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Cosmological phase transitions from the functional measure

T0 review · 3 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Gravitational-wave detectors could measure the Higgs self-coupling to collider-class precision.

desk verdict First careful GW phenomenology of the functional-measure model; the LISA reach is real but shakier than the headline because it sits on an unvalidated fit at alpha ~ 1. read the letter →

arxiv 2502.09593 v1 pith:BJNXZQLU submitted 2025-02-13 hep-ph hep-th

Bruno Berganholi , Gláuber C. Dorsch , Iberê Kuntz , Beatriz M. D. Sena , Giovanna F. do Valle This is my paper · ORCID
classification hep-phhep-th
keywords functionalmeasureelectroweakphasetransitiongravitationalwavesHiggstrilinearcouplingstochasticwavebackgroundeffectivepotentialfirst-orderscalarsector
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 argues that a nontrivial choice of integration measure in the path integral—a Riemannian metric on the space of field configurations—adds a logarithmic term to the Higgs potential, and that this term can turn the electroweak phase transition into a strong first-order transition. If so, the transition would emit a stochastic gravitational-wave background in the millihertz band, detectable by LISA, DECIGO, or BBO. The paper shows that the detectable parameter region corresponds to a Higgs trilinear coupling in the range $2.0 \le \kappa_\lambda \le 2.7$, a precision projected only for near-future lepton colliders. The central point is that gravitational-wave observatories could probe a purely scalar-sector modification of the Standard Model at a level competitive with colliders.

What carries the argument

The load-bearing object is the Riemannian functional measure and its effective-field-theory parametrization. Taking the configuration-space metric to be $G_{ij}=(A+B\phi^\dagger\phi/\Lambda^2)\delta^{(4)}(x-x')$ turns the measure determinant into an exponential of $\log(A+B\phi^\dagger\phi/\Lambda^2)$, which acts as an extra term $-\frac{\Lambda^4}{8\pi^2}\log(1-C\phi^2/\Lambda^2)$ in the effective potential, with $C=-B/2A$. For $C>0$ this term raises the potential between the origin and $\phi=\Lambda/\sqrt{C}$, lifting the broken minimum and making the electroweak phase transition progressively more strongly first order as $C$ grows. The paper feeds this potential into a standard thermal one-loop computation with daisy resummation, extracts the nucleation temperature, transition strength $\alpha$, and inverse duration $\beta/H_*$, and converts these into gravitational-wave spectra using the acoustic sound-wave fit and peak-integrated sensitivity curves. The same potential yields the Higgs trilinear coupling $\kappa_\lambda$ through the third derivative at the vacuum, which is why the gravitational-wave amplitude and $\kappa_\lambda$ are locked together.

What would settle it

A decisive check is a search for the stochastic background in the parameter region corresponding to $2.0\le\kappa_\lambda\le2.7$: a null LISA detection there, despite a future collider measuring $\kappa_\lambda$ in that window, would falsify the claimed overlap.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the minimal modified functional measure model—whose only new physics is the logarithmic term $-\frac{\Lambda^4}{8\pi^2}\log(1-C\phi^2/\Lambda^2)$ in the effective potential—produces gravitational waves from the electroweak phase transition with an amplitude tightly correlated to the Higgs trilinear self-coupling. Scanning $\Lambda \in (500,3000)$ GeV and $C>0$ under stability and symmetry-breaking constraints, the authors find that the signal-to-noise-ratio contours of LISA, DECIGO, and BBO fall inside the currently allowed range $1.1 \le \kappa_\lambda \le 4.8$ and correspond to $2.0 \le \kappa_\lambda \le 2.7$, i.e. $\delta\kappa_\lambda \lesssim 1.7$. They take this to mean that a detected spectrum would measure the trilinear coupling with an accuracy that only a 350 GeV lepton collider at 200 fb$^{-1}$, or an HL-LHC-plus-lepton-collider combination, is projected to reach. The paper concludes that gravitational-wave observatories could therefore be competitive probes of scalar-sector-only new physics.

Load-bearing premise

The quantitative reach assumes that the one-loop effective potential with daisy resummation, plus the sound-wave-only gravitational-wave template fitted for weak transitions, remains reliable for the strong transitions ($\alpha\sim1$) that LISA would need to see; beyond $\phi=\Lambda/\sqrt{C}$ the potential becomes complex, so the barrier and stability treatment there is approximate.

