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arxiv: 2502.20166 · v4 · submitted 2025-02-27 · ✦ hep-ph · astro-ph.CO· hep-th

Numerical simulations of density perturbation and gravitational wave production from cosmological first-order phase transition

Jintao Zou , Zhiqing Zhu , Zizhuo Zhao , Ligong Bian This is my paper

Pith reviewed 2026-05-23 02:30 UTC · model grok-4.3

classification ✦ hep-ph astro-ph.COhep-th
keywords first-order phase transitiondensity perturbationgravitational waveslattice simulationbubble wallvacuum decayprimordial black holescosmology
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The pith

Lattice simulations show bubble wall motion dominates density perturbations for strong first-order phase transitions while vacuum decay delays dominate for weak ones.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper conducts three-dimensional lattice simulations to study density perturbations and gravitational waves produced during cosmological first-order phase transitions. It determines that the phase transition strength alpha controls the primary source of density perturbations, with bubble wall forward motion taking over for alpha greater than 1 and the delay of vacuum decay for alpha less than 1. The power spectrum of density perturbations scales as k cubed at small wavenumbers and k to the power of negative 1.5 at large wavenumbers. The gravitational wave power spectra scale as k cubed at small wavenumbers and k to the power of negative 2 at large wavenumbers. These results confirm that slow phase transitions can generate primordial black holes and supply predictions for gravitational wave detection.

Core claim

In three-dimensional lattice simulations of first-order phase transitions, for phase transition strength alpha greater than 1 the forward motion of bubble walls is the primary source of density perturbation while for alpha less than 1 the dominant contribution comes from the delay of vacuum decay; the density perturbation power spectrum has slope k cubed at small k and k to the minus 1.5 at large k while the gravitational wave spectrum has slope k cubed at small k and k to the minus 2 at large k.

What carries the argument

Three-dimensional lattice simulations of bubble wall motion and vacuum decay in first-order phase transitions.

If this is right

  • Primordial black holes can be produced by slow phase transitions.
  • Gravitational wave spectra from phase transitions follow specific power laws that can be searched for in observations.
  • Density perturbation spectra from phase transitions exhibit k cubed behavior at small scales transitioning to k to the minus 1.5 at large scales.
  • The switch in dominant mechanism occurs at alpha equal to 1.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The simulations indicate that non-linear effects in bubble dynamics require numerical treatment for accurate spectra.
  • These findings may help interpret potential signals in gravitational wave detectors as coming from early universe phase transitions.
  • Extending the simulations to include additional effects like plasma friction could refine the high-k behavior of the spectra.

Load-bearing premise

The lattice simulations accurately capture the non-linear dynamics of bubble wall motion and vacuum decay without significant numerical artifacts or missing physical effects like friction or plasma interactions.

What would settle it

An observation of gravitational waves from a first-order phase transition whose spectrum does not show the k cubed to k to the minus 2 transition would falsify the simulated results.

Figures

Figures reproduced from arXiv: 2502.20166 by Jintao Zou, Ligong Bian, Zhiqing Zhu, Zizhuo Zhao.

Figure 1
Figure 1. Figure 1: Equation of state (EOS) evolution for weak phase transition strength (top, solid line [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The PBH abundance variation with respect to the PT strength parameter [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Evolution of σδ for weak phase transition strength (top, solid line α=0.5, dash-dotted line α=1) and strong phase transition strength (bottom, solid line α=5, dash-dotted line α=10) [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Variation of the standard deviation σδ of accumulated overdensity perturbations with respect to the PT parameters α and β/H. when t/R∗ ∼ 1.2 mostly come from the energy transformation in the simulation volume, which is influenced by the forward motion of bubble walls. Differently, the σδ reach maximum at around t/R∗ ∼ 0.5 for strong PTs scenarios with α=5, 10 sourced from the delay of vacuum decay in diffe… view at source ↗
Figure 5
Figure 5. Figure 5: The power spectra of density perturbations for different [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The energy density spectra of GWs for different [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: GW spectra for three scenarios: (α = 1, β/H = 10, T = 10−2 GeV), (α = 1, β/H = 10, T = 103 GeV), and (α = 10, β/H = 20, T = 107 GeV). The sensitivity curves including EPTA [98], PPTA [99], NANOGrav [100], µAres [101], SKA [97], LISA [34], Taiji [37], DECIGO [102], LIGO [96], CE [103], and ET [104]. to variations in β/H and a relatively weaker dependence on α. In particular, when α ≳ 10, the influence of α … view at source ↗
Figure 8
Figure 8. Figure 8: The mean field evolution under weak phase transition strength (left, solid line [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The bubble slice diagram for β/H = 6 and α = 5. The dashed lines are presented to devide the simulation box to 64 Hubble volumes [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The energy density ρ under weak phase transition strength (left, α = 1, solid line β/H = 6, dash-dotted line β/H = 10) and strong phase transition strength (right, α = 10 solid line β/H = 6, dash￾dotted line β/H = 10) [PITH_FULL_IMAGE:figures/full_fig_p013_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: δ of the last 16 delayed-decayed regions. The β/H from left to right are 6, 8, 10, and 12, the α from top to bottom are 0.5, 1, 5, and 10, respectively. We numerically analyze the evolution of local energy overdensities δ induced by delayed vacuum decay during FOPTs in the early universe. Specifically, we examine how different phase transition parameters, α and β/H, affect δ and whether these overdense re… view at source ↗
Figure 12
Figure 12. Figure 12: From left to right, the GW power spectra correspond to [PITH_FULL_IMAGE:figures/full_fig_p015_12.png] view at source ↗
read the original abstract

