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Gravitational Waves from Electroweak Phase Transitions

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Riccardo Apreda, Michele Maggiore, Alberto Nicolis, Antonio Riotto

General Relativity and Quantum Cosmology · Submitted 10 Jul 2001 · 33 references

Pith's verdict: Honest, careful computation with a solid MSSM null result; the NMSSM 10^-10 headline has an internal tension between amplitude and peak frequency that should be checked before anyone treats it as a LISA target.

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arXiv IDgr-qc/0107033 (version 1)
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CitationAlberto Nicolis, Antonio Riotto, Michele Maggiore, Riccardo Apreda. "Gravitational Waves from Electroweak Phase Transitions." arXiv:gr-qc/0107033 (2001). https://pith.science/paper/WDYJAG5G
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The paper

The paper claims

In the Next-to-Minimal Supersymmetric Standard Model, a first-order electroweak phase transition can generate a stochastic gravitational wave background at the 10^-10 to 10^-9 level, within LISA's reach, while the MSSM falls short by five orders of magnitude.

The reading

This paper asks whether a first-order phase transition at the electroweak scale (around 100 GeV, about 10^-11 seconds after the Big Bang) would have produced gravitational waves strong enough for the planned LISA space interferometer to detect. The Standard Model does not have such a transition for the measured Higgs mass, so the authors turn to supersymmetric extensions. They compute the transition in two models. In the Minimal Supersymmetric Standard Model (MSSM), the transition is driven by radiative corrections and is weak; the predicted signal is around 10^-16 to 10^-18, far below LISA's sensitivity. In the Next-to-Minimal Supersymmetric Standard Model (NMSSM), an extra singlet field makes the transition strongly first order already at tree level. There, in some corners of parameter space, the bubble-collision signal can reach 10^-10 and the turbulence signal 10^-9, both within LISA's reach at a peak frequency near 10 mHz. The calculation uses the thermal effective potential, computes the bubble nucleation rate, and feeds the resulting transition parameters into published formulas for gravitational wave production. The authors are explicit that the MSSM numbers are upper bounds because non-perturbative effects are expected to weaken the transition, and that the turbulence estimate is indicative because the underlying formula was derived for slow fluid velocities.
Extended reading notes

Core claim

In the NMSSM, a strongly first-order electroweak phase transition can produce a stochastic gravitational wave background with h0^2 Omega_gw up to about 10^-10 from bubble collisions and 10^-9 from turbulence, peaked near 10 mHz, within LISA's planned sensitivity, in parameter regions compatible with electroweak baryogenesis.

Load-bearing premise

The 10^-10 bubble-collision figure is obtained in the 'extremal' region where beta/H* approaches 1, and the paper itself states (section 3.2, discussion around fig. 20) that the production formula (4) is not valid for beta/H* below 1 because the transition would not complete in a Hubble time. The headline claim therefore rests on the computation being reliable at the boundary of its stated validity, where the transition is slow and the perturbative effective-potential treatment is least controlled.

Original abstract

Gravitational waves are generated during first-order phase transitions, either by turbolence or by bubble collisions. If the transition takes place at temperatures of the order of the electroweak scale, the frequency of these gravitational waves is today just within the band of the planned space interferometer LISA. We present a detailed analysis of the production of gravitational waves during an electroweak phase transition in different supersymmetric models where, contrary to the case of the Standard Model, the transition can be first order. We find that the stochastic background of gravitational waves generated by bubble nucleation can reach a maximum value h0^2 Omega_{gw} of order 10^{-10} - 10^{-11}, which is within the reach of the planned sensitivity of LISA, while turbolence can even produce signals at the level h0^2 Omega_{gw} \sim 10^{-9}. These values of h0^2 Omega_{gw} are obtained in the regions of the parameter space which can account for the generation of the baryon asymmetry at the electroweak scale.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,500 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.16 · human participants 0.01 · controlled comparison 0.08 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.34 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