Editorial extensions

If this is right

  • A stochastic background detected at LISA, DECIGO, or BBO in the model's parameter region would translate into a measurement of the Higgs trilinear coupling with $\delta\kappa_\lambda \lesssim 1.7$, matching the projected precision of a 350 GeV lepton collider.
  • Because the LISA sensitivity curve sits close to the metastability boundary, a LISA detection would also place a tight upper bound on the combination of $C$ and $\Lambda$ before the model becomes unphysical.
  • If no background is seen, the model would be constrained at $\kappa_\lambda \gtrsim 2$, while the smaller-$C$ region would remain consistent with current and near-future collider bounds.
  • Since LISA is scheduled to fly in the mid-2030s, a detection could tighten limits on the Higgs self-coupling before an equivalent lepton collider is available.
  • The correlation between gravitational-wave amplitude and $\kappa_\lambda$ implies that gravitational-wave observatories can serve as a scalar-sector-specific probe, complementary to collider measurements of cross sections and branching ratios.

Reading between the lines

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

  • If the Riemannian-measure mechanism is real, it should apply to any scalar sector, so analogous gravitational-wave searches could probe functional-measure corrections in dark-scalar or extended-Higgs models, not just the Standard Model Higgs.
  • The claimed $\kappa_\lambda$ window of 2.0–2.7 relies on a sound-wave template fitted for weak transitions; a template calibrated for $\alpha\sim1$ could shift the reach, so the window is a benchmark rather than a sharp boundary.
  • A detected spectrum's peak frequency and amplitude could be inverted to recover $\alpha$ and $\beta/H_*$, and from them the underlying $\Lambda$ and $C$, giving an independent cross-check on a collider-derived $\kappa_\lambda$.
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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 a scalar-sector modification of the Standard Model in which a non-trivial Riemannian functional measure adds a logarithmic term ~ Λ^4 log(1 - C φ^2/Λ^2) to the effective Higgs potential (the MMFMM). The authors compute the zero-temperature and finite-temperature effective potentials, determine the electroweak phase-transition parameters (T_n, α, β/H*) by solving the bounce equation, and evaluate the resulting gravitational-wave spectra from sound waves (with a turbulence contribution discussed but neglected in the main analysis). Using peak-integrated sensitivity curves for LISA, DECIGO, and BBO, they identify regions of (Λ, C) parameter space with SNR > 1. Their central quantitative claim is that these regions correspond to Higgs trilinear couplings in the range 2.0 ≲ κ_λ ≲ 2.7, i.e. δκ_λ ≲ 1.7, which they argue makes future gravitational-wave observatories competitive with near-future lepton colliders in probing scalar-sector-only new physics.

Significance. If the quantitative reach is robust, the paper makes an interesting and timely point: gravitational-wave observatories could probe the Higgs self-coupling in a specific BSM framework with sensitivity comparable to projected collider programs. The paper is also commendably transparent about several limitations: it explicitly notes in Sec. V that the sound-wave fit in Eq. (40) is calibrated for α ≪ 1 while the LISA-relevant points have α ~ 1; it flags the instability/complexity of the effective potential at large field values in Sec. III; and it states the v_w ≈ 1 assumption and the neglect of turbulence. The methodology (bounce computation, PISC sensitivity curves, parameter scan) is standard and internally consistent. The main value of the paper is therefore as a proof of principle that the functional-measure modification can produce strong phase transitions and detectable GW signals, with the precise numerical overlap with collider sensitivities being more fragile than the qualitative conclusion.