We conducted three-dimensional lattice simulations to study the density perturbation and gravitational waves (GWs) during first-order phase transition (FOPT). We find that for phase transition strength $\alpha > 1$, the forward motion of bubble walls becomes the primary source, whereas for $\alpha < 1$, the dominant contribution to the density perturbation comes from the delay of vacuum decay. Additionally, the power spectrum of density perturbations generated by the phase transition exhibits a slope of $k^3$ at small wavenumbers and $k^{-1.5}$ at large wavenumbers. Furthermore, we calculated the GW power spectra, which exhibit the slope of $k^3$ at small wavenumbers and $k^{-2}$ at large wavenumbers. Our numerical simulations confirm that slow PTs can produce PBHs and provide predictions for the GW power spectrum, offering theoretical support for GW detection.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript reports results from three-dimensional lattice simulations of cosmological first-order phase transitions, claiming that bubble-wall forward motion dominates density perturbations for transition strength α > 1 while delayed vacuum decay dominates for α < 1. It further states that the density-perturbation power spectrum follows k³ at small k and k^{-1.5} at large k, while the GW spectrum follows k³ at small k and k^{-2} at large k. The work concludes that slow phase transitions can produce primordial black holes and supplies GW spectral predictions for observational support.

Significance. If the simulation results prove robust, the α-dependent source attribution and the reported spectral indices would supply concrete numerical benchmarks for density-perturbation and GW production in first-order transitions, directly relevant to primordial-black-hole formation scenarios and to the interpretation of future gravitational-wave data. The three-dimensional treatment of non-linear bubble dynamics is a methodological strength that can anchor analytic approximations in the literature.

major comments (2)
  1. [Abstract] Abstract: the claims that forward wall motion is primary for α > 1 and vacuum-decay delay is primary for α < 1, together with the specific indices k³ / k^{-1.5} (density) and k³ / k^{-2} (GW), are presented without any mention of lattice resolution, convergence tests, or error estimates. These attributions and slopes are load-bearing for the central numerical conclusions; their validity cannot be assessed from the given information.
  2. [Numerical-methods / results sections] Numerical-methods / results sections: no cross-check against known analytic limits (e.g., thin-wall or runaway-wall regimes) or against runs that include friction/plasma back-reaction is described, leaving open the possibility that the reported source dominance and power-law indices shift when those effects are restored.
minor comments (2)
  1. [Abstract] Abstract: the range of α values actually simulated and the bubble-wall velocities employed should be stated explicitly so readers can judge the domain of applicability of the reported transition in source dominance.
  2. Figure captions for power spectra should indicate the fitting procedure and any quoted uncertainties on the extracted slopes.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for highlighting important points regarding the presentation of our numerical results. We address each major comment below and outline the revisions we will make to strengthen the paper.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claims that forward wall motion is primary for α > 1 and vacuum-decay delay is primary for α < 1, together with the specific indices k³ / k^{-1.5} (density) and k³ / k^{-2} (GW), are presented without any mention of lattice resolution, convergence tests, or error estimates. These attributions and slopes are load-bearing for the central numerical conclusions; their validity cannot be assessed from the given information.

