Pith. sign in

REVIEW 3 major objections 3 minor 4 cited by

Beyond the Daisy Chain: Running and the 3D EFT View of Supercooled Phase Transitions

T0 review · 3 major / 3 minor · reviewed 2026-08-04 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Consistent RG running at a tuned scale makes the simple one-loop 4D high-temperature potential agree with the two-loop 3D effective theory for supercooled phase transitions, while the one-parameter shortcut fails.

desk verdict Useful, practical cross-check of 4D vs 3D potentials for supercooled dark transitions, but the reduced-scale-dependence headline rests on a one-dimensional 3D scale scan and a benchmark with an admitted power-counting caveat. read the letter →

arxiv 2511.02910 v3 pith:K5KU3O4O submitted 2025-11-04 hep-ph

Martin Christiansen , Eric Madge , Cristina Puchades-Ibáñez , Maura E. Ramirez-Quezada , Pedro Schwaller This is my paper · ORCID
classification hep-ph
keywords supercooledphasetransitionseffectivepotentialDaisyresummationdimensionalreductionrenormalisationgrouprunningbounceactiongravitationalwavesconformalAbelianHiggsmodel
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper aims to settle how much theoretical machinery is needed to predict gravitational waves from a supercooled first-order phase transition in a classically scale-invariant dark Abelian Higgs sector. Its central claim is that a one-loop four-dimensional effective potential with high-temperature expansion and Daisy resummation, supplemented by consistent RG running of the couplings and evaluated at the renormalisation scale µ=πT, reproduces the transition parameters and gravitational-wave spectrum obtained from the more elaborate two-loop dimensionally reduced 3D effective theory. The paper also finds that the simple one-parameter approximation, when used with fixed couplings and no running, deviates strongly in the large-supercooling regime, and that the 3D EFT has markedly smaller residual renormalisation-scale dependence. The practical payoff is that a cheap and commonly used calculation can be trusted for these models once running is included, which matters for interpreting the nanohertz gravitational-wave background.

What carries the argument

The load-bearing object is the finite-temperature effective potential entering the tunnelling action S3/T, the exponent that controls bubble nucleation. The paper compares three constructions: a one-loop 4D high-temperature potential with Daisy resummation, which adds thermal screening masses to the static bosonic modes; a one-parameter polynomial approximation to that potential; and a 3D effective theory obtained by dimensional reduction, in which heavy non-static modes are integrated out and the soft potential is computed to two loops. Running of the couplings via the renormalisation-group equations connects the schemes: evaluating the 4D potential at µ=πT and matching the 3D theory at 2πT

What would settle it

Compute the tunnelling action S3/T for the same conformal dark U(1) model directly in the 3D theory with lattice Monte Carlo at T≈0.01v and gauge coupling near the strong end of the benchmark range; if the lattice result differs from the two-loop dimensionally reduced curve by more than the residual-scale bands shown in the paper, then the 4D high-temperature agreement is calibrated to a biased benchmark.

Watch

Extended reading notes

Core claim

Central claim: once RG running is included and the scale chosen deliberately, the standard 4D one-loop high-temperature potential with Daisy resummation closes the gap to the two-loop dimensionally reduced 3D effective theory. For a conformal dark U(1) scalar-gauge model, the authors compute the tunnelling action S3/T and extract T_n, α, β/H and the gravitational-wave spectrum. The 4D potential at µ=πT reproduces the 3D two-loop benchmark; the one-parameter approximation without running deviates sharply at large supercooling. The 3D theory has much smaller residual scale dependence; NLO inputs on a one-loop 3D potential can worsen agreement.

Load-bearing premise

The load-bearing premise is that the two-loop dimensionally reduced 3D effective theory is an accurate benchmark in the supercooled regime, even though the paper itself notes that its power counting assumes λ∼g^2 and is not valid where λ≲g^4; if that benchmark carries uncontrolled truncation errors, the calibrated scale µ=πT and the claimed 4D–3D consistency lose their anchor.

Editorial extensions

If this is right

  • For a classically scale-invariant dark Abelian Higgs sector, the one-loop 4D high-temperature potential with Daisy resummation and RG running at µ=πT is a reliable, cheap substitute for the two-loop 3D EFT when predicting T_n, α, β/H and gravitational-wave spectra.
  • Neglecting RG running, as in the one-parameter approximation used here, shifts T_n and β/H substantially at small gauge coupling, produces an unphysical peak in β/H, and gives a gravitational-wave spectrum that lies away from the RG-improved predictions.
  • The two-loop dimensionally reduced EFT is the most scale-stable scheme; the residual µ-dependence visible in its potential and spectra is much narrower than in the 4D high-temperature scheme.
  • Including NLO masses and couplings in a one-loop 3D potential can increase scale dependence; partial higher-order corrections do not automatically improve predictions.
  • The authors caution that the calibrated agreement is not automatically valid for other models or the non-conformal regime.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A testable extension: repeat the same scale-calibration exercise in a non-Abelian or multi-field conformal model; if µ=πT remains the agreement point, the recipe becomes a general shortcut for scanning supercooled transitions.
  • Because the 3D two-loop benchmark itself assumes λ∼g^2, which the paper notes is violated when λ≲g^4, the agreement between 4D and 3D could partly reflect shared truncation errors; a lattice computation of the 3D theory would expose this.
  • The close agreement between 4D and 3D at low temperature suggests the thermal barrier is controlled by the same resummed physics in both schemes; if confirmed, simpler 4D codes can be used for pulsar-timing-array interpretation without two-loop machinery.
  • The paper's OPA comparison is deliberately unfavourable (no running, no Daisy terms); since the appendix shows OPA with running approaches the 4D high-temperature result, the substantive lesson is not that the parameterisation is inherently bad but that running is essential.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 3 minor

Summary. The paper studies supercooled first-order phase transitions in a classically scale-invariant dark Abelian Higgs model, comparing three theoretical treatments of the finite-temperature effective potential: the one-parameter approximation (OPA), the 4D high-temperature one-loop potential with Daisy resummation, and a dimensionally reduced 3D effective field theory computed with DRalgo at one and two loops. The authors argue that consistent RG running is essential, that the OPA without running deviates significantly, that the 3D two-loop EFT has substantially reduced renormalisation-scale dependence, and that, once the 4D HT scale is chosen as µ=πT, the 4D HT result agrees with the 3D two-loop benchmark. The bounce action is computed semi-analytically and validated against shooting, and the phase-transition parameters (Tn, α, β/H) are propagated to gravitational-wave spectra and compared with NANOGrav 15-year data.

Significance. If the conclusions hold, the paper provides a practically useful simplification: a one-loop 4D Daisy-resummed potential with RG running and µ=πT may be sufficient for supercooled conformal dark sectors, and the more expensive 3D two-loop DR calculation can be reserved as a benchmark. The use of DRalgo makes the 3D matching a parameter-free application of a public automated framework, and the explicit comparison of four 3D schemes is a useful systematic study. The paper is also candid about several limitations, including the validity of the NLO power counting in the deep supercooled regime. However, the central claim of reduced scale dependence in the 3D EFT is not fully established because only one of the two independent scales of the 3D EFT is varied, and the agreement between 4D HT and 3D is partly a calibration rather than an independent prediction.