major comments (3)
  1. [Sec. V, Eq. (40), Fig. 6] The LISA detectability claim rests on the sound-wave peak amplitude fit of Eq. (40), which is taken from simulations [63] valid for α ≪ 1. The paper's own footnote 12 acknowledges this, and Fig. 6 shows that the LISA-relevant points require α ~ 1 and β/H* of a few hundred. Since Eq. (40) also uses the bag-model efficiency fit Eq. (41) for κ_sw, both ingredients are extrapolations at the LISA points. Because Fig. 5 shows that a few-percent change in C shifts the peak amplitude by about two orders of magnitude, an O(1) correction to the strong-transition spectrum can move the LISA SNR = 1 contour substantially in the (Λ, C) plane and therefore change the claimed κ_λ interval 2.0–2.7. This is the load-bearing quantitative step in the abstract and Sec. VI. I ask the authors to quantify this uncertainty: for example, by showing the effect of using alternative sound-wave fits or a conservative envelope, or by restricting the collider-competitiveness claim to the less-extrapolated BBO/DECIGO regions where α is smaller.
  2. [Sec. IV and Sec. VI, Figs. 3 and 6] The thermal parameters α, β/H*, and T_n are computed from a 1-loop daisy-resummed effective potential. The LISA-relevant points sit close to the metastability boundary, where the perturbative loop expansion is least reliable and where the potential is only defined up to φ = Λ/√C before becoming complex. The paper requires stability only up to φ_stab ~ 1 TeV and notes in Sec. VII that a fuller decay-time analysis is beyond its scope. Since the nucleation criterion S_3/T_n ≈ 140 is exponentially sensitive to the potential barrier, even moderate higher-order corrections to the potential can shift T_n, α, and β/H* enough to matter at the quoted precision. I request a concrete robustness test, such as a comparison with a two-loop or high-temperature-resummed potential, or at least an explicit estimate of the shift in the SNR = 1 contours from varying the renormalization scale and daisy prescription.
  3. [Sec. VI, Fig. 7 and Conclusions] The comparison between gravitational-wave reach and collider precision is not apples-to-apples. The SNR = 1 contours give model-dependent discovery reach: they delimit parameters for which a detectable GW background is produced, and converting that background to κ_λ requires the MMFMM relation Eq. (28) and the full effective-potential calculation. A detected GW spectrum directly constrains (α, β/H*, T*), not κ_λ itself. The paper acknowledges this model dependence in Sec. VII, but the abstract and the phrase 'competitive with the projected sensitivity of near-future colliders' overstate the case, since discovery reach is not the same as a precision measurement of κ_λ with δκ_λ ≲ 1.7. I recommend softening the wording to 'detection reach' and making the model-dependent mapping explicit in the abstract and the main quantitative summary.
minor comments (3)
  1. [Throughout] There are several typographical issues: 'Incidentaly' in Sec. II, 'V acuum stability' as a section heading in Sec. III, and the notation '1e − 2 mHz' in Eqs. (45) and (49), which should be written consistently as '1.9 × 10^{-2} mHz' and '2.7 × 10^{-2} mHz'.
  2. [Sec. V] The distinction between PLISCs and PISCs is explained verbally, but the figure captions for Figs. 4 and 5 could be clearer: Fig. 4 uses PLISCs while Fig. 5 uses PISCs, and the reader must infer that the peak-integrated method is the reliable one for broken power-law spectra. A sentence in each caption restating this would help.
  3. [Sec. III, Eq. (28)] Equation (28) gives κ_λ as a function of C and Λ, but the text does not discuss the numerical size of the Standard Model 1-loop contribution relative to the measure contribution. A brief sentence quantifying these two terms in the allowed parameter region would aid the interpretation of Fig. 7.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the GW amplitudes and kappa_lambda values are independent outputs of a stated potential, not fitted to one another.

full rationale

The paper's derivation chain is not circular. The central potential (Eq. 20) is not merely imported: Section II derives the logarithmic measure correction explicitly from the Riemannian measure (Eqs. 4, 13, 17), so the citation to [13] is attribution for a construction reproduced in the text rather than an unverified load-bearing premise. The Higgs-mass/VEV conditions fix mu and lambda, and kappa_lambda is then computed as the third derivative of the same potential (Eqs. 27-28); it is an output, not a fitted input. The phase-transition parameters (alpha, beta/H*, Tn) and the GW spectrum are obtained by numerically solving the bounce and applying standard external fits (Eqs. 36-40), so the claimed LISA/BBO/DECIGO sensitivity is an independent dynamical consequence of the model, not a restatement of kappa_lambda. The comparison with collider sensitivity uses an external ATLAS constraint (1.1 <= kappa_lambda <= 4.8), not a value derived from the GW calculation. The paper's own footnote admits that the sound-wave fit [63] was calibrated for alpha << 1 while some LISA-detectable points have alpha ~ 1; this is a validity/robustness concern about the strong-transition regime, not a circularity. No equation in the paper is identical by construction to the result it is said to predict, and no fitted parameter is renamed as a prediction. Therefore the circularity score is 0.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new particles or forces. The new physics is a two-parameter (C, Lambda) logarithmic deformation of the Higgs potential inherited from the authors' earlier functional-measure work. The main unproven inputs are the EFT truncation, the perturbative effective potential, and the GW spectral templates used in the strong-transition regime.