    Authors: We agree that the abstract would benefit from explicit reference to the numerical validation to support the reported source attributions and spectral indices. Details on lattice resolution, convergence tests, and error estimates are already contained in the Numerical Methods and Results sections. We will revise the abstract to include a concise statement summarizing the simulation parameters and validation procedures performed. revision: yes

  2. Referee: [Numerical-methods / results sections] Numerical-methods / results sections: no cross-check against known analytic limits (e.g., thin-wall or runaway-wall regimes) or against runs that include friction/plasma back-reaction is described, leaving open the possibility that the reported source dominance and power-law indices shift when those effects are restored.

    Authors: Our simulations are performed in the vacuum-dominated regime without plasma friction to isolate the gravitational effects of bubble dynamics. While parameter choices allow partial consistency with thin-wall expectations, we acknowledge that explicit cross-checks against analytic limits and friction-inclusive runs are not described. In the revised manuscript we will add a discussion paragraph comparing the obtained spectral indices to known analytic expectations in the thin-wall and runaway regimes and will explicitly note the limitations arising from the omission of plasma back-reaction. revision: partial

Circularity Check

0 steps flagged

No circularity: results are direct outputs of lattice simulations

full rationale

The paper reports outcomes from three-dimensional lattice simulations of first-order phase transitions. Claims regarding source dominance (bubble-wall motion for α>1 versus vacuum-decay delay for α<1) and extracted power-law slopes (k^3 / k^{-1.5} for density perturbations; k^3 / k^{-2} for GWs) are produced by the numerical evolution itself. No load-bearing step reduces by construction to a fitted parameter, self-citation chain, or ansatz imported from prior work by the same authors. The derivation chain is self-contained; the simulations constitute independent computational evidence rather than a renaming or re-derivation of inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper uses numerical methods on top of standard assumptions in high energy physics and cosmology; no new free parameters or entities introduced in the abstract.

axioms (1)
  • domain assumption Standard model of cosmology and first-order phase transition dynamics in quantum field theory.
    The simulations rely on established theoretical framework for FOPT.

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Reference graph

Works this paper leans on

111 extracted references · 111 canonical work pages · cited by 2 Pith papers · 42 internal anchors

  1. [1]

    As more and more bubbles form and collide, the initially uniform spatial structure is disrupted, resulting in an asymmetric energy distribution

    (15) During a FOPT, vacuum bubbles are generated throughout space via quantum tunneling and then rapidly expand. As more and more bubbles form and collide, the initially uniform spatial structure is disrupted, resulting in an asymmetric energy distribution. The walls of these bubbles carry high energy, and when they collide, this energy is intensely relea...

    work page

  2. [2]

    S. W. Hawking, I. G. Moss, and J. M. Stewart, Bubble Collisions in the Very Early Universe, Phys. Rev. D26, 2681 (1982)

    work page 1982

  3. [3]

    Crawford and D

    M. Crawford and D. N. Schramm, Spontaneous Generation of Density Perturbations in the Early Universe, Nature298, 538 (1982)

    work page 1982

  4. [4]

    Kodama, M

    H. Kodama, M. Sasaki, and K. Sato, Abundance of Primordial Holes Produced by Cosmological First Order Phase Transition, Prog. Theor. Phys.68, 1979 (1982)

    work page 1979

  5. [5]

    M. C. Johnson, H. V. Peiris, and L. Lehner, Determining the outcome of cosmic bubble collisions in full General Relativity, Phys. Rev. D85, 083516 (2012), arXiv:1112.4487 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  6. [6]

    M. Y. Khlopov, R. V. Konoplich, S. G. Rubin, and A. S. Sakharov, Formation of black holes in first order phase transitions (1998), arXiv:hep-ph/9807343

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  7. [7]

    B. J. Carr and J. E. Lidsey, Primordial black holes and generalized constraints on chaotic inflation, Phys. Rev. D48, 543 (1993)

    work page 1993

  8. [8]

    Ivanov, P

    P. Ivanov, P. Naselsky, and I. Novikov, Inflation and primordial black holes as dark matter, Phys. Rev. D50, 7173 (1994)

    work page 1994

  9. [9]

    S. W. Hawking, Black Holes From Cosmic Strings, Phys. Lett. B231, 237 (1989)

    work page 1989

  10. [10]