Gravitational waves are generated during first-order phase transitions, either by turbolence or by bubble collisions. If the transition takes place at temperatures of the order of the electroweak scale, the frequency of these gravitational waves is today just within the band of the planned space interferometer LISA. We present a detailed analysis of the production of gravitational waves during an electroweak phase transition in different supersymmetric models where, contrary to the case of the Standard Model, the transition can be first order. We find that the stochastic background of gravitational waves generated by bubble nucleation can reach a maximum value h0^2 Omega_{gw} of order 10^{-10} - 10^{-11}, which is within the reach of the planned sensitivity of LISA, while turbolence can even produce signals at the level h0^2 Omega_{gw} \sim 10^{-9}. These values of h0^2 Omega_{gw} are obtained in the regions of the parameter space which can account for the generation of the baryon asymmetry at the electroweak scale.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Gravitational waves are generated during first-order phase transitions, either by turbolence or by bubble collisions. background 1.00 0.05 0.03 asserted 1.00
s1 If the transition takes place at temperatures of the order of the electroweak scale, the frequency of these gravitational waves is today just within the band of the planned space interferometer LISA. background 0.92 0.11 0.07 asserted 0.94
s2 We present a detailed analysis of the production of gravitational waves during an electroweak phase transition in different supersymmetric models where, contrary to the case of the Standard Model, the transition can be first order. contribution 1.00 0.89 0.07 asserted 0.97
s3 We find that the stochastic background of gravitational waves generated by bubble nucleation can reach a maximum value h0^2 Omega_{gw} of order 10^{-10} - 10^{-11}, which is within the reach of the planned sensitivity of LISA, while turbolence can even produce signals at the level h0^2 Omega_{gw} \sim 10^{-9}. result 1.00 0.96 0.71 asserted 0.98
s4 These values of h0^2 Omega_{gw} are obtained in the regions of the parameter space which can account for the generation of the baryon asymmetry at the electroweak scale. interpretation 0.70 0.81 0.13 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.85 · evidence from simulation or computation 0.71.

What was examined, and what was not

sources read
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not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. Typical temperature-dependence of the potential
Figure 2. Typical profile of a bubble solution φb(r) interpolating from the true vacuum phase at r = 0 to the false vacuum at r = ∞. The solution shown corresponds to the critical bubble at the transition temperature in the MSSM for sin2 βMSSM = 0.8, mHiggs = 110 GeV and mstop = 140 GeV, see section II. supplemented by the boundary conditions dφb dr
Figure 3. Overshooting-undershooting method. 9
Figure 4. The plot shows the rapid variation experienced by the actio
Figure 5. Contour plot of α as a function of the Higgs and stop masses for sin2 β = 0.8. The shaded region is forbidden by the request of absence of color breaking minima. α is computed at the destabilization temperature, as discussed in the text. 14
Figure 6. Contour plot of α as a function of the Higgs and stop masses for sin2 β = 0.3 . stop. We see also that the strength of the transition is very slightly dependent on sin2 β; thus for definiteness we fix sin2 β = 0.8. The most important two loop corrections are of the form φ 2 log(φ) and are induced by the Standard Model weak gauge bosons as well as by the stop and gluon loops [28, 31] V2(φ, T) ≃ φ 2T 2 32π 2   51 16 g 2 − 3h 4 t sin4 β
Figure 7. Plot for α (dashed line), β/H∗ (dotted line), and h 2 0Ωgw from bubble collisions (solid line), as functions of the stop mass, for a 110 GeV Higgs and sin2 β = 0.8. The intensity of the produced gravity waves rapidly grows as the masses decrease. This is explained by the fact that h 2 0Ωgw scales roughly as α 3 /β2 and a linear increase of α and decrease of β leads to a rapid power-law growth of the intensity. Unfortunately, the general prediction is that the intensity of the gravitational waves produced during the MSSM phase transition is too small for LISA. For instance, taking
Figure 8. Plot for α (dashed line), β/H∗ (dotted line), and h 2 0Ωgw from bubble collisions (solid line), as functions of the Higgs mass, for a 155 GeV stop and sin2 β = 0.8. We have also estimated what happens if we lower the Higgs mass down to (the already excluded value of) 80 GeV, setting the right-handed stop mass at 155 GeV – which is the lower value compatible with the absence of color breaking minima – and sin2 β = 0.8, see fig. 8. We obtain α ≃ 0.1 and β/H∗≃ 2 × 103 , giving h 2 0Ωgw ≃ 2 × 10−16, a signal still not relevant. The situation does not improve when both Higgses are invol
Figure 9. h 2 0Ωgw from turbolence, as a function of the stop mass for a 110 GeV Higgs and sin2 β = 0.8. 90 110 130 150 -18 -20 -22 -24 2 0 Ω h gw Log [ ] mHiggs (GeV)
Figure 10. h 2 0Ωgw from turbolence, as a function of the Higgs mass for a 155 GeV stop and sin2 β = 0.8. 3 The Next-to-Minimal Supersymmetric Standard Model 18
Figure 11. Rough contour plot for α as a function of Aλ and Ak, for x = 350 GeV, λ = 0.83, k = 0.67 and tan β = 2 . T∗ is evaluated, one easily computes α from its definition (18). A result of this procedure is summarized in fig. 11, which shows a rapid growth of α versus Aλ and Ak, with x = 350 GeV, λ = 0.83, k = 0.67 and tan β = 2. Next we moved to computing exactly T∗, α and β/H∗ for specific values of Aλ and Ak in the region of fig. 11, again by numerically computing the temperature for which the bubble solution φb(r) gives S3(T)/T ≃ 140 and then applying relations (18) and (10). Typical…
Figure 12. Plot for α (dashed line), β/H∗ (dotted line), and h 2 0Ωgw (solid line) from bubble collisions, as functions of Aλ for Ak = 480 GeV, x = 350 GeV, λ = 0.83, k = 0.67 and tan β = 2. 410 420 430 440 -13 -11 -9 2 0 Ω h gw Log [ ] A λ (GeV)
Figure 13. h 2 0Ωgw from turbolence as a function of Aλ for Ak = 480 GeV, x = 350 GeV, λ = 0.83, k = 0.67 and tan β = 2. 23
Figure 14. Plot for α (dashed line), β/H∗ (dotted line), and h 2 0Ωgw (solid line) from bubble collisions, as functions of Ak for Aλ = 450 GeV, x = 350 GeV, λ = 0.83, k = 0.67 and tan β = 2. 445 455 465 475 -16 -14 -12 -10 2 0 Ω h gw Log [ ] A (GeV) k
Figure 15. h 2 0Ωgw from turbolence as a function of Ak for Aλ = 450 GeV, x = 350 GeV, λ = 0.83, k = 0.67 and tan β = 2. 24
Figure 16. Plot for α (dashed line), β/H∗ (dotted line), and h 2 0Ωgw from bubble collisions (solid line), as functions of Aλ for Ak = 167 GeV, x = 300 GeV, λ = 0.523, k = 0.37 and tan β = 2. 240 260 280 300 -17 -15 -13 -11 -9 A λ (GeV) 2 0 Ω h gw Log [ ]
Figure 17. h 2 0Ωgw from turbolence as a function of Aλ for Ak = 167 GeV, x = 300 GeV, λ = 0.523, k = 0.37 and tan β = 2. 25
Figure 18. Plot for α (dashed line), β/H∗ (dotted line), and h 2 0Ωgw from bubble collisions, (solid line) as functions of Ak for Aλ = 450 GeV, x = 350 GeV, λ = 0.83, k = 0.67 and tan β = 2. 340 350 360 370 380 390 -16 -15 -13 -14 A λ (GeV) 2 0 Ω h gw Log [ ]
Figure 19. h 2 0Ωgw from turbolence as a function of Ak for Aλ = 450 GeV, x = 350 GeV, λ = 0.83, k = 0.67 and tan β = 2. 26
Figure 20. Behaviour of the action S3/T versus T in an “extremal” region of the parameters-space. The horizontal dashed line indicates the value S3/T ≃ 140 that must be reached in order that the transition takes place. In figs. 16 and 14 the intensity of the produced gravity waves grows rapidly as a function of the soft breaking mass terms. However, Aλ and Ak cannot be extended beyond the values shown in the figures: beyond such values the condition S3(T∗)/T∗ ≃ 140 is never satisfied and the transition does not take place. This is due to the presence of a potential barrier also at zero…