free parameters (4)
  • C = Positive, scanned up to about 4 at Lambda = 1000 GeV
    Dimensionless coefficient of the logarithmic measure term; controls transition strength and effective Higgs trilinear. Scanned, not fitted to data.
  • Lambda = 500 to 3000 GeV
    Cutoff scale of the EFT measure correction; second scanned model parameter.
  • phi_stab = 2 TeV or 500-1000 TeV in different stability scenarios
    Field value up to which the potential is required to remain real or stable; chosen by hand, and the paper notes the 2 TeV choice excludes most of the interesting parameter space.
  • v_w = 1
    Assumed bubble wall velocity; affects GW amplitude linearly and peak frequency. The paper states this is an approximation within the correct order of magnitude.
assumptions (5)
  • domain assumption The functional measure contributes V_measure = -Lambda^4/(8 pi^2) ln(A + B phi-dagger phi / Lambda^2) to the effective potential (Eq. 17).
    Taken from ref. [13] with overlapping authorship; the present paper does not rederive it. If dimensional regularization neutralizes the measure or the metric choice is different, the model's new term disappears.
  • domain assumption The configuration-space metric G_IJ is truncated at O(Lambda^-2) with coefficients c1 to c8 (Eq. 12).
    Higher-order terms in the metric would generate additional operators beyond the logarithmic term, altering the phase transition and the GW spectrum.
  • domain assumption The 1-loop zero-temperature plus finite-temperature effective potential with daisy resummation is reliable (Eqs. 19 and 29).
    No estimate of two-loop or non-perturbative corrections is given; the strongest transitions are near the boundary where perturbativity is questionable.
  • domain assumption The GW signal is dominated by the sound-wave contribution (Eq. 40) with suppression Ysup; turbulence is set to zero and the scalar field contribution is neglected.
    The chosen sound-wave fit is validated only for alpha much less than 1, as the paper notes, while LISA-detectable points have alpha ~ 1.
  • standard math The phase transition completes when S3/T is about 140 (Eq. 36), and standard Hubble-redshifted GW formulas apply.
    Standard semiclassical nucleation and cosmology; footnote 13 notes that the one-bubble-per-horizon criterion is an approximation.

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Cite this review

Pith. "Pith review of Cosmological phase transitions from the functional measure." pith.science (2026). https://pith.science/paper/BJNXZQLU

@misc{pith2026250209593,
  author       = {Pith},
  title        = {Pith review of: Cosmological phase transitions from the functional measure},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BJNXZQLU}},
  note         = {Machine review of arXiv:2502.09593}
}
read the original abstract

We investigate how competitive gravitational wave detectors can be to current and near-future colliders in probing a model where the new physics is completely encapsulated in a modified scalar sector. For this, we study a model where an additional logarithmic term arises in the scalar potential due to a non-trivial path integral measure, which is constructed using effective field theory. This new term alters the dynamics of cosmological phase transitions and could lead to potentially detectable gravitational waves from the early Universe. Our results confirm the expectation that the intensity of such spectrum is highly correlated to the scalar field's self-coupling, and that gravitational wave experiments could therefore be used to probe these couplings with an accuracy competitive with the projected sensitivity of near-future colliders.

Figures

Figures reproduced from arXiv: 2502.09593 by the authors.

Figure 1
Figure 1. Normalized shapes of the potential at zero temper [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Normalized shape of the potential at zero temper [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Contour plot for the values of the transition strength [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Typical gravitational wave spectra for a fixed value [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 6
Figure 6. Figure 6: Contour plot for (above) α and (below) β in the (Λ, C) plane. Colored regions correspond to points within detectability range of different interferometers (obtained by requiring the theoretical amplitude to be larger than the PISC at the peak frequency). The gray regio…
Figure 7
Figure 7. Figure 7: Dashed curves show the values of C and Λ for which this model predicts a cosmological gravitational wave background with SNR = 1 at LISA, DECIGO and BBO. The gray dotted lines represent the values of ϕstab at which the potential becomes complex-valued. Also shown are t…

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

72 extracted references · 26 canonical work pages

  1. [63]

    A. J. Long and J. Turner, Thermal pressure on ultrarela- tivistic bubbles from a semiclassical formalism , JCAP 11 (2024) 024 [ 2407.18196]

    arXiv 2024

  2. [1]