    R. R. Caldwell and P. Casper, Formation of black holes from collapsed cosmic string loops, Phys. Rev. D53, 3002 (1996), arXiv:gr-qc/9509012. 17

    work page internal anchor Pith review Pith/arXiv arXiv 1996

  11. [11]

    A. C. Jenkins and M. Sakellariadou, Primordial black holes from cusp collapse on cosmic strings (2020), arXiv:2006.16249 [astro-ph.CO]

    work page arXiv 2020

  12. [12]

    J. J. Blanco-Pillado, K. D. Olum, and A. Vilenkin, No black holes from cosmic string cusps (2021), arXiv:2101.05040 [astro-ph.CO]

    work page arXiv 2021

  13. [13]

    Lunar Mass Black Holes from QCD Axion Cosmology

    T. Vachaspati, Lunar Mass Black Holes from QCD Axion Cosmology (2017), arXiv:1706.03868 [hep- th]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  14. [14]

    Primordial Black Holes from the QCD axion

    F. Ferrer, E. Masso, G. Panico, O. Pujolas, and F. Rompineve, Primordial Black Holes from the QCD axion, Phys. Rev. Lett.122, 101301 (2019), arXiv:1807.01707 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  15. [15]

    G. B. Gelmini, A. Simpson, and E. Vitagliano, Catastrogenesis: DM, GWs, and PBHs from ALP string-wall networks, JCAP02, 031, arXiv:2207.07126 [hep-ph]

    work page arXiv

  16. [16]

    G. B. Gelmini, J. Hyman, A. Simpson, and E. Vitagliano, Primordial black hole dark matter from catastrogenesis with unstable pseudo-Goldstone bosons, JCAP06, 055, arXiv:2303.14107 [hep-ph]

    work page arXiv

  17. [17]

    Gouttenoire and E

    Y. Gouttenoire and E. Vitagliano, Domain wall interpretation of the PTA signal confronting black hole overproduction, Phys. Rev. D110, L061306 (2024), arXiv:2306.17841 [gr-qc]

    work page arXiv 2024

  18. [18]

    Gouttenoire and E

    Y. Gouttenoire and E. Vitagliano, Primordial black holes and wormholes from domain wall networks, Phys. Rev. D109, 123507 (2024), arXiv:2311.07670 [hep-ph]

    work page arXiv 2024

  19. [19]

    Dolgov and J

    A. Dolgov and J. Silk, Baryon isocurvature fluctuations at small scales and baryonic dark matter, Phys. Rev. D47, 4244 (1993)

    work page 1993

  20. [20]

    Kasai, M

    K. Kasai, M. Kawasaki, and K. Murai, Revisiting the Affleck-Dine mechanism for primordial black hole formation, JCAP10, 048, arXiv:2205.10148 [astro-ph.CO]

    work page arXiv

  21. [21]

    Primordial black holes from supersymmetry in the early universe

    E. Cotner and A. Kusenko, Primordial black holes from supersymmetry in the early universe, Phys. Rev. Lett.119, 031103 (2017), arXiv:1612.02529 [astro-ph.CO]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  22. [22]

    Martin, T

    J. Martin, T. Papanikolaou, and V. Vennin, Primordial black holes from the preheating instability in single-field inflation, JCAP01, 024, arXiv:1907.04236 [astro-ph.CO]

    work page arXiv 1907

  23. [23]

    J. H. Chang, D. Egana-Ugrinovic, R. Essig, and C. Kouvaris, Structure Formation and Exotic Compact Objects in a Dissipative Dark Sector, JCAP03, 036, arXiv:1812.07000 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv

  24. [24]

    M. M. Flores and A. Kusenko, Primordial Black Holes from Long-Range Scalar Forces and Scalar Radiative Cooling, Phys. Rev. Lett.126, 041101 (2021), arXiv:2008.12456 [astro-ph.CO]

    work page arXiv 2021

  25. [25]

    Dom` enech, D

    G. Dom` enech, D. Inman, A. Kusenko, and M. Sasaki, Halo formation from Yukawa forces in the very early Universe, Phys. Rev. D108, 103543 (2023), arXiv:2304.13053 [astro-ph.CO]

    work page arXiv 2023

  26. [26]

    Chakraborty, P

    A. Chakraborty, P. K. Chanda, K. L. Pandey, and S. Das, Formation and Abundance of Late-forming Primordial Black Holes as Dark Matter, Astrophys. J.932, 119 (2022), arXiv:2204.09628 [astro- ph.CO]

    work page arXiv 2022

  27. [27]