major comments (3)
  1. [§II.c, Eq. (A16), Fig. 2 (right), Fig. 6 (right)] The reduced scale dependence of the 3D EFT is demonstrated by varying only the matching scale µ_Match around 2πT, while the soft scale µ3 is fixed at gT (Table I). However, Eq. (A16) explicitly depends on log(µ3/µ), and the 3D loop potential itself depends on µ3. Thus the purple bands in Figs. 2 and 6 cover only one of the two independent renormalisation scales of the 3D EFT. The conclusions even state that the 3D 1-L (NLO) scheme 'introduces additional scale dependence through the matching of the soft scale µ3', but this dependence is never quantified. To support the headline claim, the authors should repeat the variation with µ3 varied (e.g., µ3 ∈ [gT/4, 4gT]) and propagate this to S3/T, Tn, β/H and the GW spectrum. If the 3D band widens substantially, the conclusion should be moderated to 'reduced matching-scale dependence' rather than 'reduced scale dependence'.
  2. [§II.c] The paper itself notes that the NLO power counting assumes λ∼g^2 and therefore 'is, strictly speaking, not valid in the supercooled regime where λ≲g^4'. The benchmark model has λ(µ0)=0 and negative running, so the 3D two-loop result is used precisely in the regime where its systematic expansion may break down. Since the 4D HT scale µ=πT is then calibrated against this same 3D benchmark, any uncontrolled truncation error in the 3D potential propagates directly into the central consistency claim. A quantitative estimate of this truncation uncertainty — for example by studying the size of the available higher-order terms, by comparing LO/NLO/mixed schemes more carefully, or by varying the power-counting assumptions — is needed before the 3D result can serve as the benchmark that anchors the µ=πT choice.
  3. [II (last paragraph), III.A] The choice µ=πT is not an independent prediction but is fixed by requiring the 4D HT bounce action to agree with the 3D two-loop result: 'we take the 3D two-loop NLO potential as a benchmark and fix the 4D HT renormalisation scale by requiring the good agreement with the bounce action'. Therefore the agreement between 4D HT and 3D two-loop is partly by construction. The abstract's phrase 'with a suitable choice of RGE scale' is accurate, but the stronger statement that the 4D HT calculation 'yields results consistent with the 3D calculations' should be framed as a calibration statement. An independent validation of µ=πT — for example by a principal-of-minimal-sensitivity condition on the 4D potential itself, or by showing that the calibrated scale remains stable when µ3 is varied — would substantially strengthen the argument.
minor comments (3)
  1. [Fig. 2 and Fig. 6 captions] The captions should explicitly state that for the 3D bands the soft scale µ3 is held fixed at gT and only µ_Match is varied. Currently this information is only in Table I, and the caption wording 'varied analogously' could mislead readers into thinking the full scale variation of the 3D EFT is shown.
  2. [Appendix B, Eq. (B1)] It would be useful to state explicitly that the beta functions and anomalous dimensions are at one-loop order, and to clarify that the RGEs are solved in the 4D MS scheme with the same reference scale µ0 used for the input parameters. This would make the matching between the 4D running and the 3D matching scale µ_Match easier to follow.
  3. [III.B, Fig. 5 (right)] The text says that at larger couplings 'both approximations show comparably sized deviations relative to the two-loop curve', but the figure is on a logarithmic axis and no numerical deviation is quoted. A quantitative statement, even for one benchmark point, would make the comparison more informative.

Circularity Check

1 steps flagged · score 5.0 of 10

4D-HT 'consistency' with the 3D benchmark is obtained by first tuning the 4D scale to match the 3D bounce action; the core 3D scale-reduction result is independent.

  1. fitted input called prediction [Section II, discussion of Fig. 2 (right), before Sec. III; used throughout Secs. III.A-C]
    "Concretely, we take the 3D two-loop NLO potential as a benchmark and fix the 4D HT renormalisation scale by requiring the good agreement with the bounce action, which is presented in the following section. This criterion sets the renormalisation scale to µ=πT in our 4D HT calculations; we use this as our reference hereafter."

    All the phase-transition observables (T_n, α, β/H) and the GW spectra are derived from S3(T)/T (Sec. III). The paper first tunes the single free scale µ of the 4D HT scheme so that its bounce action matches the 3D two-loop benchmark, and then presents agreement with the 3D results in Figs. 3-6 as evidence. This is a one-parameter calibration, not an independent confirmation: the 4D-HT 'consistency' highlighted in the abstract is partly enforced by construction. The disclosure mitigates the issue, and the separate 3D scale-reduction claim retains independent content, so the circularity is partial rather than total.

full rationale

The derivation chain is mostly non-circular. The 3D DR calculation is a parameter-free application of the external DRalgo matching to the stated Abelian Higgs model, and the reduced residual scale dependence of the 3D two-loop scheme is an independent technical result. The OPA comparison is a controlled benchmark, and no load-bearing self-citation appears: DRalgo [110] and Ref. [118] are by other author groups, while the present authors' own Ref. [72] is only used for phenomenological context. The main circular element is the 4D-HT vs 3D consistency claim: Section II explicitly fixes µ=πT by requiring the 4D HT bounce action to agree with the 3D two-loop benchmark, and since S3/T is the single input from which T_n, α, β/H, and the GW spectra are derived, the subsequent 'excellent agreement' of 4D HT with 3D is a calibrated result rather than a blind prediction. This is disclosed in the text, and the fit is only a single global scale, so the 3D scale-reduction claim does not rest on it. I also note two non-circular validity concerns that the reviewing rules ask to be flagged: the paper itself states that the DR power-counting 'assumes λ~g^2 and therefore is, strictly speaking, not valid in the supercooled regime where λ≲g^4', and the displayed 3D scale-dependence band varies only µ_Match around 2πT while keeping the soft scale µ3 fixed, rather than exploring the full two-scale dependence. These weaken the strength of the benchmark comparison but are not definitional circularity, so they do not raise the score beyond the calibration issue.

Assumptions & free parameters 3 free parameters · 7 assumptions · 0 invented entities

The central comparison rests on the 3D DR matching being a trustworthy benchmark, on one-loop RG running being sufficient, on the perturbative expansion surviving the supercooled regime despite the power-counting caveat, and on the no-running OPA being a fair illustration of the running failure. No new particles or forces are introduced; the model is the standard conformal Abelian Higgs dark sector.

free parameters (3)
  • g(µ0), the dark gauge coupling at reference scale = scanned over 0.5–1; key plots at 0.5, 0.6, 0.7, 0.8
    Chosen by hand as benchmark inputs; the phase-transition parameters and GW spectra depend on this coupling, but it is not fitted to external data.
  • µ = πT, the 4D HT renormalisation scale = πT
    Fixed in Section II by requiring the 4D HT bounce action to agree with the 3D two-loop NLO benchmark, so part of the reported agreement is calibrated rather than derived.
  • Conformal boundary conditions λ(µ0)=0, m²(µ0)=0, µ0=1 GeV = λ=0, m²=0, µ0=1 GeV
    Hand-chosen initial conditions defining the classically scale-invariant benchmark model; not fitted to data.
assumptions (7)
  • domain assumption The 3D dimensionally reduced EFT correctly captures the thermal IR physics and remains applicable in the supercooled regime despite λ≲g^4 violating the formal power-counting λ∼g^2.
    Section II, where the scheme list notes the power-counting 'strictly speaking, not valid in the supercooled regime'; the paper relies on the 3D two-loop result as its accuracy benchmark.
  • domain assumption One-loop RG running (Appendix B) is sufficient to resum the large logarithms and remove the dominant µ-dependence of the phase-transition parameters.
    The paper uses one-loop β-functions and anomalous dimensions throughout and does not quantify the effect of two-loop running on the conclusions.
  • domain assumption The thermal barrier region is correctly described by the high-temperature/Daisy or 3D EFT potentials, while the broken-phase minimum is correctly described by the zero-temperature Coleman-Weinberg potential, with the two regimes stitched together.
    Section II and Ref. [118] motivate this split; the paper assumes it when extracting T_n and α.
  • standard math The Espinosa semi-analytic method accurately approximates the exact bounce action in the parameter range of interest.
    Section III A; validated against a direct shooting method, but the validation is only described qualitatively, not quantified.
  • domain assumption The nucleation rate prefactor can be neglected at the precision required, and T*=T_n with percolation effects neglected.
    Equation (4) and discussion in Section III B; the authors explicitly leave a more systematic fluctuation-determinant treatment to future work.
  • domain assumption The bulk-flow (relativistic) model with wall velocity v_w=1 is adequate for the GW spectrum in strongly supercooled transitions.
    Section III C; smaller wall velocities are said not to change the qualitative conclusions, but no quantitative check is shown.
  • domain assumption The Landau gauge choice ξ=0 and DR-based gauge control leave the extracted phase-transition parameters unaffected at the quoted precision.
    Used for the 4D CW potential and Daisy term; the paper cites gauge-dependence studies but does not compute a gauge-bracket error.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Beyond the Daisy Chain: Running and the 3D EFT View of Supercooled Phase Transitions." pith.science (2026). https://pith.science/paper/K5KU3O4O