    (33) and (32) to find the nucleation rate per unit volume, which then allows us to compute the number of nucleated bubbles per Hubble horizon

    Having found a solution, one can plug it back into Eqs. (33) and (32) to find the nucleation rate per unit volume, which then allows us to compute the number of nucleated bubbles per Hubble horizon. The nucleation temperature Tn is defined as the temperature at which this number is unity. For phase transitions occurring at the electroweak scale O(100 GeV)...

    work page

  3. [2]

    LIGO Scientific, Virgocollaboration, Observation of Gravitational Waves from a Binary Black Hole Merger , Phys. Rev. Lett. 116 (2016) 061102 [ 1602.03837]

    arXiv 2016

  4. [3]

    LIGO Scientific, Virgo collaboration, GWTC-1: A Gravitational-Wave Transient Catalog of Compact Bi- nary Mergers Observed by LIGO and Virgo during the First and Second Observing Runs, Phys. Rev. X 9 (2019) 031040 [1811.12907]

    arXiv 2019

  5. [4]

    LIGO Scientific, VIRGO collaboration, GWTC-2.1: Deep extended catalog of compact binary coalescences ob- served by LIGO and Virgo during the first half of the third observing run , Phys. Rev. D 109 (2024) 022001 [2108.01045]

    arXiv 2024

  6. [5]

    KAGRA, VIRGO, LIGO Scientific collaboration, GWTC-3: Compact Binary Coalescences Observed by LIGO and Virgo during the Second Part of the Third Observing Run , Phys. Rev. X 13 (2023) 041039 [2111.03606]

    arXiv 2023

  7. [6]

    NANOGrav collaboration, The NANOGrav 15 yr Data Set: Evidence for a Gravitational-wave Background , As- trophys. J. Lett. 951 (2023) L8 [ 2306.16213]

    arXiv 2023

  8. [7]

    Search for gravitational wave signals, Astron

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

    arXiv 2023

Show all 72 references
  1. [8]

    Arcadi, G

    G. Arcadi, G. C. Dorsch, J. P. Neto, F. S. Queiroz and Y. M. Oviedo-Torres, Probing a dark sector with collider physics, direct detection, and gravitational waves , Phys. Lett. B 848 (2024) 138382 [ 2307.06376]

    2024 arXiv

  2. [9]

    , Nature 607 (2022) 60 [ 2207.00043]

    CMS collaboration, A portrait of the Higgs boson by the CMS experiment ten years after the discovery. , Nature 607 (2022) 60 [ 2207.00043]

    2022 arXiv

  3. [10]

    Di Vita, C

    S. Di Vita, C. Grojean, G. Panico, M. Riembau and T. Vantalon, A global view on the Higgs self-coupling , JHEP 09 (2017) 069 [ 1704.01953]

    2017 arXiv

  4. [11]

    Di Vita, G

    S. Di Vita, G. Durieux, C. Grojean, J. Gu, Z. Liu, G. Panico et al., A global view on the Higgs self-coupling at lepton colliders , JHEP 02 (2018) 178 [ 1711.03978]

    2018 arXiv

  5. [12]

    M. L. Mangano, G. Ortona and M. Selvaggi, Measur- ing the Higgs self-coupling via Higgs-pair production at a 13 100 TeV p-p collider , Eur. Phys. J. C 80 (2020) 1030 [2004.03505]

    2020 arXiv

  6. [13]

    Kuntz and R

    I. Kuntz and R. da Rocha, Transport coefficients in AdS/CFT and quantum gravity corrections due to a functional measure , Nucl. Phys. B 993 (2023) 116258 [2211.11913]

    2023 arXiv

  7. [14]

    Kuntz and A

    I. Kuntz and A. Malagi, Constraining the UV with the electroweak effective action , JHEP 12 (2025) 210 [2404.06987]

    2025 arXiv

  8. [15]

    K. G. Wilson, The renormalization group and critical phenomena, Rev. Mod. Phys. 55 (1983) 583

    1983

  9. [16]

    Costello, Renormalization and Effective Field Theory, vol

    K. Costello, Renormalization and Effective Field Theory, vol. 170. American Mathematical Society, 2022

    2022

  10. [17]

    R. K. Unz, Path Integration and the Functional Measure, Nuovo Cim. A 92 (1986) 397

    1986

  11. [18]

