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Phase transitions in an expanding medium -- hot remnants

T0 review · 3 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read In an expanding medium, a first-order phase transition can leave hot remnants of the high-energy phase that remain at the transition temperature while shrinking, and then heat up further as they dissolve.

desk verdict A credible model-based discovery of hot remnants in expanding plasma, with robust scaling laws in the shrinking stage and a load-bearing zero-dissipation assumption in the dissolution stage. read the letter →

arxiv 2502.07879 v2 pith:DO6OWUU2 submitted 2025-02-11 hep-th astro-ph.COhep-ph

Romuald A. Janik , Matti Jarvinen , Jacob Sonnenschein This is my paper · ORCID
classification hep-thastro-ph.COhep-ph
keywords first-orderphasetransitiondeconfinementhotremnantsholographicconfinementmodelboost-invariantexpansionFRWcosmologyentropyconservationdomainwall
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

The paper argues that the standard picture of a first-order phase transition in an expanding medium — supercool, then nucleate bubbles of the low-temperature phase — is incomplete. Using a holography-inspired effective model of the confinement/deconfinement transition, it shows that regions of the high-energy phase can persist as 'hot remnants' whose temperature stays pinned near the critical temperature $T_c$ even as the medium keeps expanding. These remnants shrink rather than nucleate bubbles; once small enough, they dissolve, and the remaining plasma heats up above $T_c$. The same qualitative stages appear in boost-invariant flat-space expansion and in cosmological FRW expansion, suggesting a generic mechanism rather than a peculiarity of a specific geometry. The paper's conclusion matters because hot remnants would alter predictions for heavy-ion collisions and for first-order phase transitions in the early universe.

What carries the argument

The central object is an effective boundary model in which the total energy-momentum tensor is a mixture of a perfect-fluid deconfined phase and a trivial confined phase, controlled by a dynamical order parameter $\gamma$ (the 'mixing fraction'), plus a domain-wall contribution. The dynamics of $\gamma$ come from a Lagrangian with double-well potential $V_{TOT}(\gamma,T)$, whose explicit temperature dependence couples the order parameter to hydrodynamics through the formalism of [13]. This coupling is such that the model needs no dissipation terms and still reproduces holographic domain-wall velocities [16]. The argument is carried by the exactly conserved entropy current $j^\mu = -\partial_T V_{TOT}(\gamma,T)\,u^\mu$: entropy conservation plus the virial approximation $c\,\gamma'^2=2V_{TOT}(\gamma,T)$ yields the shrinking and dissolution scalings and the late-time heating.

What would settle it

Rerun the boost-invariant simulation with the dissipative or friction terms that the paper deliberately omits, using the same initial conditions: if a blob of deconfined matter near $T_c$ then cools below $T_c$, nucleates bubbles, or dissolves without reheating above $T_c$, the claim of generic hot remnants is falsified; a weaker test is to measure the entropy production rate, since the scaling laws require exact conservation.

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Extended reading notes

Core claim

The central discovery is that in an expanding medium, first-order confinement/deconfinement transitions do not simply proceed by supercooling and bubble nucleation. Instead, trapped regions of the deconfined high-energy phase remain at a temperature near the critical temperature $T_c$ while the rest of the system cools: latent heat released as the surrounding confined phase expands keeps the remnant hot. These hot remnants shrink, and only when their width reaches the domain-wall scale do they dissolve; during dissolution the residual deconfined plasma is heated further, moving above $T_c$. The paper derives scaling laws for each stage — $L(\tau)\sim 1/\tau$ for boost-invariant expansion and $L(t)\sim 1/a(t)^3$ in an expanding FRW universe, then $1-\gamma\sim1/\sqrt{\tau}$ with rising temperature — and shows that all of it follows from exact entropy conservation in a dissipation-free hydrodynamic model with a dynamical mixing fraction.

Load-bearing premise

The load-bearing premise is that the model has no dissipation or friction, so the entropy current is exactly conserved and the remnant's behavior is derived from that conservation; if real plasma viscosity is significant, the remnants could cool, nucleate bubbles, or dissolve away without reheating.