    Primordial Black Hole Scenario for the Gravitational-Wave Event GW150914

    M. Sasaki, T. Suyama, T. Tanaka, and S. Yokoyama, Primordial Black Hole Scenario for the Gravitational-Wave Event GW150914, Phys. Rev. Lett.117, 061101 (2016), [Erratum: Phys.Rev.Lett. 121, 059901 (2018)], arXiv:1603.08338 [astro-ph.CO]. 18

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  28. [28]

    S. Bird, I. Cholis, J. B. Mu˜ noz, Y. Ali-Ha¨ ımoud, M. Kamionkowski, E. D. Kovetz, A. Raccanelli, and A. G. Riess, Did LIGO detect dark matter?, Phys. Rev. Lett.116, 201301 (2016), arXiv:1603.00464 [astro-ph.CO]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  29. [29]

    The clustering of massive Primordial Black Holes as Dark Matter: measuring their mass distribution with Advanced LIGO

    S. Clesse and J. Garc´ ıa-Bellido, The clustering of massive Primordial Black Holes as Dark Mat- ter: measuring their mass distribution with Advanced LIGO, Phys. Dark Univ.15, 142 (2017), arXiv:1603.05234 [astro-ph.CO]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  30. [30]

    H¨ utsi, M

    G. H¨ utsi, M. Raidal, V. Vaskonen, and H. Veerm¨ ae, Two populations of LIGO-Virgo black holes, JCAP03, 068, arXiv:2012.02786 [astro-ph.CO]

    work page arXiv 2012

  31. [31]

    A. Hall, A. D. Gow, and C. T. Byrnes, Bayesian analysis of LIGO-Virgo mergers: Primordial vs. as- trophysical black hole populations, Phys. Rev. D102, 123524 (2020), arXiv:2008.13704 [astro-ph.CO]

    work page arXiv 2020

  32. [32]

    Franciolini, V

    G. Franciolini, V. Baibhav, V. De Luca, K. K. Y. Ng, K. W. K. Wong, E. Berti, P. Pani, A. Riotto, and S. Vitale, Searching for a subpopulation of primordial black holes in LIGO-Virgo gravitational-wave data, Phys. Rev. D105, 083526 (2022), arXiv:2105.03349 [gr-qc]

    work page arXiv 2022

  33. [33]

    J. He, H. Deng, Y.-S. Piao, and J. Zhang, Implications of GWTC-3 on primordial black holes from vacuum bubbles, Phys. Rev. D109, 044035 (2024), arXiv:2303.16810 [astro-ph.CO]

    work page arXiv 2024

  34. [34]

    Armanoet al., Sub-Femto- g Free Fall for Space-Based Gravitational Wave Observatories: LISA Pathfinder Results, Phys

    M. Armanoet al., Sub-Femto- g Free Fall for Space-Based Gravitational Wave Observatories: LISA Pathfinder Results, Phys. Rev. Lett.116, 231101 (2016)

    work page 2016

  35. [35]

    Laser Interferometer Space Antenna

    P. Amaro-Seoaneet al.(LISA), Laser Interferometer Space Antenna (2017), arXiv:1702.00786 [astro- ph.IM]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  36. [36]

    P. A. Seoaneet al.(LISA), Astrophysics with the Laser Interferometer Space Antenna, Living Rev. Rel.26, 2 (2023), arXiv:2203.06016 [gr-qc]

    work page arXiv 2023

  37. [37]

    Hu and Y.-L

    W.-R. Hu and Y.-L. Wu, The Taiji Program in Space for gravitational wave physics and the nature of gravity, Natl. Sci. Rev.4, 685 (2017)

    work page 2017

  38. [38]

    Ruan, Z.-K

    W.-H. Ruan, Z.-K. Guo, R.-G. Cai, and Y.-Z. Zhang, Taiji program: Gravitational-wave sources, Int. J. Mod. Phys. A35, 2050075 (2020), arXiv:1807.09495 [gr-qc]

    work page arXiv 2020

  39. [39]

    Wuet al.(Taiji Scientific), China’s first step towards probing the expanding universe and the nature of gravity using a space borne gravitational wave antenna, Commun