Figures reproduced from arXiv:gr-qc/0107033 by the authors.

References
How later papers used it

107 later papers cite it

Evidence: substantial. 107 of the citing papers share no author with it, and 12 of those carry a confident label from Pith's citation classifier.

Pith read 24 of the 138 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

Uses its method or tool 3 of 9 papers with this label

  • Used this paper's formula for the time scale, taking the slope of a linear fit to S_3(T)/T around the nucleation temperature.

    Gravitational waves from conformal symmetry breaking (2018) · 1.00 · paraphrased: the licence does not allow quoting

  • Used an undershooting-overshooting algorithm, citing this paper, to solve the equation of motion for the canonical field with respect to the radial coordinate.

    Hysteresis of Axionic Charge Density Waves (2024) · 1.00 · paraphrased: the licence does not allow quoting

  • Following this paper, set the parameter to zero for simplicity in the numerical calculation.

    Velocity of electroweak bubble walls (2009) · 0.83 · paraphrased: the licence does not allow quoting

How the literature uses it

102 arXiv papers cite this in 130 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 72 sentences without a confident label

Contests it 1 paper
  • However, the authors of Ref. [ref] used a crude method to determine the bubble configurations, which overestimates the strength of the phase transition and the GW signal.

    Production of Gravitational Waves in the nMSSM · Introduction · contest 0.92

Uses its method or tool 8 papers
Uses its data or values 1 paper
Compares with it 1 paper
Cites it as background 41 papers

Discussion

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