@misc{pith2026251102910,
  author       = {Pith},
  title        = {Pith review of: Beyond the Daisy Chain: Running and the 3D EFT View of Supercooled Phase Transitions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/K5KU3O4O}},
  note         = {Machine review of arXiv:2511.02910}
}
read the original abstract

Pulsar timing arrays have recently observed a stochastic gravitational wave background at nano-Hertz frequencies. This raises the question whether the signal can be of primordial origin. Supercooled first-order phase transitions are among the few early Universe scenarios that can successfully explain it. To further scrutinise this possibility, a precise theoretical understanding of the dynamics of the phase transition is required. Here we perform such an analysis for a dark sector with an Abelian Higgs model in the conformal limit, which is known to admit large supercooling. We compare simple analytic parametrisations of the bounce action, one-loop finite temperature calculations including Daisy resummation, and results of a dimensionally reduced (3D) effective theory including up to two-loop corrections using the DRalgo framework. Consistent renormalisation group evolution (RGE) of the couplings is essential for a meaningful interpretation of the results. We find that the 3D EFT with consistent expansion in the 4D parameters gives a significantly reduced scale dependence of the phase transition parameters. With a suitable choice of RGE scale, the 4D high temperature expanded effective potential yields results consistent with the 3D calculations, while the analytic parametrisation deviates significantly in the limit of large supercooling.

Figures

Figures reproduced from arXiv: 2511.02910 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Comparison of different prescriptions for the effective potential at [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Comparison of the tunnelling action [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Nucleation temperature [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Domain Walls From Confining Bubbles: $SU(N_{c})$ Yang Mills at Finite $\theta$

    hep-ph 2026-07 conditional novelty 6.0 of 10

    A nonzero theta angle weakens supercooling in SU(Nc) Yang-Mills confinement and makes any resulting domain-wall gravitational-wave signal invisible except under severe fine-tuning.

  2. Theoretical uncertainties in reconstructing model parameters with gravitational waves from supercooled phase transitions

    hep-ph 2026-07 conditional novelty 6.0 of 10

    In the conformal U(1)_X model, the daisy-resummed nucleation scheme shifts LISA-reconstructed gauge couplings by O(10%) relative to the NLO effective-field-theory result, an order of magnitude above the Fisher-matrix ...

  3. Hard thermal contributions to phase transition observables at NNLO

    hep-ph 2026-02 conditional novelty 6.0 of 10

    Three-loop thermal masses and two-loop quartic couplings complete the O(g^6) high-temperature EFT of U(1) and SU(N) gauge-Higgs models, with a missing contribution identified in a known three-loop master integral.

  4. The price for monopole dark matter

    hep-ph 2026-07 conditional novelty 5.0 of 10

    Dark matter can consist of 't Hooft-Polyakov monopoles in a dark SU(2) sector, but only in a narrow window with m_M ≳ 10^8 GeV, near-bound dark radiation, and possibly detectable gravitational waves.

Reference graph

Works this paper leans on

156 extracted references · 127 linked inside Pith · cited by 4 Pith papers

  1. [1]

    Daisy (ring) resummation: 4D potential 13

  2. [2]

    Full” potential, where the thermal function Jb(m2 i /T 2) is evaluated numerically (see Eq. (A7)), and the 4D “High-Temperature

    Dimensional Reduction: 3D EFT Parameters 16 B. Running Couplings and Anomalous Dimensions 18 References 18 I. INTRODUCTION The discovery of gravitational waves (GWs) by the LIGO/Virgo Collaboration [1] has opened a new obser- vational window into the Universe. GWs carry information not only about their astrophysical sources, but also about fundamental int...

  3. [3]

    (1) as Φ = (ϕ+iχ)/ √ 2, we restrict to the real directionϕ, which is the relevant background field for the effective potential

    Daisy (ring) resummation: 4D potential Expanding the complex scalar field of Eq. (1) as Φ = (ϕ+iχ)/ √ 2, we restrict to the real directionϕ, which is the relevant background field for the effective potential. In the 4D approach this can be written schematically as Veff (ϕ, T;µ) =V tree(ϕ) +V CW(ϕ;µ) +V T (ϕ, T) +VDaisy(ϕ, T),(A1) whereV CW(ϕ;µ) denotes th...

  4. [4]

    In the limitλ→0, this reproduces the corresponding expression in Ref

    Reλ 3 2 4π T , λ T =λ+ 3g 4 8π2 log T Mg2 + 5λ 2 4π2 log T Mλ ,(A12) whereM a = µ 4π exp(na −γ E),a∈ {g 2, λ}, withn g2 = 1/3 andn λ = 2/3. In the limitλ→0, this reproduces the corresponding expression in Ref. [127]. If the renormalisation scale is set independent of the field value, it is straightforward to incorporate the RG running in the OPA, replacin...

  5. [5]

    The corresponding Lagrangian is L3D = 1 4 F 3D ij 2 + 1 2 ∂iA3D 0 2 + 1 2 ∂i −ig 3DA3D i Φ3D 2 −m 2 3D Φ3D 2 + 1 2 µ2 DA3D 0 2 −λ 3D Φ3D 4 + λA 4! A3D 0 2 + λAΦ 2 A3D 0 2 Φ3D 2

    Dimensional Reduction: 3D EFT Parameters Dimensional reduction integrates out the non-zero Matsubara modes, leading to a three-dimensional EFT for the soft modes. The corresponding Lagrangian is L3D = 1 4 F 3D ij 2 + 1 2 ∂iA3D 0 2 + 1 2 ∂i −ig 3DA3D i Φ3D 2 −m 2 3D Φ3D 2 + 1 2 µ2 DA3D 0 2 −λ 3D Φ3D 4 + λA 4! A3D 0 2 + λAΦ 2 A3D 0 2 Φ3D 2 . (A13) We follow...

  6. [6]

    B. P. Abbottet al.(LIGO Scientific, Virgo), Observation of Gravitational Waves from a Binary Black Hole Merger, Phys. Rev. Lett.116, 061102 (2016), arXiv:1602.03837 [gr-qc]

    arXiv 2016

  7. [7]

    M. A. McLaughlin, The North American Nanohertz Observatory for Gravitational Waves, Class. Quant. Grav.30, 224008 (2013), arXiv:1310.0758 [astro-ph.IM]

    arXiv 2013

  8. [8]

    Ransomet al.(NANOGrav), The NANOGrav Program for Gravitational Waves and Fundamental Physics, Bull

    S. Ransomet al.(NANOGrav), The NANOGrav Program for Gravitational Waves and Fundamental Physics, Bull. Am. Astron. Soc.51(2019), arXiv:1908.05356 [astro-ph.IM]

    arXiv 2019

Show all 156 references
  1. [9]

    Kramer and D

    M. Kramer and D. J. Champion, The European Pulsar Timing Array and the Large European Array for Pulsars, Class. Quant. Grav.30, 224009 (2013)

    2013

  2. [10]

    Desvigneset al., High-precision timing of 42 millisecond pulsars with the European Pulsar Timing Array, Mon

    G. Desvigneset al., High-precision timing of 42 millisecond pulsars with the European Pulsar Timing Array, Mon. Not. Roy. Astron. Soc.458, 3341 (2016), arXiv:1602.08511 [astro-ph.HE]

    2016 arXiv

  3. [11]

    R. N. Manchesteret al., The Parkes Pulsar Timing Array Project, Publ. Astron. Soc. Austral.30, 17 (2013), arXiv:1210.6130 [astro-ph.IM]

    2013 arXiv

  4. [12]

    Chandra Joshiet al., Nanohertz gravitational wave astronomy during SKA era: An InPTA perspective, J

    B. Chandra Joshiet al., Nanohertz gravitational wave astronomy during SKA era: An InPTA perspective, J. Astrophys. Astron.43, 98 (2022), arXiv:2207.06461 [astro-ph.HE]

    2022

  5. [13]

    K. J. Lee, Prospects of Gravitational Wave Detection Using Pulsar Timing Array for Chinese Future Telescopes, in Frontiers in Radio Astronomy and F AST Early Sciences Symposium 2015, Astronomical Society of the Pacific Conference Series, Vol. 502, edited by L. Qain and D. Li (...