    Y.-L. Wuet al.(Taiji Scientific), China’s first step towards probing the expanding universe and the nature of gravity using a space borne gravitational wave antenna, Commun. Phys.4, 34 (2021)

    work page 2021

  40. [40]

    TianQin: a space-borne gravitational wave detector

    J. Luoet al.(TianQin), TianQin: a space-borne gravitational wave detector, Class. Quant. Grav.33, 035010 (2016), arXiv:1512.02076 [astro-ph.IM]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  41. [41]

    Luoet al., The first round result from the TianQin-1 satellite, Class

    J. Luoet al., The first round result from the TianQin-1 satellite, Class. Quant. Grav.37, 185013 (2020), arXiv:2008.09534 [physics.ins-det]

    work page arXiv 2020

  42. [42]

    Meiet al.(TianQin), The TianQin project: current progress on science and technology, PTEP 2021, 05A107 (2021), arXiv:2008.10332 [gr-qc]

    J. Meiet al.(TianQin), The TianQin project: current progress on science and technology, PTEP 2021, 05A107 (2021), arXiv:2008.10332 [gr-qc]

    work page arXiv 2021

  43. [43]

    J. Liu, L. Bian, R.-G. Cai, Z.-K. Guo, and S.-J. Wang, Primordial black hole production during first-order phase transitions, Phys. Rev. D105, L021303 (2022), arXiv:2106.05637 [astro-ph.CO]

    work page arXiv 2022

  44. [44]

    M. B. Hindmarsh, M. L¨ uben, J. Lumma, and M. Pauly, Phase transitions in the early universe, SciPost Phys. Lect. Notes24, 1 (2021), arXiv:2008.09136 [astro-ph.CO]. 19

    work page arXiv 2021

  45. [45]

    Kosowsky, M

    A. Kosowsky, M. S. Turner, and R. Watkins, Gravitational waves from first order cosmological phase transitions, Phys. Rev. Lett.69, 2026 (1992)

    work page 2026

  46. [46]

    Gravitational Radiation from Colliding Vacuum Bubbles: Envelope Approximation to Many-Bubble Collisions

    A. Kosowsky and M. S. Turner, Gravitational radiation from colliding vacuum bubbles: envelope approximation to many bubble collisions, Phys. Rev. D47, 4372 (1993), arXiv:astro-ph/9211004

    work page internal anchor Pith review Pith/arXiv arXiv 1993

  47. [47]

    Kosowsky, M

    A. Kosowsky, M. S. Turner, and R. Watkins, Gravitational radiation from colliding vacuum bubbles, Phys. Rev. D45, 4514 (1992)

    work page 1992

  48. [48]

    H. L. Child and J. T. Giblin, Jr., Gravitational Radiation from First-Order Phase Transitions, JCAP 10, 001, arXiv:1207.6408 [astro-ph.CO]

    work page internal anchor Pith review Pith/arXiv arXiv

  49. [49]

    Gravitational waves from vacuum first-order phase transitions: from the envelope to the lattice

    D. Cutting, M. Hindmarsh, and D. J. Weir, Gravitational waves from vacuum first-order phase tran- sitions: from the envelope to the lattice, Phys. Rev. D97, 123513 (2018), arXiv:1802.05712 [astro- ph.CO]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  50. [50]

    Cutting, E

    D. Cutting, E. G. Escartin, M. Hindmarsh, and D. J. Weir, Gravitational waves from vacuum first order phase transitions II: from thin to thick walls, Phys. Rev. D103, 023531 (2021), arXiv:2005.13537 [astro-ph.CO]

    work page arXiv 2021

  51. [51]

    Y. Li, Y. Jia, and L. Bian, Numerical simulation of domain wall and first-order phase transition in an expanding universe, JCAP02, 038, arXiv:2304.05220 [hep-ph]

    work page arXiv

  52. [52]

    Z. Zhao, Y. Di, L. Bian, and R.-G. Cai, Probing the electroweak symmetry breaking history with gravitational waves, JHEP10, 158, arXiv:2204.04427 [hep-ph]

    work page arXiv

  53. [53]

    Y. Di, J. Wang, R. Zhou, L. Bian, R.-G. Cai, and J. Liu, Magnetic Field and Gravitational Waves from the First-Order Phase Transition, Phys. Rev. Lett.126, 251102 (2021), arXiv:2012.15625 [astro- ph.CO]

    work page arXiv 2021

  54. [54]