Editorial extensions

If this is right

  • In an expanding medium that passes through a first-order transition, bubble nucleation can be suppressed because trapped supercooled plasma is reheated to the transition temperature by latent heat from the shrinking boundary.
  • The remnant size shrinks as $L(\tau)\sim 1/\tau$ in boost-invariant flow and $L(t)\sim 1/a(t)^3$ in an FRW universe, giving clean scaling predictions for numerical and possibly experimental searches.
  • During the final dissolution stage, the residual deconfined plasma heats up above $T_c$ while $1-\gamma\sim 1/\sqrt{\tau}$, a counterintuitive signature that distinguishes this mechanism from ordinary supercooled droplet collapse.
  • Because the same stages appear in flat and cosmological geometries, the effect is expected in any slowly expanding medium with a first-order transition and negligible dissipation.

Reading between the lines

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

  • If the mechanism is as generic as claimed, first-order transitions in the early universe could leave long-lived hot regions whose subsequent decay produces gravitational waves and particle abundances different from the standard spherical-bubble scenario; the paper identifies this as the most promising application but does not compute it.
  • In the holographic dual, the dissolving remnant should correspond to a shrinking black hole that eventually disappears into the confining geometry; the paper notes this is an open problem, and a bulk simulation would provide a sharp test.
  • Adding a small bulk or shear viscosity to the same model would test whether the heating stage persists; the paper explicitly works in the dissipation-free limit, so this is an extension, not a claim.
  • The virial-theorem explanation ties the reheating to the shape of the Landau potential, so a different potential shape could reverse the effect; the claimed robustness may therefore be limited to models fitted to the same domain-wall data.
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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 / 5 minor

Summary. The paper studies a first-order confinement/deconfinement phase transition in an expanding medium using an effective boundary model fitted to the holographic Witten model. The model couples boost-invariant (or FRW) hydrodynamics to a scalar order parameter γ that interpolates between deconfined and confined phases. The authors report a generic dynamical outcome: after supercooling and bubble nucleation, trapped regions of deconfined plasma form 'hot remnants' that cool to Tc, shrink while releasing latent heat, and then dissolve, with the remaining deconfined fraction decreasing as 1−γ ~ 1/√τ while the temperature rises above Tc. The shrinking and dissolution stages are reproduced in both flat-space boost-invariant expansion and expanding FRW spacetime. The paper interprets the shrinking stage via energy conservation and the dissolution stage via the exactly conserved entropy current combined with a virial approximation for the scalar profile.

Significance. If the findings are robust, they challenge the standard picture in which a first-order transition in an expanding medium proceeds purely via bubble nucleation and cooling below Tc. The hot-remnant phenomenon and the reheating during dissolution could be relevant for heavy-ion collisions and for early-universe phase transitions. The paper provides transparent physical arguments, two independent numerical setups, and analytic scalings (L ~ 1/τ, 1−γ ~ 1/√τ) that are directly confronted with the numerics. The authors also supply a detailed supplemental description of the numerical scheme, including regularization, grid settings, and convergence checks for several runs. These are concrete strengths. The main limitations are the zero-dissipation idealization, the reliance on a potential fitted only near Tc, and the sensitivity of the dissolution onset to the numerical filter; these affect the headline claim that the phenomenon is 'very generic'.