    2015

  6. [14]

    M. T. Mileset al., The MeerKAT Pulsar Timing Array: first data release, Mon. Not. Roy. Astron. Soc.519, 3976 (2023), arXiv:2212.04648 [astro-ph.HE]

    2023 arXiv

  7. [15]

    J. P. W. Verbiestet al., Timing stability of millisecond pulsars and prospects for gravitational-wave detection, Mon. Not. Roy. Astron. Soc.400, 951 (2009), arXiv:0908.0244 [astro-ph.GA]

    2009 arXiv

  8. [16]

    R. N. Manchester, The International Pulsar Timing Array, Class. Quant. Grav.30, 224010 (2013), arXiv:1309.7392 [astro-ph.IM]

    2013 arXiv

  9. [17]

    Agazieet al.(NANOGrav), The NANOGrav 15 yr Data Set: Evidence for a Gravitational-wave Background, Astrophys

    G. Agazieet al.(NANOGrav), The NANOGrav 15 yr Data Set: Evidence for a Gravitational-wave Background, Astrophys. J. Lett.951, L8 (2023), arXiv:2306.16213 [astro-ph.HE]

    2023 arXiv

  10. [18]

    Antoniadiset al.(EPTA, InPTA:), The second data release from the European Pulsar Timing Array - III

    J. Antoniadiset al.(EPTA, InPTA:), The second data release from the European Pulsar Timing Array - III. Search for gravitational wave signals, Astron. Astrophys.678, A50 (2023), arXiv:2306.16214 [astro-ph.HE]

    2023 arXiv

  11. [19]

    D. J. Reardonet al., Search for an Isotropic Gravitational-wave Background with the Parkes Pulsar Timing Array, Astrophys. J. Lett.951, L6 (2023), arXiv:2306.16215 [astro-ph.HE]

    2023 arXiv

  12. [20]

    Xuet al., Searching for the Nano-Hertz Stochastic Gravitational Wave Background with the Chinese Pulsar Timing Array Data Release I, Res

    H. Xuet al., Searching for the Nano-Hertz Stochastic Gravitational Wave Background with the Chinese Pulsar Timing Array Data Release I, Res. Astron. Astrophys.23, 075024 (2023), arXiv:2306.16216 [astro-ph.HE]

    2023 arXiv

  13. [21]

    M. T. Mileset al., The MeerKAT Pulsar Timing Array: the first search for gravitational waves with the MeerKAT radio telescope, Mon. Not. Roy. Astron. Soc.536, 1489 (2024), arXiv:2412.01153 [astro-ph.HE]. 19

    2024 arXiv

  14. [22]

    Rajagopal and R

    M. Rajagopal and R. W. Romani, Ultralow frequency gravitational radiation from massive black hole binaries, Astrophys. J.446, 543 (1995), arXiv:astro-ph/9412038

    1995 arXiv

  15. [23]

    A. H. Jaffe and D. C. Backer, Gravitational waves probe the coalescence rate of massive black hole binaries, Astrophys. J.583, 616 (2003), arXiv:astro-ph/0210148

    2003 arXiv

  16. [24]

    J. S. B. Wyithe and A. Loeb, Low - frequency gravitational waves from massive black hole binaries: Predictions for LISA and pulsar timing arrays, Astrophys. J.590, 691 (2003), arXiv:astro-ph/0211556

    2003 arXiv

  17. [25]

    Sesana, F

    A. Sesana, F. Haardt, P. Madau, and M. Volonteri, Low - frequency gravitational radiation from coalescing massive black hole binaries in hierarchical cosmologies, Astrophys. J.611, 623 (2004), arXiv:astro-ph/0401543

    2004 arXiv

  18. [26]

    S. T. McWilliams, J. P. Ostriker, and F. Pretorius, Gravitational waves and stalled satellites from massive galaxy mergers atz≤1, Astrophys. J.789, 156 (2014), arXiv:1211.5377 [astro-ph.CO]

    2014 arXiv

  19. [27]

    Burke-Spolaoret al., The Astrophysics of Nanohertz Gravitational Waves, Astron

    S. Burke-Spolaoret al., The Astrophysics of Nanohertz Gravitational Waves, Astron. Astrophys. Rev.27, 5 (2019), arXiv:1811.08826 [astro-ph.HE]

    2019 arXiv

  20. [28]

    B´ ecsy, N

    B. B´ ecsy, N. J. Cornish, and L. Z. Kelley, Exploring Realistic Nanohertz Gravitational-wave Backgrounds, Astrophys. J. 941, 119 (2022), arXiv:2207.01607 [astro-ph.HE]

    2022 arXiv

  21. [29]

    Ellis, M

    J. Ellis, M. Fairbairn, G. H¨ utsi, M. Raidal, J. Urrutia, V. Vaskonen, and H. Veerm¨ ae, Prospects for future binary black hole gravitational wave studies in light of PTA measurements, Astron. Astrophys.676, A38 (2023), arXiv:2301.13854 [astro-ph.CO]

    2023 arXiv

  22. [30]

    Kosowsky, M

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

    1992

  23. [31]

    Schwaller, Gravitational Waves from a Dark Phase Transition, Phys

    P. Schwaller, Gravitational Waves from a Dark Phase Transition, Phys. Rev. Lett.115, 181101 (2015), arXiv:1504.07263 [hep-ph]

    2015 arXiv

  24. [32]

    Breitbach, J

    M. Breitbach, J. Kopp, E. Madge, T. Opferkuch, and P. Schwaller, Dark, Cold, and Noisy: Constraining Secluded Hidden Sectors with Gravitational Waves, J. Cosmol. Astropart. Phys.07(2019), 007, arXiv:1811.11175 [hep-ph]

    2019 arXiv

  25. [33]

    Fairbairn, E

    M. Fairbairn, E. Hardy, and A. Wickens, Hearing without seeing: gravitational waves from hot and cold hidden sectors, J. High Energy Phys.07(2019), 044, arXiv:1901.11038 [hep-ph]

    2019 arXiv

  26. [34]

    Addazi, Y.-F

    A. Addazi, Y.-F. Cai, Q. Gan, A. Marciano, and K. Zeng, NANOGrav results and dark first order phase transitions, Sci. China Phys. Mech. Astron.64, 290411 (2021), arXiv:2009.10327 [hep-ph]

    2021 arXiv

  27. [35]

    Nakai, M

    Y. Nakai, M. Suzuki, F. Takahashi, and M. Yamada, Gravitational Waves and Dark Radiation from Dark Phase Transition: Connecting NANOGrav Pulsar Timing Data and Hubble Tension, Phys. Lett. B816, 136238 (2021), arXiv:2009.09754 [astro-ph.CO]

    2021 arXiv

  28. [36]

    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]

    2021 arXiv

  29. [37]

    H.-H. Li, G. Ye, and Y.-S. Piao, Is the NANOGrav signal a hint of dS decay during inflation?, Phys. Lett. B816, 136211 (2021), arXiv:2009.14663 [astro-ph.CO]

    2021 arXiv

  30. [38]

    Borah, A

    D. Borah, A. Dasgupta, and S. K. Kang, A first order dark SU(2) D phase transition with vector dark matter in the light of NANOGrav 12.5 yr data, JCAP12(12), 039, arXiv:2109.11558 [hep-ph]

    arXiv

  31. [39]

    Morgante, N

    E. Morgante, N. Ramberg, and P. Schwaller, Gravitational waves from dark SU(3) Yang-Mills theory, Phys. Rev. D107, 036010 (2023), arXiv:2210.11821 [hep-ph]

    2023 arXiv

  32. [40]

    Bringmann, P

    T. Bringmann, P. F. Depta, T. Konstandin, K. Schmidt-Hoberg, and C. Tasillo, Does NANOGrav observe a dark sector phase transition?, J. Cosmol. Astropart. Phys.11(2023), 053, arXiv:2306.09411 [astro-ph.CO]

    2023 arXiv

  33. [41]

    Megias, G

    E. Megias, G. Nardini, and M. Quiros, Pulsar timing array stochastic background from light Kaluza-Klein resonances, Phys. Rev. D108, 095017 (2023), arXiv:2306.17071 [hep-ph]

    2023 arXiv

  34. [42]