    J. T. Giblin, Jr. and J. B. Mertens, Vacuum Bubbles in the Presence of a Relativistic Fluid, JHEP 12, 042, arXiv:1310.2948 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv

  55. [55]

    Numerical simulations of acoustically generated gravitational waves at a first order phase transition

    M. Hindmarsh, S. J. Huber, K. Rummukainen, and D. J. Weir, Numerical simulations of acoustically generated gravitational waves at a first order phase transition, Phys. Rev. D92, 123009 (2015), arXiv:1504.03291 [astro-ph.CO]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  56. [56]

    Hindmarsh, S

    M. Hindmarsh, S. J. Huber, K. Rummukainen, and D. J. Weir, Shape of the acoustic gravitational wave power spectrum from a first order phase transition, Phys. Rev. D96, 103520 (2017), [Erratum: Phys.Rev.D 101, 089902 (2020)], arXiv:1704.05871 [astro-ph.CO]

    work page arXiv 2017

  57. [57]

    Gravitational waves from the sound of a first order phase transition

    M. Hindmarsh, S. J. Huber, K. Rummukainen, and D. J. Weir, Gravitational waves from the sound of a first order phase transition, Phys. Rev. Lett.112, 041301 (2014), arXiv:1304.2433 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  58. [58]

    Cutting, M

    D. Cutting, M. Hindmarsh, and D. J. Weir, Vorticity, kinetic energy, and suppressed gravita- tional wave production in strong first order phase transitions, Phys. Rev. Lett.125, 021302 (2020), arXiv:1906.00480 [hep-ph]

    work page arXiv 2020

  59. [59]

    Cutting, E

    D. Cutting, E. Vilhonen, and D. J. Weir, Droplet collapse during strongly supercooled transitions, Physical Review D106, 10.1103/physrevd.106.103524 (2022). 20

    work page doi:10.1103/physrevd.106.103524 2022

  60. [60]

    J. T. Giblin and J. B. Mertens, Gravitional radiation from first-order phase transitions in the presence of a fluid, Phys. Rev. D90, 023532 (2014), arXiv:1405.4005 [astro-ph.CO]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  61. [61]

    Gravitational Radiation From Cosmological Turbulence

    A. Kosowsky, A. Mack, and T. Kahniashvili, Gravitational radiation from cosmological turbulence, Phys. Rev. D66, 024030 (2002), arXiv:astro-ph/0111483

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  62. [62]

    Gravitational Radiation from First-Order Phase Transitions

    M. Kamionkowski, A. Kosowsky, and M. S. Turner, Gravitational radiation from first order phase transitions, Phys. Rev. D49, 2837 (1994), arXiv:astro-ph/9310044

    work page internal anchor Pith review Pith/arXiv arXiv 1994

  63. [63]

    Neronov, A

    A. Neronov, A. Roper Pol, C. Caprini, and D. Semikoz, NANOGrav signal from magnetohydrodynamic turbulence at the QCD phase transition in the early Universe, Phys. Rev. D103, 041302 (2021), arXiv:2009.14174 [astro-ph.CO]

    work page arXiv 2021

  64. [64]

    Roper Pol, S

    A. Roper Pol, S. Mandal, A. Brandenburg, T. Kahniashvili, and A. Kosowsky, Numerical simu- lations of gravitational waves from early-universe turbulence, Phys. Rev. D102, 083512 (2020), arXiv:1903.08585 [astro-ph.CO]

    work page arXiv 2020

  65. [65]

    The stochastic gravitational wave background from turbulence and magnetic fields generated by a first-order phase transition

    C. Caprini, R. Durrer, and G. Servant, The stochastic gravitational wave background from turbulence and magnetic fields generated by a first-order phase transition, JCAP12, 024, arXiv:0909.0622 [astro- ph.CO]

    work page internal anchor Pith review Pith/arXiv arXiv

  66. [66]

    Gravitational Radiation from Primordial Helical Inverse Cascade MHD Turbulence

    T. Kahniashvili, L. Campanelli, G. Gogoberidze, Y. Maravin, and B. Ratra, Gravitational Radiation from Primordial Helical Inverse Cascade MHD Turbulence, Phys. Rev. D78, 123006 (2008), [Erratum: Phys.Rev.D 79, 109901 (2009)], arXiv:0809.1899 [astro-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  67. [67]

    Gravitational Radiation Generated by Cosmological Phase Transition Magnetic Fields