major comments (3)
  1. [Supplemental Material, 'Entropy current' and main text, p. 2] The dissolution-stage scaling 1−Γ ~ 1/τ (equivalently 1−γ ~ 1/√τ) and the associated temperature rise are derived from the exactly conserved entropy current j^μ = −∂_T V_TOT u^μ (Eq. A.14 in the Supplemental Material). This conservation law requires the absence of dissipation, as the authors explicitly state: 'we do not have any dissipation terms which are usually added'. Real plasmas possess shear and bulk viscosity and order-parameter friction, which would make ∂_μ j^μ > 0. The paper does not quantify how the reported scalings and the reheating would be modified by a dissipative term of physically motivated size. Since the abstract and discussion claim that hot remnants appear 'very generically', this idealization is load-bearing; a test with a small dissipative term, or at least an order-of-magnitude estimate, is needed to support the extrapolation beyond the zero-dissipation model.
  2. [p. 2, paragraph on the potential; Figs. 6 and 9] The 'virial theorem prediction' for the temperature rise during dissolution reads off the zeros of V_TOT(γ*, T)=0, but V_TOT is the same fitted potential whose form is used 'also away from Tc' even though it 'was fitted only in the neighbourhood of Tc = 1 up to its first T-derivative'. The reheating above Tc is therefore not an independent prediction: it follows from an extrapolation of the fitted potential. The paper should either test the sensitivity of the dissolution-stage temperature rise to the form of V_TOT away from Tc (e.g., by varying β(γ) or adding higher-order terms) or soften the claim that the reheating is a generic consequence of the holographic model rather than of the chosen extrapolation.
  3. [Supplemental Material, 'Some details on the numerical simulations'] The dissolution-onset timescale is reported to be sensitive to the numerical filter: 'the results in Figs. 6, 7, 8 and 9 suffer from a mild uncertainty in the timescale where the γ field profile starts to dissolve. This timescale turns out to be sensitive to the filter we are using.' Figures 6 and 9 are exactly the figures that establish the counterintuitive reheating stage. Since the filter is a numerical artifact, the uncertainty in the onset time propagates to the duration and quantitative details of the dissolution stage. The paper should provide a quantitative estimate of how much this uncertainty affects the extracted scalings and the temperature-rise curve, rather than only stating that the predictions of the virial theorem are reproduced.
minor comments (5)
  1. [p. 3, Eq. (3)] The potential V_TOT(γ, T) is written with a term '+1' at the end; it would help to clarify that this constant shifts the overall zero of the potential and does not affect the equations of motion.
  2. [p. 4, Eq. (12)] The profile ansatz 1−γ ∼ b(τ)/cosh^4(d(τ) q_* x / 4) is introduced with parameters b(τ) and d(τ); the rationale for the specific power 4 and the factor 1/4 in the argument would be clearer if tied directly to the potential approximation in Eq. (14).
  3. [p. 4, after Eq. (14)] The text says 'expanding s = −∂_T V_TOT around γ = 1 gives 1−Γ ∼ 1/τ' but does not show the intermediate steps; for a letter this is acceptable, but a one-line derivation in the Supplemental Material would improve reproducibility.
  4. [p. 5, FRW section] The statement 'we checked that similar results hold also e.g. for a(t) ∼ e^{Ht} with small H' is not accompanied by any quantitative detail; adding a brief comment on the range of H and the observed scalings would strengthen the genericity claim.
  5. [References] The reference list would benefit from the page numbers or article numbers for the arXiv:2411.17806 paper (Ref. [23]) and for the published versions of Refs. [27] and [28].

Circularity Check

0 steps flagged · score 2.0 of 10

Hot-remnant phenomenon is an emergent dynamical result of a model fitted to external static-domain-wall data; no load-bearing circular reduction found, only minor self-citations.

full rationale

The model (Eqs. (1)-(3)) is fitted to the external AMW static domain-wall data [3], not to the remnant dynamics or to the dissolution-stage heating; the central claims—hot remnant formation, L ~ 1/τ shrinking, 1−γ ~ 1/√τ dissolution, and T rising above Tc—are obtained by solving the model in two different expanding backgrounds. The virial-theorem and entropy-conservation arguments are analytic approximations or exact consequences of the same Lagrangian, so their agreement with the numerics is an internal self-consistency check rather than a test of the model against new data; this reduces the epistemic weight of those 'predictions' but does not make them circular, because the fitted inputs (c, q*, β) do not already contain the target outcomes. The no-dissipation assumption is the physically fragile point: entropy-current conservation (Supplement) and the resulting 1−γ ~ 1/√τ scaling rely on it, and real plasmas have viscosity and order-parameter friction. The paper justifies this assumption in part by self-citation to [16], but the mathematical derivation is self-contained and does not import a conclusion from that citation. Thus no load-bearing circularity is present; the score of 2 reflects the minor self-citations and the internal-consistency nature of the analytic 'predictions,' not a definitional reduction.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The paper's central claim rests on a calibrated effective model. The free parameters are the model's fitted coefficients and the extrapolated potential. The most important axioms are the validity of the model as a proxy for holographic dynamics and the absence of dissipation, both of which are explicit but not independently verified. No new entities are postulated.