    Addazi, Y.-F

    A. Addazi, Y.-F. Cai, A. Marciano, and L. Visinelli, Have pulsar timing array methods detected a cosmological phase transition?, Phys. Rev. D109, 015028 (2024), arXiv:2306.17205 [astro-ph.CO]

    2024 arXiv

  35. [43]

    Han, K.-P

    C. Han, K.-P. Xie, J. M. Yang, and M. Zhang, Self-interacting dark matter implied by nano-Hertz gravitational waves, Phys. Rev. D109, 115025 (2024), arXiv:2306.16966 [hep-ph]

    2024 arXiv

  36. [44]

    Arzoumanianet al.(NANOGrav), Searching for Gravitational Waves from Cosmological Phase Transitions with the NANOGrav 12.5-Year Dataset, Phys

    Z. Arzoumanianet al.(NANOGrav), Searching for Gravitational Waves from Cosmological Phase Transitions with the NANOGrav 12.5-Year Dataset, Phys. Rev. Lett.127, 251302 (2021), arXiv:2104.13930 [astro-ph.CO]

    2021 arXiv

  37. [45]

    Freese and M

    K. Freese and M. W. Winkler, Have pulsar timing arrays detected the hot big bang: Gravitational waves from strong first order phase transitions in the early Universe, Phys. Rev. D106, 103523 (2022), arXiv:2208.03330 [astro-ph.CO]

    2022 arXiv

  38. [46]

    Fujikura, S

    K. Fujikura, S. Girmohanta, Y. Nakai, and M. Suzuki, NANOGrav signal from a dark conformal phase transition, Phys. Lett. B846, 138203 (2023), arXiv:2306.17086 [hep-ph]

    2023 arXiv

  39. [47]

    Athron, A

    P. Athron, A. Fowlie, C.-T. Lu, L. Morris, L. Wu, Y. Wu, and Z. Xu, Can Supercooled Phase Transitions Explain the Grav- itational Wave Background Observed by Pulsar Timing Arrays?, Phys. Rev. Lett.132, 221001 (2024), arXiv:2306.17239 [hep-ph]

    2024 arXiv

  40. [48]

    Jiang, A

    S. Jiang, A. Yang, J. Ma, and F. P. Huang, Implication of nano-Hertz stochastic gravitational wave on dynamical dark matter through a dark first-order phase transition, Class. Quant. Grav.41, 065009 (2024), arXiv:2306.17827 [hep-ph]

    2024 arXiv

  41. [49]

    Gon¸ calves, D

    J. Gon¸ calves, D. Marfatia, A. P. Morais, and R. Pasechnik, Supercooled phase transitions in conformal dark sectors explain NANOGrav data, Phys. Lett. B869, 139829 (2025), arXiv:2501.11619 [hep-ph]

    2025 arXiv

  42. [50]

    Costa, J

    F. Costa, J. Hoefken Zink, M. Lucente, S. Pascoli, and S. Rosauro-Alcaraz, Supercooled dark scalar phase transitions explanation of NANOGrav data, Phys. Lett. B868, 139634 (2025), arXiv:2501.15649 [hep-ph]

    2025

  43. [51]

    Vachaspati and A

    T. Vachaspati and A. Vilenkin, Gravitational Radiation from Cosmic Strings, Phys. Rev. D31, 3052 (1985)

    1985

  44. [52]

    Sakellariadou, Gravitational waves emitted from infinite strings, Phys

    M. Sakellariadou, Gravitational waves emitted from infinite strings, Phys. Rev. D42, 354 (1990), [Erratum: Phys.Rev.D 43, 4150 (1991)]. 20

    1990

  45. [53]

    Ellis and M

    J. Ellis and M. Lewicki, Cosmic String Interpretation of NANOGrav Pulsar Timing Data, Phys. Rev. Lett.126, 041304 (2021), arXiv:2009.06555 [astro-ph.CO]

    2021 arXiv

  46. [54]

    Blasi, V

    S. Blasi, V. Brdar, and K. Schmitz, Has NANOGrav found first evidence for cosmic strings?, Phys. Rev. Lett.126, 041305 (2021), arXiv:2009.06607 [astro-ph.CO]

    2021 arXiv

  47. [55]

    Buchmuller, V

    W. Buchmuller, V. Domcke, and K. Schmitz, From NANOGrav to LIGO with metastable cosmic strings, Phys. Lett. B 811, 135914 (2020), arXiv:2009.10649 [astro-ph.CO]

    2020 arXiv

  48. [56]

    Quelquejay Leclereet al.(European Pulsar Timing Array, EPTA), Practical approaches to analyzing PTA data: Cosmic strings with six pulsars, Phys

    H. Quelquejay Leclereet al.(European Pulsar Timing Array, EPTA), Practical approaches to analyzing PTA data: Cosmic strings with six pulsars, Phys. Rev. D108, 123527 (2023), arXiv:2306.12234 [gr-qc]

    2023 arXiv

  49. [57]

    Z. Wang, L. Lei, H. Jiao, L. Feng, and Y.-Z. Fan, The nanohertz stochastic gravitational wave background from cos- mic string loops and the abundant high redshift massive galaxies, Sci. China Phys. Mech. Astron.66, 120403 (2023), arXiv:2306.17150 [astro-ph.HE]

    2023 arXiv

  50. [58]

    Ellis, M

    J. Ellis, M. Lewicki, C. Lin, and V. Vaskonen, Cosmic superstrings revisited in light of NANOGrav 15-year data, Phys. Rev. D108, 103511 (2023), arXiv:2306.17147 [astro-ph.CO]

    2023 arXiv

  51. [59]

    Bai, T.-K

    Y. Bai, T.-K. Chen, and M. Korwar, QCD-collapsed domain walls: QCD phase transition and gravitational wave spec- troscopy, J. High Energy Phys.12(2023), 194, arXiv:2306.17160 [hep-ph]

    2023 arXiv

  52. [60]

    Kitajima and K

    N. Kitajima and K. Nakayama, Nanohertz gravitational waves from cosmic strings and dark photon dark matter, Phys. Lett. B846, 138213 (2023), arXiv:2306.17390 [hep-ph]

    2023 arXiv

  53. [61]

    Eichhorn, R

    A. Eichhorn, R. R. Lino dos Santos, and J. L. Miqueleto, From quantum gravity to gravitational waves through cosmic strings, Phys. Rev. D109, 026013 (2024), arXiv:2306.17718 [gr-qc]

    2024 arXiv

  54. [62]

    Hiramatsu, M

    T. Hiramatsu, M. Kawasaki, and K. Saikawa, On the estimation of gravitational wave spectrum from cosmic domain walls, J. Cosmol. Astropart. Phys.02(2014), 031, arXiv:1309.5001 [astro-ph.CO]

    2014 arXiv

  55. [63]

    R. Z. Ferreira, A. Notari, O. Pujolas, and F. Rompineve, Gravitational waves from domain walls in Pulsar Timing Array datasets, J. Cosmol. Astropart. Phys.02(2023), 001, arXiv:2204.04228 [astro-ph.CO]

    2023 arXiv

  56. [64]

    S.-Y. Guo, M. Khlopov, X. Liu, L. Wu, Y. Wu, and B. Zhu, Footprints of axion-like particle in pulsar timing array data and James Webb Space Telescope observations, Sci. China Phys. Mech. Astron.67, 111011 (2024), arXiv:2306.17022 [hep-ph]

    2024 arXiv

  57. [65]

    Kitajima, J

    N. Kitajima, J. Lee, K. Murai, F. Takahashi, and W. Yin, Gravitational waves from domain wall collapse, and application to nanohertz signals with QCD-coupled axions, Phys. Lett. B851, 138586 (2024), arXiv:2306.17146 [hep-ph]

    2024 arXiv

  58. [66]

    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]

    2024 arXiv

  59. [67]

    Lazarides, R

    G. Lazarides, R. Maji, and Q. Shafi, Superheavy quasistable strings and walls bounded by strings in the light of NANOGrav 15 year data, Phys. Rev. D108, 095041 (2023), arXiv:2306.17788 [hep-ph]

    2023 arXiv

  60. [68]

    Blasi, A

    S. Blasi, A. Mariotti, A. Rase, and A. Sevrin, Axionic domain walls at Pulsar Timing Arrays: QCD bias and particle friction, J. High Energy Phys.11(2023), 169, arXiv:2306.17830 [hep-ph]