    D. J. Toms, The Functional Measure for Quantum Field Theory in Curved Space-time , Phys. Rev. D 35 (1987) 3796

    1987

  12. [19]

    Moretti, Direct zeta function approach and renormal- ization of one loop stress tensors in curved space-times , Phys

    V. Moretti, Direct zeta function approach and renormal- ization of one loop stress tensors in curved space-times , Phys. Rev. D 56 (1997) 7797 [ hep-th/9705060]

    1997 arXiv

  13. [20]

    Hatsuda, P

    M. Hatsuda, P. van Nieuwenhuizen, W. Troost and A. Van Proeyen, The Regularized Phase Space Path Inte- gral Measure for a Scalar Field Coupled to Gravity , Nucl. Phys. B 335 (1990) 166

    1990

  14. [21]

    van Nieuwenhuizen, Consistent anomalies from Hamil- tonian path-integrals , Nucl

    P. van Nieuwenhuizen, Consistent anomalies from Hamil- tonian path-integrals , Nucl. Phys. B Proc. Suppl. 16 (1990) 605

    1990

  15. [22]

    Armendariz-Picon, J

    C. Armendariz-Picon, J. T. Neelakanta and R. Penco, General Covariance Constraints on Cosmological Corre- lators, JCAP 01 (2015) 035 [ 1411.0036]

    2015 arXiv

  16. [23]

    Becker and M

    M. Becker and M. Reuter, Background Independent Field Quantization with Sequences of Gravity-Coupled Approx- imants, Phys. Rev. D 102 (2020) 125001 [ 2008.09430]

    2020 arXiv

  17. [24]

    I. L. Buchbinder and S. L. Lyakhovich, Canonical Quan- tization and Local Measure of R**2 Gravity , Class. Quant. Grav. 4 (1987) 1487

    1987

  18. [25]

    Hamamoto and M

    S. Hamamoto and M. Nakamura, Path integral measures in higher derivative gravities , Prog. Theor. Phys. 104 (2000) 691 [ hep-th/0005131]

    2000 arXiv

  19. [26]

    B. S. DeWitt, The global approach to quantum field the- ory. Vol. 1, 2nd ed. , vol. 114. 2003

    2003

  20. [27]

    K. Finn, S. Karamitsos and A. Pilaftsis, Frame Covari- ance in Quantum Gravity , Phys. Rev. D 102 (2020) 045014 [1910.06661]

    2020 arXiv

  21. [28]

    G. A. Vilkovisky, The Unique Effective Action in Quan- tum Field Theory , Nucl. Phys. B 234 (1984) 125

    1984

  22. [29]

    E. S. Fradkin and G. A. Vilkovisky, On Renormalization of Quantum Field Theory in Curved Space-Time , Lett. Nuovo Cim. 19 (1977) 47

    1977

  23. [30]

    E. S. Fradkin and G. A. Vilkovisky, S matrix for gravita- tional field. ii. local measure, general relations, elements of renormalization theory , Phys. Rev. D 8 (1973) 4241

    1973

  24. [31]

    Fujikawa, Path Integral Measure for Gauge Invariant Fermion Theories, Phys

    K. Fujikawa, Path Integral Measure for Gauge Invariant Fermion Theories, Phys. Rev. Lett. 42 (1979) 1195

    1979

  25. [32]

    Fujikawa, Path Integral for Gauge Theories with Fermions, Phys

    K. Fujikawa, Path Integral for Gauge Theories with Fermions, Phys. Rev. D 21 (1980) 2848

    1980

  26. [33]

    Fujikawa, Comment on Chiral and Conformal Anoma- lies, Phys

    K. Fujikawa, Comment on Chiral and Conformal Anoma- lies, Phys. Rev. Lett. 44 (1980) 1733

    1980

  27. [34]

    J. a. M. L. de Freitas and I. Kuntz,Massive graviton from diffeomorphism invariance, 2307.13803

    arXiv

  28. [35]

    Casadio, I

    R. Casadio, I. Kuntz and R. da Rocha, When grav- itational decoupling and quantum gravity (re)unite , 2403.13099

    arXiv

  29. [36]

    G. C. Dorsch, S. J. Huber, K. Mimasu and J. M. No, The Higgs Vacuum Uplifted: Revisiting the Electroweak Phase Transition with a Second Higgs Doublet , JHEP 12 (2017) 086 [ 1705.09186]

    2017 arXiv

  30. [37]