free parameters (4)
  • c (domain-wall tension coefficient) = ≈5.892
    Fitted to the AMW domain-wall solution in ref [3]; controls the surface tension and enters all dynamics through the potential and kinetic terms.
  • β(γ) potential exponent = 2(1+Γ) in 3d; 4 in 4d
    Determined by fit to AMW data in ref [1] for the 2+1-dimensional case; in the 3+1-dimensional FRW case it is chosen as a simpler option because no strict fit is available.
  • Extrapolation of VTOT away from Tc
    The double-well potential was fitted only in the neighborhood of Tc up to its first T-derivative, but is used 'for definiteness' at all temperatures; this extrapolation directly controls the dissolution-stage heating.
  • Initial seed amplitude and width = amplitude 0.75, width 4.4 (Fig. 1 case)
    Chosen by hand to mimic fluctuations; the claimed robustness across scenarios reduces dependence, but specific dynamics in Fig. 1 depends on these choices.
assumptions (6)
  • standard math Energy-momentum conservation in time-dependent geometries, Eq. (5) of ref [26].
    Used to derive the integrated conservation equations (6) and (16) that yield the L~1/τ and L~1/a^3 scalings.
  • domain assumption The effective boundary model is a valid proxy for the holographic Witten model in time-dependent, expanding backgrounds.
    The model is fitted to a static domain wall (AMW solution) from ref [3]; extrapolation to dynamics is assumed, since bulk numerical relativity is 'extremely challenging'.
  • domain assumption The confined phase has a trivial, temperature-independent stress tensor (T=η).
    This is a property of the Witten model used in Eq. (1); it means the cold phase does not participate in hydrodynamics.
  • domain assumption The mixed phase has no dissipation, so the entropy current is exactly conserved.
    No dissipation terms are added (p.2); this conservation is used to derive 1−γ ~ 1/√τ and the virial-theorem heating.
  • domain assumption No back-reaction of the phase transition on the FRW scale factor.
    The paper states 'the back-reaction of the matter system on the cosmological expansion is not taken into account' (p.4), so the a(t)~t^{1/2} background is fixed.
  • ad hoc to paper The mixing fraction Γ(γ)=γ^2(3−2γ) and the form of the potential used beyond the fitted region.
    The functional form is motivated by the AMW profile but extended ad hoc, especially for γ<0 and γ>1 where the map is clipped to 0/1, affecting the equations in those regions.

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Pith. "Pith review of Phase transitions in an expanding medium -- hot remnants." pith.science (2026). https://pith.science/paper/DO6OWUU2

@misc{pith2026250207879,
  author       = {Pith},
  title        = {Pith review of: Phase transitions in an expanding medium -- hot remnants},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DO6OWUU2}},
  note         = {Machine review of arXiv:2502.07879}
}
read the original abstract

We analyze the dynamics of a first order confinement/deconfinement phase transition in an expanding medium using an effective boundary description fitted to the holographic Witten model. We observe and analyze hot plasma remnants, which do not cool down or nucleate bubbles despite the expansion of the system. The appearance of the hot remnants, the dynamics of their shrinking and subsequent dissolution and further heating up is very robust and persists in such diverse scenarios as boost-invariant expansion with a flat Minkowski metric and cosmological expansion in a Friedmann-Robertson-Walker spacetime.

Figures

Figures reproduced from arXiv: 2502.07879 by the authors.

