than 1% to the dark matter density.

Methodology

• The framework incorporates a light dilaton field alongside the Standard Model fields, through a consistent realization of hidden scale invariance.
• The dynamics of the QCD-triggered electroweak phase transition are modeled using a linear σ-model coupled to the Higgs field.
• Primordial black hole formation is analyzed considering two scenarios: collapse of overdensities generated during the phase transition, and gravitational collapse of pre-existing inflationary overdensities.
• Gravitational wave production is estimated using the envelope approximation, accounting for the dominant contributions from sound waves and turbulence in the plasma.
• The formation and stability of quark-lepton nuggets are studied by minimizing their constrained free energy.

Results

Electroweak Phase Transition

• The electroweak phase transition is delayed until after the QCD chiral phase transition, which occurs at around T^(χ)_c ≈ 28 MeV.
• The chiral phase transition destabilizes the Higgs potential, allowing the Higgs field to roll down to its electroweak-breaking minimum.

Primordial Black Holes

• PBHs could form from the gravitational collapse of pre-existing inflationary overdensities during the QCD hadronization (mPBH ~ 3 M☉) or chiral-electroweak (mPBH ~ 40 M☉) phase transitions.
• The PBH abundance is highly sensitive to the details of the inflationary model, with the most favorable scenario yielding ΩPBH h^2 ~ 0.05-1 for the hadronization transition and ΩPBH h^2 ~ 10^-7 - 10^-4 for the chiral-electroweak transition.

Gravitational Waves

• Gravitational waves from the chiral-electroweak phase transition have a peak frequency of fSW ≈ 2×10^-5 Hz and a peak amplitude ΩSW(f_SW) h^2 ≈ 1.2×10^-16, too small to be detected by current or planned observatories.

Quark-Lepton Nuggets

• Metastable quark-lepton nuggets with typical masses around 10^9 kg could form during the hadronization transition.
• However, their contribution to the dark matter density is estimated to be less than 1%.

Limitations & Uncertainties

• The results rely on several simplifying assumptions in modeling the QCD-triggered electroweak phase transition dynamics.
• The PBH abundance estimates depend exponentially on the details of the inflationary model, which are not fully constrained.
• The formation and stability of quark-lepton nuggets involve non-perturbative QCD dynamics that are difficult to model precisely.

What Comes Next

• More comprehensive numerical analysis of the phase transition dynamics using advanced computational tools would help refine the quantitative predictions.
• Extending the analysis to beyond-the-Standard-Model frameworks that address issues like neutrino masses, electroweak vacuum stability, or the strong CP problem could yield different cosmological implications.