    E. J. Weinberg and A.-q. Wu, Understanding complex perturbative effective potentials, Phys. Rev. D 36 (1987) 2474

    1987

  31. [38]

    ATLAS collaboration, Constraints on the Higgs boson self-coupling from single- and double-Higgs production with the ATLAS detector using pp collisions at s=13 TeV, Phys. Lett. B 843 (2023) 137745 [ 2211.01216]

    2023 arXiv

  32. [39]

    Laine and A

    M. Laine and A. Vuorinen, Basics of Thermal Field The- ory, vol. 925. Springer, 2016, 10.1007/978-3-319-31933-9, [1701.01554]

    2016 arXiv

  33. [40]

    M. B. Hindmarsh, M. L¨ uben, J. Lumma and M. Pauly, Phase transitions in the early universe , SciPost Phys. Lect. Notes 24 (2021) 1 [ 2008.09136]

    2021 arXiv

  34. [41]

    J. R. Espinosa, M. Quiros and F. Zwirner, On the nature of the electroweak phase transition , Phys. Lett. B 314 (1993) 206 [ hep-ph/9212248]

    1993 arXiv

  35. [42]

    D’Onofrio and K

    M. D’Onofrio and K. Rummukainen, Standard model cross-over on the lattice , Phys. Rev. D 93 (2016) 025003 [1508.07161]

    2016 arXiv

  36. [43]

    Caprini et al., Science with the space-based interfer- ometer eLISA

    C. Caprini et al., Science with the space-based interfer- ometer eLISA. II: Gravitational waves from cosmological phase transitions, JCAP 04 (2016) 001 [ 1512.06239]

    2016 arXiv

  37. [44]

    Caprini et al., Detecting gravitational waves from cos- mological phase transitions with LISA: an update , JCAP 03 (2020) 024 [ 1910.13125]

    C. Caprini et al., Detecting gravitational waves from cos- mological phase transitions with LISA: an update , JCAP 03 (2020) 024 [ 1910.13125]

    2020 arXiv

  38. [45]

    Mukhanov, Physical Foundations of Cosmol- ogy

    V. Mukhanov, Physical Foundations of Cosmol- ogy. Cambridge University Press, Oxford, 2005, 10.1017/CBO9780511790553

    2005 doi

  39. [46]

    Coleman, Fate of the false vacuum: Semiclassical the- ory, Phys

    S. Coleman, Fate of the false vacuum: Semiclassical the- ory, Phys. Rev. D 15 (1977) 2929

    1977

  40. [47]

    Athron, C

    P. Athron, C. Bal´ azs, A. Fowlie, L. Morris and L. Wu, Cosmological phase transitions: From perturbative par- ticle physics to gravitational waves , Progress in Particle and Nuclear Physics 135 (2024) 104094

    2024

  41. [48]

    Giese, T

    F. Giese, T. Konstandin and J. van de Vis, Model- independent energy budget of cosmological first-order phase transitions—A sound argument to go beyond the bag model, JCAP 07 (2020) 057 [ 2004.06995]

    2020 arXiv

  42. [49]

    Schmitz, New Sensitivity Curves for Gravitational- Wave Signals from Cosmological Phase Transitions , JHEP 01 (2021) 097 [ 2002.04615]

    K. Schmitz, New Sensitivity Curves for Gravitational- Wave Signals from Cosmological Phase Transitions , JHEP 01 (2021) 097 [ 2002.04615]

    2021 arXiv

  43. [50]

    Enqvist, J

    K. Enqvist, J. Ignatius, K. Kajantie and K. Rum- mukainen, Nucleation and bubble growth in a first order cosmological electroweak phase transition , Phys. Rev. D 45 (1992) 3415

    1992

  44. [51]

    De Curtis, L

    S. De Curtis, L. D. Rose, A. Guiggiani, A. G. Muyor and G. Panico, Bubble wall dynamics at the electroweak phase transition, JHEP 03 (2022) 163 [ 2201.08220]

    2022 arXiv

  45. [52]

    Laurent and J

    B. Laurent and J. M. Cline, First principles determina- tion of bubble wall velocity , Phys. Rev. D 106 (2022) 023501 [2204.13120]

    2022 arXiv

  46. [53]

    G. D. Moore and T. Prokopec, How fast can the wall move? A Study of the electroweak phase transition dy- namics, Phys. Rev. D 52 (1995) 7182 [hep-ph/9506475]

    1995 arXiv

  47. [54]