Figure 1
Figure 1. FIG. 1. Evolution of the confined ( [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The potential at [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The central plasma region of Fig. 1. Hydrodynamic [PITH_FULL_IMAGE:figures/full_fig_p002_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: FIG. 4. The evolution of the size of the shrinking remnant [PITH_FULL_IMAGE:figures/full_fig_p003_4.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Evolution of [PITH_FULL_IMAGE:figures/full_fig_p004_7.png]
Figure 6
Figure 6. Figure 6: (right)). It is important to note, that T in our model denotes the (hydrodynamic) temperature of the deconfined phase, as the confined phase in the Witten model is temperature independent. For the description of the profile ( [PITH_FULL_IMAGE:figures/full_fig_p004_6.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Evolution of the temperature in the middle of the [PITH_FULL_IMAGE:figures/full_fig_p005_9.png]

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Forward citations

Cited by 1 Pith paper

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

  1. Testing the effective action approach to bubble nucleation in holography

    hep-th 2025-07 conditional novelty 5.0 of 10

    A two-derivative holographic effective action reproduces critical bubble solutions from the full gravity theory to within a few percent across thin-wall and thick-wall regimes.

Reference graph

Works this paper leans on

38 extracted references · 10 canonical work pages · cited by 1 Pith paper

  1. [1]

    R. A. Janik, M. Järvinen, and J. Sonnenschein, A sim- ple description of holographic domain walls in confin- ing theories — extended hydrodynamics, JHEP09, 129, arXiv:2106.02642 [hep-th]

    arXiv

  2. [2]

    E.Witten,Anti-deSitterspace, thermalphasetransition, and confinement in gauge theories, Adv. Theor. Math. Phys. 2, 505 (1998), arXiv:hep-th/9803131

    arXiv 1998

  3. [3]

    high energy

    To this end, we can make a rough estimate by considering the conservation of energy with the shrink- ing of the transverse size of the blob releasing latent heat which can keep the temperature constant. In view of a similar computation in the FRW space- time, it is convenient to use an integrated version of the conservation equation in a general time-depe...

    work page

  4. [4]

    Aharony, S

    O. Aharony, S. Minwalla, and T. Wiseman, Plasma-balls in large N gauge theories and localized black holes, Class. Quant. Grav.23, 2171 (2006), arXiv:hep-th/0507219

    arXiv 2006

  5. [5]

    Aharony, J

    O. Aharony, J. Sonnenschein, and S. Yankielowicz, A Holographic model of deconfinement and chiral sym- metry restoration, Annals Phys. 322, 1420 (2007), arXiv:hep-th/0604161

    arXiv 2007

  6. [6]

    Gursoy, E

    U. Gursoy, E. Kiritsis, L. Mazzanti, and F. Nitti, De- confinement and Gluon Plasma Dynamics in Improved Holographic QCD, Phys. Rev. Lett.101, 181601 (2008), arXiv:0804.0899 [hep-th]

    arXiv 2008

  7. [7]

    Gursoy, E

    U. Gursoy, E. Kiritsis, L. Mazzanti, and F. Nitti, Holography and Thermodynamics of 5D Dilaton-gravity, JHEP 05, 033, arXiv:0812.0792 [hep-th]

    arXiv

  8. [8]

    T. Alho, M. Jarvinen, K. Kajantie, E. Kiritsis, and K. Tuominen, Quantum and stringy corrections to the equation of state of holographic QCD matter and the nature of the chiral transition, Phys. Rev. D91, 055017 (2015), arXiv:1501.06379 [hep-ph]

    arXiv 2015

Show all 38 references
  1. [9]

    Aref’eva and K

    I. Aref’eva and K. Rannu, Holographic Anisotropic Back- ground with Confinement-Deconfinement Phase Transi- tion, JHEP05, 206, arXiv:1802.05652 [hep-th]

    arXiv

  2. [10]

    Y. Bea, J. Casalderrey-Solana, T. Giannakopoulos, D. Mateos, M. Sanchez-Garitaonandia, and M. Zilhão, Bubble wall velocity from holography, Phys. Rev. D104, L121903 (2021), arXiv:2104.05708 [hep-th]

    2021 arXiv

  3. [11]

    Bantilan, P

    H. Bantilan, P. Figueras, and D. Mateos, Real-time Dy- namics of Plasma Balls from Holography, Phys. Rev. Lett. 124, 191601 (2020), arXiv:2001.05476 [hep-th]