    2023 arXiv

  61. [69]

    M. S. Turner, Detectability of inflation produced gravitational waves, Phys. Rev. D55, R435 (1997), arXiv:astro- ph/9607066

    1997

  62. [70]

    De Luca, G

    V. De Luca, G. Franciolini, and A. Riotto, NANOGrav Data Hints at Primordial Black Holes as Dark Matter, Phys. Rev. Lett.126, 041303 (2021), arXiv:2009.08268 [astro-ph.CO]

    2021 arXiv

  63. [71]

    Vagnozzi, Implications of the NANOGrav results for inflation, Mon

    S. Vagnozzi, Implications of the NANOGrav results for inflation, Mon. Not. Roy. Astron. Soc.502, L11 (2021), arXiv:2009.13432 [astro-ph.CO]

    2021 arXiv

  64. [72]

    Ashoorioon, K

    A. Ashoorioon, K. Rezazadeh, and A. Rostami, NANOGrav signal from the end of inflation and the LIGO mass and heavier primordial black holes, Phys. Lett. B835, 137542 (2022), arXiv:2202.01131 [astro-ph.CO]

    2022 arXiv

  65. [73]

    S. Vagnozzi, Inflationary interpretation of the stochastic gravitational wave background signal detected by pulsar timing array experiments, JHEAp39, 81 (2023), arXiv:2306.16912 [astro-ph.CO]

    2023 arXiv

  66. [74]

    Cai, X.-C

    Y.-F. Cai, X.-C. He, X.-H. Ma, S.-F. Yan, and G.-W. Yuan, Limits on scalar-induced gravitational waves from the stochastic background by pulsar timing array observations, Sci. Bull.68, 2929 (2023), arXiv:2306.17822 [gr-qc]

    2023 arXiv

  67. [75]

    Ratzinger and P

    W. Ratzinger and P. Schwaller, Whispers from the dark side: Confronting light new physics with NANOGrav data, SciPost Phys.10, 047 (2021), arXiv:2009.11875 [astro-ph.CO]

    2021 arXiv

  68. [76]

    Bian, R.-G

    L. Bian, R.-G. Cai, J. Liu, X.-Y. Yang, and R. Zhou, Evidence for different gravitational-wave sources in the NANOGrav dataset, Phys. Rev. D103, L081301 (2021), arXiv:2009.13893 [astro-ph.CO]

    2021 arXiv

  69. [77]

    Madge, E

    E. Madge, E. Morgante, C. Puchades-Ib´ a˜ nez, N. Ramberg, W. Ratzinger, S. Schenk, and P. Schwaller, Primordial gravi- tational waves in the nano-Hertz regime and PTA data — towards solving the GW inverse problem, J. High Energy Phys. 10(2023), 171, arXiv:2306.14856 [hep-ph]

    2023 arXiv

  70. [78]

    Afzalet al.(NANOGrav), The NANOGrav 15 yr Data Set: Search for Signals from New Physics, Astrophys

    A. Afzalet al.(NANOGrav), The NANOGrav 15 yr Data Set: Search for Signals from New Physics, Astrophys. J. Lett. 951, L11 (2023), [Erratum: Astrophys.J.Lett. 971, L27 (2024), Erratum: Astrophys.J. 971, L27 (2024)], arXiv:2306.16219 [astro-ph.HE]

    2023 arXiv

  71. [79]

    Franciolini, D

    G. Franciolini, D. Racco, and F. Rompineve, Footprints of the QCD Crossover on Cosmological Gravitational Waves at Pulsar Timing Arrays, Phys. Rev. Lett.132, 081001 (2024), [Erratum: Phys.Rev.Lett. 133, 189901 (2024)], arXiv:2306.17136 [astro-ph.CO]

    2024 arXiv

  72. [80]

    Ellis, M

    J. Ellis, M. Fairbairn, G. Franciolini, G. H¨ utsi, A. Iovino, M. Lewicki, M. Raidal, J. Urrutia, V. Vaskonen, and H. Veerm¨ ae, What is the source of the PTA GW signal?, Phys. Rev. D109, 023522 (2024), arXiv:2308.08546 [astro-ph.CO]

    2024 arXiv

  73. [81]

    Kajantie, M

    K. Kajantie, M. Laine, K. Rummukainen, and M. E. Shaposhnikov, Is there a hot electroweak phase transition at mH ≳m W ?, Phys. Rev. Lett.77, 2887 (1996), arXiv:hep-ph/9605288

    1996 arXiv

  74. [82]

    Y. Aoki, G. Endrodi, Z. Fodor, S. D. Katz, and K. K. Szabo, The Order of the quantum chromodynamics transition 21 predicted by the standard model of particle physics, Nature443, 675 (2006), arXiv:hep-lat/0611014

    2006 arXiv

  75. [83]

    Witten, Cosmic Separation of Phases, Phys

    E. Witten, Cosmic Separation of Phases, Phys. Rev. D30, 272 (1984)

    1984

  76. [84]

    C. J. Hogan, Gravitational radiation from cosmological phase transitions, Mon. Not. Roy. Astron. Soc.218, 629 (1986)

    1986

  77. [85]

    Kamionkowski, A

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

    1994 arXiv

  78. [86]

    Ellis, M

    J. Ellis, M. Lewicki, J. M. No, and V. Vaskonen, Gravitational wave energy budget in strongly supercooled phase transi- tions, J. Cosmol. Astropart. Phys.06(2019), 024, arXiv:1903.09642 [hep-ph]

    2019 arXiv

  79. [87]

    Ellis, M

    J. Ellis, M. Lewicki, and V. Vaskonen, Updated predictions for gravitational waves produced in a strongly supercooled phase transition, J. Cosmol. Astropart. Phys.11(2020), 020, arXiv:2007.15586 [astro-ph.CO]

    2020 arXiv

  80. [88]

    Lewicki and V

    M. Lewicki and V. Vaskonen, Gravitational waves from bubble collisions and fluid motion in strongly supercooled phase transitions, Eur. Phys. J. C83, 109 (2023), arXiv:2208.11697 [astro-ph.CO]

    2023 arXiv

  81. [89]

    Kierkla, A

    M. Kierkla, A. Karam, and B. Swiezewska, Conformal model for gravitational waves and dark matter: a status update, J. High Energy Phys.03(2023), 007, arXiv:2210.07075 [astro-ph.CO]

    2023 arXiv

  82. [90]

    Jinno and M

    R. Jinno and M. Takimoto, Probing a classically conformal B-L model with gravitational waves, Phys. Rev. D95, 015020 (2017), arXiv:1604.05035 [hep-ph]

    2017 arXiv

  83. [91]

    S. Iso, P. D. Serpico, and K. Shimada, QCD-Electroweak First-Order Phase Transition in a Supercooled Universe, Phys. Rev. Lett.119, 141301 (2017), arXiv:1704.04955 [hep-ph]

    2017 arXiv

  84. [92]

    Marzola, A

    L. Marzola, A. Racioppi, and V. Vaskonen, Phase transition and gravitational wave phenomenology of scalar conformal extensions of the Standard Model, Eur. Phys. J. C77, 484 (2017), arXiv:1704.01034 [hep-ph]

    2017 arXiv

  85. [93]

    Azatov, D

    A. Azatov, D. Barducci, and F. Sgarlata, Gravitational traces of broken gauge symmetries, J. Cosmol. Astropart. Phys. 07(2020), 027, arXiv:1910.01124 [hep-ph]

    2020 arXiv

  86. [94]

    S. R. Coleman and E. J. Weinberg, Radiative Corrections as the Origin of Spontaneous Symmetry Breaking, Phys. Rev. D7, 1888 (1973)

    1973

  87. [95]

    Balan, T

    S. Balan, T. Bringmann, F. Kahlhoefer, J. Matuszak, and C. Tasillo, Sub-GeV dark matter and nano-Hertz gravitational waves from a classically conformal dark sector, J. Cosmol. Astropart. Phys.08(2025), 062, arXiv:2502.19478 [hep-ph]

    2025

  88. [96]

    A. D. Linde, Phase Transitions in Gauge Theories and Cosmology, Rept. Prog. Phys.42, 389 (1979)

    1979

  89. [97]