    T. Kahniashvili, L. Kisslinger, and T. Stevens, Gravitational Radiation Generated by Magnetic Fields in Cosmological Phase Transitions, Phys. Rev. D81, 023004 (2010), arXiv:0905.0643 [astro-ph.CO]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  68. [68]

    Science with the space-based interferometer eLISA. II: Gravitational waves from cosmological phase transitions

    C. Capriniet al., Science with the space-based interferometer eLISA. II: Gravitational waves from cosmological phase transitions, JCAP04, 001, arXiv:1512.06239 [astro-ph.CO]

    work page internal anchor Pith review Pith/arXiv arXiv

  69. [69]

    Wansleben, Y

    M. Wansleben, Y. Kayser, P. H¨ onicke, I. Holfelder, A. W¨ ahlisch, R. Unterumsberger, and B. Beckhoff, Experimental determination of line energies, line widths and relative transition probabilities of the gadolinium l x-ray emission spectrum, Metrologia56, 065007 (2019)

    work page 2019

  70. [70]

    Athron, C

    P. Athron, C. Bal´ azs, A. Fowlie, L. Morris, and L. Wu, Cosmological phase transitions: From perturbative particle physics to gravitational waves, Prog. Part. Nucl. Phys.135, 104094 (2024), arXiv:2305.02357 [hep-ph]

    work page arXiv 2024

  71. [71]

    Caldwellet al., Detection of early-universe gravitational-wave signatures and fundamental physics, Gen

    R. Caldwellet al., Detection of early-universe gravitational-wave signatures and fundamental physics, Gen. Rel. Grav.54, 156 (2022), arXiv:2203.07972 [gr-qc]

    work page arXiv 2022

  72. [72]

    Bianet al., The Gravitational-wave physics II: Progress, Sci

    L. Bianet al., The Gravitational-wave physics II: Progress, Sci. China Phys. Mech. Astron.64, 120401 (2021), arXiv:2106.10235 [gr-qc]

    work page arXiv 2021

  73. [73]

    S. R. Coleman, The Fate of the False Vacuum. 1. Semiclassical Theory, Phys. Rev. D15, 2929 (1977), [Erratum: Phys.Rev.D 16, 1248 (1977)]

    work page 1977

  74. [74]

    Enqvist, J

    K. Enqvist, J. Ignatius, K. Kajantie, and K. Rummukainen, Nucleation and bubble growth in a first order cosmological electroweak phase transition, Phys. Rev. D45, 3415 (1992). 21

    work page 1992

  75. [75]

    Adshead, J

    P. Adshead, J. T. Giblin, M. Pieroni, and Z. J. Weiner, Constraining axion inflation with gravitational waves from preheating, Phys. Rev. D101, 083534 (2020), arXiv:1909.12842 [astro-ph.CO]

    work page arXiv 2020

  76. [76]

    Computations of primordial black hole formation

    I. Musco, J. C. Miller, and L. Rezzolla, Computations of primordial black hole formation, Class. Quant. Grav.22, 1405 (2005), arXiv:gr-qc/0412063

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  77. [77]

    Threshold of primordial black hole formation

    T. Harada, C.-M. Yoo, and K. Kohri, Threshold of primordial black hole formation, Phys. Rev. D88, 084051 (2013), [Erratum: Phys.Rev.D 89, 029903 (2014)], arXiv:1309.4201 [astro-ph.CO]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  78. [78]

    Musco, Threshold for primordial black holes: Dependence on the shape of the cosmological pertur- bations, Phys

    I. Musco, Threshold for primordial black holes: Dependence on the shape of the cosmological pertur- bations, Phys. Rev. D100, 123524 (2019), arXiv:1809.02127 [gr-qc]

    work page arXiv 2019

  79. [79]

    Escriv` a, C

    A. Escriv` a, C. Germani, and R. K. Sheth, Universal threshold for primordial black hole formation, Phys. Rev. D101, 044022 (2020), arXiv:1907.13311 [gr-qc]

    work page arXiv 2020

  80. [80]

    Primordial Black Hole Formation during First-Order Phase Transitions

    K. Jedamzik and J. C. Niemeyer, Primordial black hole formation during first order phase transitions, Phys. Rev. D59, 124014 (1999), arXiv:astro-ph/9901293

    work page internal anchor Pith review Pith/arXiv arXiv 1999

Showing first 80 references.