    2020 arXiv

  4. [12]

    Bigazzi, A

    F. Bigazzi, A. Caddeo, A. L. Cotrone, and A. Paredes, Fate of false vacua in holographic first-order phase tran- sitions, JHEP12, 200, arXiv:2008.02579 [hep-th]

    2008 arXiv

  5. [13]

    Bigazzi, A

    F. Bigazzi, A. Caddeo, A. L. Cotrone, and A. Paredes, Dark Holograms and Gravitational Waves, JHEP 04, 094, arXiv:2011.08757 [hep-ph]

    2011 arXiv

  6. [14]

    F. M. Haehl, R. Loganayagam, and M. Ranga- mani, Adiabatic hydrodynamics: the eightfold way to dissipation, Journal of High Energy Physics 2015, 10.1007/jhep05(2015)060 (2015)

    2015 doi

  7. [15]

    Ignatius, K

    J. Ignatius, K. Kajantie, H. Kurki-Suonio, and M. Laine, The growth of bubbles in cosmological phase transitions, Phys. Rev. D49, 3854 (1994), arXiv:astro-ph/9309059

    1994 arXiv

  8. [16]

    Hindmarsh, S

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

    2015 arXiv

  9. [17]

    R. A. Janik, M. Jarvinen, H. Soltanpanahi, and J. Son- nenschein, Perfect Fluid Hydrodynamic Picture of Do- mainWallVelocitiesatStrongCoupling,Phys.Rev.Lett. 129, 081601 (2022), arXiv:2205.06274 [hep-th]

    2022 arXiv

  10. [18]

    J. D. Bjorken, Highly Relativistic Nucleus-Nucleus Col- lisions: The Central Rapidity Region, Phys. Rev. D27, 140 (1983)

    1983

  11. [19]

    R. A. Janik and R. B. Peschanski, Asymptotic perfect fluid dynamics as a consequence of Ads/CFT, Phys. Rev. D 73, 045013 (2006), arXiv:hep-th/0512162

    2006 arXiv

  12. [20]

    P. M. Chesler and L. G. Yaffe, Boost invariant flow, black hole formation, and far-from-equilibrium dynamics in N = 4 supersymmetric Yang-Mills theory, Phys. Rev. D82, 026006 (2010), arXiv:0906.4426 [hep-th]

    2010 arXiv

  13. [21]

    M. P. Heller, R. A. Janik, and P. Witaszczyk, The char- acteristics of thermalization of boost-invariant plasma from holography, Phys. Rev. Lett.108, 201602 (2012), arXiv:1103.3452 [hep-th]

    2012 arXiv

  14. [22]

    M. P. Heller, R. A. Janik, and P. Witaszczyk, Hydrody- namic Gradient Expansion in Gauge Theory Plasmas, Phys. Rev. Lett. 110, 211602 (2013), arXiv:1302.0697 [hep-th]

    2013 arXiv

  15. [23]

    Gursoy, M

    U. Gursoy, M. Jarvinen, and G. Policastro, Late time behavior of non-conformal plasmas, JHEP 01, 134, arXiv:1507.08628 [hep-th]

    arXiv

  16. [24]

    Aragonès Fontboté, D

    M. Aragonès Fontboté, D. Mateos, G. P. Martín, W. van der Schee, and J. G. Subils, Cosmic censorship in a (dual) collider, (2024), arXiv:2411.17806 [hep-th]

    2024 arXiv

  17. [25]

    Li, S.-J

    L. Li, S.-J. Wang, and Z.-Y. Yuwen, Bubble expansion at strong coupling, Phys. Rev. D 108, 096033 (2023), arXiv:2302.10042 [hep-th]. 7

    2023 arXiv

  18. [26]

    Wang, Z.-Y

    J.-C. Wang, Z.-Y. Yuwen, Y.-S. Hao, and S.-J. Wang, General bubble expansion at strong coupling, Phys. Rev. D 109, 096012 (2024), arXiv:2311.07347 [hep-ph]

    2024 arXiv

  19. [27]

    Clough, Continuity equations for general matter: ap- plications in numerical relativity, Class