    A. D. Linde, Infrared Problem in Thermodynamics of the Yang-Mills Gas, Phys. Lett. B96, 289 (1980)

    1980

  90. [98]

    Weinberg, Gauge and Global Symmetries at High Temperature, Phys

    S. Weinberg, Gauge and Global Symmetries at High Temperature, Phys. Rev. D9, 3357 (1974)

    1974

  91. [99]

    Dolan and R

    L. Dolan and R. Jackiw, Symmetry Behavior at Finite Temperature, Phys. Rev. D9, 3320 (1974)

    1974

  92. [100]

    R. R. Parwani, Resummation in a hot scalar field theory, Phys. Rev. D45, 4695 (1992), [Erratum: Phys.Rev.D 48, 5965 (1993)], arXiv:hep-ph/9204216

    1992 arXiv

  93. [101]

    P. B. Arnold and O. Espinosa, The Effective potential and first order phase transitions: Beyond leading-order, Phys. Rev. D47, 3546 (1993), [Erratum: Phys.Rev.D 50, 6662 (1994)], arXiv:hep-ph/9212235

    1993 arXiv

  94. [102]

    Gould and T

    O. Gould and T. V. I. Tenkanen, On the perturbative expansion at high temperature and implications for cosmological phase transitions, J. High Energy Phys.06(2021), 069, arXiv:2104.04399 [hep-ph]

    2021 arXiv

  95. [103]

    C. G. Boyd, D. E. Brahm, and S. D. H. Hsu, Resummation methods at finite temperature: The Tadpole way, Phys. Rev. D48, 4963 (1993), arXiv:hep-ph/9304254

    1993 arXiv

  96. [104]

    Curtin, P

    D. Curtin, P. Meade, and H. Ramani, Thermal Resummation and Phase Transitions, Eur. Phys. J. C78, 787 (2018), arXiv:1612.00466 [hep-ph]

    2018 arXiv

  97. [105]

    Curtin, J

    D. Curtin, J. Roy, and G. White, Gravitational waves and tadpole resummation: Efficient and easy convergence of finite temperature QFT, Phys. Rev. D109, 116001 (2024), arXiv:2211.08218 [hep-ph]

    2024 arXiv

  98. [106]

    H. Bahl, M. Carena, A. Ireland, and C. E. M. Wagner, Improved thermal resummation for multi-field potentials, J. High Energy Phys.09(2024), 153, arXiv:2404.12439 [hep-ph]

    2024 arXiv

  99. [107]

    Bittar, S

    P. Bittar, S. Roy, and C. E. M. Wagner, Self consistent thermal resummation: a case study of the phase transition in 2HDM, J. High Energy Phys.12(2025), 021, arXiv:2504.02024 [hep-ph]

    2025

  100. [108]

    Navarrete, R

    P. Navarrete, R. Paatelainen, K. Sepp¨ anen, and T. V. I. Tenkanen, Cosmological phase transitions without high- temperature expansions, arXiv:2507.07014 [hep-ph] (2025)

    2025 arXiv

  101. [109]

    D. V. Vassilevich, Heat kernel expansion: User’s manual, Phys. Rept.388, 279 (2003), arXiv:hep-th/0306138

    2003 arXiv

  102. [110]

    Chakrabortty and S

    J. Chakrabortty and S. Mohanty, One Loop Thermal Effective Action, Nucl. Phys. B1020, 117165 (2025), arXiv:2411.14146 [hep-th]

    2025 arXiv

  103. [111]

    Balui, T

    D. Balui, T. Biswas, J. Chakrabortty, D. Dey, C. Englert, and S. Mohanty, Gauge choices, infrared pitfalls, and thermal effects in effective potentials, Phys. Rev. D112, 056022 (2025), arXiv:2507.22706 [hep-th]

    2025 arXiv

  104. [112]

    Farakos, K

    K. Farakos, K. Kajantie, K. Rummukainen, and M. E. Shaposhnikov, 3-D physics and the electroweak phase transition: Perturbation theory, Nucl. Phys. B425, 67 (1994), arXiv:hep-ph/9404201

    1994 arXiv

  105. [113]

    Kajantie, M

    K. Kajantie, M. Laine, K. Rummukainen, and M. E. Shaposhnikov, Generic rules for high temperature dimensional reduction and their application to the standard model, Nucl. Phys. B458, 90 (1996), arXiv:hep-ph/9508379

    1996 arXiv

  106. [114]

    Braaten and A

    E. Braaten and A. Nieto, Effective field theory approach to high temperature thermodynamics, Phys. Rev. D51, 6990 (1995), arXiv:hep-ph/9501375

    1995 arXiv

  107. [115]

    Ekstedt, P

    A. Ekstedt, P. Schicho, and T. V. I. Tenkanen, DRalgo: A package for effective field theory approach for thermal phase transitions, Comput. Phys. Commun.288, 108725 (2023), arXiv:2205.08815 [hep-ph]

    2023 arXiv

  108. [116]

    Gould and J

    O. Gould and J. Hirvonen, Effective field theory approach to thermal bubble nucleation, Phys. Rev. D104, 096015 (2021), arXiv:2108.04377 [hep-ph]

    2021 arXiv

  109. [117]

    L¨ ofgren, M

    J. L¨ ofgren, M. J. Ramsey-Musolf, P. Schicho, and T. V. I. Tenkanen, Nucleation at Finite Temperature: A Gauge-Invariant Perturbative Framework, Phys. Rev. Lett.130, 251801 (2023), arXiv:2112.05472 [hep-ph]. 22

    2023 arXiv

  110. [118]

    Hirvonen, J

    J. Hirvonen, J. L¨ ofgren, M. J. Ramsey-Musolf, P. Schicho, and T. V. I. Tenkanen, Computing the gauge-invariant bubble nucleation rate in finite temperature effective field theory, J. High Energy Phys.07(2022), 135, arXiv:2112.08912 [hep-ph]

    2022 arXiv

  111. [119]

    Schicho, T

    P. Schicho, T. V. I. Tenkanen, and G. White, Combining thermal resummation and gauge invariance for electroweak phase transition, J. High Energy Phys.11(2022), 047, arXiv:2203.04284 [hep-ph]

    2022 arXiv

  112. [120]

    Ekstedt, O

    A. Ekstedt, O. Gould, and J. L¨ ofgren, Radiative first-order phase transitions to next-to-next-to-leading order, Phys. Rev. D106, 036012 (2022), [Erratum: Phys.Rev.D 110, 019901 (2024)], arXiv:2205.07241 [hep-ph]

    2022 arXiv

  113. [121]

    L¨ ofgren, Stop comparing resummation methods, J

    J. L¨ ofgren, Stop comparing resummation methods, J. Phys. G50, 125008 (2023), arXiv:2301.05197 [hep-ph]

    2023 arXiv

  114. [122]

    Gould and T

    O. Gould and T. V. I. Tenkanen, Perturbative effective field theory expansions for cosmological phase transitions, J. High Energy Phys.01(2024), 048, arXiv:2309.01672 [hep-ph]

    2024 arXiv

  115. [123]

    Kierkla, B

    M. Kierkla, B. Swiezewska, T. V. I. Tenkanen, and J. van de Vis, Gravitational waves from supercooled phase transitions: dimensional transmutation meets dimensional reduction, J. High Energy Phys.02(2024), 234, arXiv:2312.12413 [hep-ph]

    2024 arXiv

  116. [124]

    Bodeker and G

    D. Bodeker and G. D. Moore, Can electroweak bubble walls run away?, J. Cosmol. Astropart. Phys.05(2009), 009, arXiv:0903.4099 [hep-ph]

    2009 arXiv

  117. [125]

    Bodeker and G

    D. Bodeker and G. D. Moore, Electroweak Bubble Wall Speed Limit, J. Cosmol. Astropart. Phys.05(2017), 025, arXiv:1703.08215 [hep-ph]

    2017 arXiv

  118. [126]

    Konstandin and G

    T. Konstandin and G. Servant, Cosmological Consequences of Nearly Conformal Dynamics at the TeV scale, J. Cosmol. Astropart. Phys.12(2011), 009, arXiv:1104.4791 [hep-ph]