    G. C. Dorsch and D. A. Pinto, Bubble wall velocities with an extended fluid Ansatz , JCAP 04 (2024) 027 [2312.02354]

    2024 arXiv

  48. [55]

    G. C. Dorsch, T. Konstandin, E. Perboni and D. A. Pinto, Non-singular solutions to the Boltzmann equation with a fluid Ansatz , 2412.09266. 14

    arXiv

  49. [56]

    G. C. Dorsch, S. J. Huber and T. Konstandin,On the wall velocity dependence of electroweak baryogenesis , JCAP 08 (2021) 020 [ 2106.06547]

    2021 arXiv

  50. [57]

    J. R. Espinosa, T. Konstandin, J. M. No and G. Ser- vant, Energy Budget of Cosmological First-order Phase Transitions, JCAP 06 (2010) 028 [ 1004.4187]

    2010 arXiv

  51. [58]

    Ekstedt, O

    A. Ekstedt, O. Gould, J. Hirvonen, B. Laurent, L. Niemi, P. Schicho et al., How fast does the WallGo? A package for computing wall velocities in first-order phase transi- tions, 2411.04970

    arXiv

  52. [59]

    Athron, C

    P. Athron, C. Bal´ azs and L. Morris, Supercool subtleties of cosmological phase transitions , JCAP 03 (2023) 006 [2212.07559]

    2023

  53. [60]

    H¨ oche, J

    S. H¨ oche, J. Kozaczuk, A. J. Long, J. Turner and Y. Wang, Towards an all-orders calculation of the electroweak bubble wall velocity , JCAP 03 (2021) 009 [2007.10343]

    2021 arXiv

  54. [61]

    Azatov and M

    A. Azatov and M. Vanvlasselaer, Bubble wall velocity: heavy physics effects, JCAP 01 (2021) 058 [2010.02590]

    2021 arXiv

  55. [62]

    W.-Y. Ai, X. Nagels and M. Vanvlasselaer, Criterion for ultra-fast bubble walls: the impact of hydrodynamic ob- struction, JCAP 03 (2024) 037 [ 2401.05911]

    2024 arXiv

  56. [64]

    Hindmarsh, S

    M. Hindmarsh, S. J. Huber, K. Rummukainen and D. J. Weir, Shape of the acoustic gravitational wave power spec- trum from a first order phase transition, Phys. Rev. D 96 (2017) 103520 [ 1704.05871]

    2017 arXiv

  57. [65]

    Hindmarsh, S

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

    2014 arXiv

  58. [66]

    Ellis, M

    J. Ellis, M. Lewicki and J. M. No, On the Maximal Strength of a First-Order Electroweak Phase Transition and its Gravitational Wave Signal , JCAP 04 (2019) 003 [1809.08242]

    2019 arXiv

  59. [67]

    Ellis, M

    J. Ellis, M. Lewicki and J. M. No, Gravitational waves from first-order cosmological phase transitions: life- time of the sound wave source , JCAP 07 (2020) 050 [2003.07360]

    2020 arXiv

  60. [68]

    Alves, D

    A. Alves, D. Gon¸ calves, T. Ghosh, H.-K. Guo and K. Sinha, Di-Higgs Production in the 4b Channel and Gravitational Wave Complementarity , JHEP 03 (2020) 053 [1909.05268]

    2020 arXiv

  61. [69]

    Ellis, M

    J. Ellis, M. Lewicki, J. M. No and V. Vaskonen, Gravi- tational wave energy budget in strongly supercooled phase transitions, JCAP 06 (2019) 024 [ 1903.09642]

    2019 arXiv

  62. [70]

    Thrane and J

    E. Thrane and J. D. Romano, Sensitivity curves for searches for gravitational-wave backgrounds , Phys. Rev. D 88 (2013) 124032 [ 1310.5300]

    2013 arXiv

  63. [71]

    Gowling, M

    C. Gowling, M. Hindmarsh, D. C. Hooper and J. Tor- rado, Reconstructing physical parameters from template gravitational wave spectra at LISA: first order phase tran- sitions, JCAP 04 (2023) 061 [ 2209.13551]

    2023 arXiv

  64. [72]

    LISA Cosmology Working Group collaboration, Gravitational waves from first-order phase transitions in LISA: reconstruction pipeline and physics interpretation , JCAP 10 (2024) 020 [ 2403.03723]

    2024 arXiv

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