    K. Clough, Continuity equations for general matter: ap- plications in numerical relativity, Class. Quant. Grav.38, 167001 (2021), arXiv:2104.13420 [gr-qc]

    2021 arXiv

  20. [28]

    Ecker, E

    C. Ecker, E. Kiritsis, and W. van der Schee, Dynamical inflaton coupled to strongly interacting matter, Physi- cal Review Letters130, 10.1103/physrevlett.130.251001 (2023)

    2023 doi

  21. [29]

    Cutting, M

    D. Cutting, M. Hindmarsh, and D. J. Weir, Vorticity, ki- netic energy, and suppressed gravitational wave produc- tion in strong first order phase transitions, Phys. Rev. Lett. 125, 021302 (2020), arXiv:1906.00480 [hep-ph]

    2020 arXiv

  22. [30]

    Cutting, E

    D. Cutting, E. Vilhonen, and D. J. Weir, Droplet col- lapse during strongly supercooled transitions, Phys. Rev. D 106, 103524 (2022), arXiv:2204.03396 [astro-ph.CO]

    2022 arXiv

  23. [31]

    domain wall

    M. B. Hindmarsh, M. Lüben, J. Lumma, and M. Pauly, Phase transitions in the early universe, SciPost Phys. Lect. Notes 24, 1 (2021), arXiv:2008.09136 [astro- ph.CO]. 8 SUPPLEMENTAL MATERIAL Details on the simplified model We discuss here some details of the extended hydro- dyna...

    2021 arXiv

  24. [32]

    We replaced factors of1/(1 − γ) in the equations of motion by(1−γ)/[(1−γ)2 +εγ]

    The pointγ = 1. We replaced factors of1/(1 − γ) in the equations of motion by(1−γ)/[(1−γ)2 +εγ]

  25. [33]

    We replaced non-integer powers of Ts by regulated expressions, e.g.,T f (γ) s by (T 2 s + εT )f (γ)/2

    The pointTs = 0. We replaced non-integer powers of Ts by regulated expressions, e.g.,T f (γ) s by (T 2 s + εT )f (γ)/2

  26. [34]

    We replaced factors of1/J by J/(J 2 + εJ )

    The point J = 0, where J is the Jacobian arising when solving for the leading time derivatives in the hydrodynamic sector. We replaced factors of1/J by J/(J 2 + εJ ). Moreover, the map Γ(γ) = γ2(3 − 2γ) for the mixing fraction actually only makes sense for0 < γ <1. Nu- merical...

  27. [35]

    The velocity and the time derivative ofγ were set to zero initially

    Initial condition atτ = 300 with T = 0 .95Tc and γ = 0 (deconfined phase) except for two Gaussian 10 perturbations in γ, centered atx ≈ ±10 with am- plitude 0.75 and width 4.4. The velocity and the time derivative ofγ were set to zero initially. This simulation was used to gen...

  28. [36]

    That is, γ was chosen to be nonzero only for|x| ≲ 30, with the blob limited by smooth tanh steps hav- ing widths equal to one

    Initial condition for a blob of deconfined matter with T = 0 .95Tc and width 60 at τ = 300. That is, γ was chosen to be nonzero only for|x| ≲ 30, with the blob limited by smooth tanh steps hav- ing widths equal to one. To be precise, we chose the rescaled temperature to beTs =...

  29. [37]

    Profiles were chosen similarly to the second simulation, but we added also an initial velocity profile (i.e.,θ) corre- sponding to a uniformly shrinking blob

    Initial condition for a blob of deconfined matter with T = Tc and width 40 at τ = 50. Profiles were chosen similarly to the second simulation, but we added also an initial velocity profile (i.e.,θ) corre- sponding to a uniformly shrinking blob. This was necessarytodamptheiniti...

  30. [38]

    This width was chosen to see better the effect of the decaying γ profile

    Initial conditions as in the third simulation, but with a narrower width20. This width was chosen to see better the effect of the decaying γ profile. The simulation was used in Fig. 6. We also ran a simulation in the FRW scenario with the following initial condition. We chosea...

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