    2011 arXiv

  119. [127]

    Konstandin and G

    T. Konstandin and G. Servant, Natural Cold Baryogenesis from Strongly Interacting Electroweak Symmetry Breaking, J. Cosmol. Astropart. Phys.07(2011), 024, arXiv:1104.4793 [hep-ph]

    2011 arXiv

  120. [128]

    von Harling and G

    B. von Harling and G. Servant, QCD-induced Electroweak Phase Transition, J. High Energy Phys.01(2018), 159, arXiv:1711.11554 [hep-ph]

    2018 arXiv

  121. [129]

    Marzo, L

    C. Marzo, L. Marzola, and V. Vaskonen, Phase transition and vacuum stability in the classically conformal B–L model, Eur. Phys. J. C79, 601 (2019), arXiv:1811.11169 [hep-ph]

    2019 arXiv

  122. [130]

    Prokopec, J

    T. Prokopec, J. Rezacek, and B. ´Swie˙ zewska, Gravitational waves from conformal symmetry breaking, J. Cosmol. As- tropart. Phys.02(2019), 009, arXiv:1809.11129 [hep-ph]

    2019 arXiv

  123. [131]

    Baratella, A

    P. Baratella, A. Pomarol, and F. Rompineve, The Supercooled Universe, J. High Energy Phys.03(2019), 100, arXiv:1812.06996 [hep-ph]

    2019 arXiv

  124. [132]

    N. Levi, T. Opferkuch, and D. Redigolo, The supercooling window at weak and strong coupling, J. High Energy Phys. 02(2023), 125, arXiv:2212.08085 [hep-ph]

    2023 arXiv

  125. [133]

    Kierkla, N

    M. Kierkla, N. Ramberg, P. Schicho, and D. Schmitt, Theoretical uncertainties for primordial black holes from cosmological phase transitions, arXiv:2506.15496 [hep-ph] (2025)

    2025 arXiv

  126. [134]

    Croon, O

    D. Croon, O. Gould, P. Schicho, T. V. I. Tenkanen, and G. White, Theoretical uncertainties for cosmological first-order phase transitions, J. High Energy Phys.04(2021), 055, arXiv:2009.10080 [hep-ph]

    2021 arXiv

  127. [135]

    Matsubara, A New approach to quantum statistical mechanics, Prog

    T. Matsubara, A New approach to quantum statistical mechanics, Prog. Theor. Phys.14, 351 (1955)

    1955

  128. [136]

    Laine and A

    M. Laine and A. Vuorinen,Basics of Thermal Field Theory, Vol. 925 (Springer, 2016) arXiv:1701.01554 [hep-ph]

    2016 arXiv

  129. [137]

    Chala, J

    M. Chala, J. C. Criado, L. Gil, and J. L. Miras, Higher-order-operator corrections to phase-transition parameters in dimensional reduction, J. High Energy Phys.10(2024), 025, arXiv:2406.02667 [hep-ph]

    2024 arXiv

  130. [138]

    Bernardo, P

    F. Bernardo, P. Klose, P. Schicho, and T. V. I. Tenkanen, Higher-dimensional operators at finite temperature affect gravitational-wave predictions, J. High Energy Phys.08(2025), 109, arXiv:2503.18904 [hep-ph]

    2025 arXiv

  131. [139]

    Chala, L

    M. Chala, L. Gil, and Z. Ren, Phase transitions in dimensional reduction up to three loops*, Chin. Phys.49, 123105 (2025), arXiv:2505.14335 [hep-ph]

    2025 arXiv

  132. [140]

    Chala, A

    M. Chala, A. Dashko, and G. Guedes, Running Couplings in High-Temperature Effective Field Theory, arXiv:2510.26878 [hep-ph] (2025)

    2025

  133. [141]

    Kierkla, P

    M. Kierkla, P. Schicho, B. Swiezewska, T. V. I. Tenkanen, and J. van de Vis, Finite-temperature bubble nucleation with shifting scale hierarchies, J. High Energy Phys.07(2025), 153, arXiv:2503.13597 [hep-ph]

    2025

  134. [142]

    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]

    2024 arXiv

  135. [143]

    J. R. Espinosa, A Fresh Look at the Calculation of Tunneling Actions, J. Cosmol. Astropart. Phys.07(2018), 036, arXiv:1805.03680 [hep-th]

    2018 arXiv

  136. [144]

    Athron, C

    P. Athron, C. Bal´ azs, and L. Morris, Supercool subtleties of cosmological phase transitions, J. Cosmol. Astropart. Phys. 03(2023), 006, arXiv:2212.07559 [hep-ph]

    2023

  137. [145]

    Yamada, Maximal GW amplitude from bubble collisions in supercooled phase transitions, arXiv:2509.13402 [gr-qc] (2025)

    M. Yamada, Maximal GW amplitude from bubble collisions in supercooled phase transitions, arXiv:2509.13402 [gr-qc] (2025)

    2025

  138. [146]

    Yamada, Analytic derivation of GW spectrum from bubble collisions in FLR W Universe, arXiv:2509.16073 [astro- ph.CO] (2025)

    M. Yamada, Analytic derivation of GW spectrum from bubble collisions in FLR W Universe, arXiv:2509.16073 [astro- ph.CO] (2025)

    2025

  139. [147]

    Jinno and M

    R. Jinno and M. Takimoto, Gravitational waves from bubble dynamics: Beyond the Envelope, J. Cosmol. Astropart. Phys.01(2019), 060, arXiv:1707.03111 [hep-ph]

    2019 arXiv

  140. [148]

    Konstandin, Gravitational radiation from a bulk flow model, J

    T. Konstandin, Gravitational radiation from a bulk flow model, J. Cosmol. Astropart. Phys.03(2018), 047, arXiv:1712.06869 [astro-ph.CO]

    2018 arXiv

  141. [149]

    Baldes, M

    I. Baldes, M. Dichtl, Y. Gouttenoire, and F. Sala, Particle shells from relativistic bubble walls, J. High Energy Phys.07 (2024), 231, arXiv:2403.05615 [hep-ph]

    2024 arXiv

  142. [150]

    A. D. Johnsonet al.(NANOGrav), NANOGrav 15-year gravitational-wave background methods, Phys. Rev. D109, 103012 (2024), arXiv:2306.16223 [astro-ph.HE]. 23

    2024 arXiv

  143. [151]

    Ramberg, W

    N. Ramberg, W. Ratzinger, and P. Schwaller, Oneµto rule them all: CMB spectral distortions can probe domain walls, cosmic strings and low scale phase transitions, J. Cosmol. Astropart. Phys.02(2023), 039, arXiv:2209.14313 [hep-ph]

    2023 arXiv

  144. [152]

    Gouttenoire, WIMPs and new physics interpretations of the PTA signal are incompatible, arXiv:2503.03857 [hep-ph] (2025)

    Y. Gouttenoire, WIMPs and new physics interpretations of the PTA signal are incompatible, arXiv:2503.03857 [hep-ph] (2025)

    2025 arXiv

  145. [153]

    Bringmann, D

    T. Bringmann, D. Croon, and S. Sevillano Mu˜ noz, Updated constraints on the primordial power spectrum at sub-Mpc scales, arXiv:2506.20704 [astro-ph.CO] (2025)

    2025 arXiv

  146. [154]

    Caprini, R

    C. Caprini, R. Jinno, T. Konstandin, A. Roper Pol, H. Rubira, and I. Stomberg, Gravitational waves from first-order phase transitions: from weak to strong, J. High Energy Phys.07(2025), 217, arXiv:2409.03651 [gr-qc]

    2025 arXiv

  147. [155]

    Correia, M

    J. Correia, M. Hindmarsh, K. Rummukainen, and D. J. Weir, Gravitational waves from strong first order phase transitions, arXiv:2505.17824 [astro-ph.CO] (2025)

    2025

  148. [156]

    Gouttenoire,Beyond the Standard Model Cocktail, Springer Theses (Springer, Cham, 2022) arXiv:2207.01633 [hep-ph]

    Y. Gouttenoire,Beyond the Standard Model Cocktail, Springer Theses (Springer, Cham, 2022) arXiv:2207.01633 [hep-ph]

    2022 arXiv

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.