pith. sign in

arxiv: 2410.01892 · v2 · pith:2VJT5URRnew · submitted 2024-10-02 · ✦ hep-ph · cond-mat.quant-gas· hep-th

Superfluids in expanding backgrounds and attractor times

Guri K. Buza , Toshali Mitra , Alexander Soloviev This is my paper

Pith reviewed 2026-05-23 20:02 UTC · model grok-4.3

classification ✦ hep-ph cond-mat.quant-gashep-th
keywords superfluidhydrodynamic attractorsexpanding backgroundsBjorken flowGubser flowspontaneous symmetry breakingattractor timeMüller-Israel-Stewart theory
0
0 comments X

The pith

In expanding backgrounds, hydrodynamic evolution of a superfluid can spontaneously break U(1) symmetry and then follow attractor dynamics for a finite attractor time set by initial conditions.

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

The paper shows that in a U(1) superfluid coupled to Müller-Israel-Stewart hydrodynamics, the fluid variables in expanding flows can reshape the Goldstone mode potential. This triggers spontaneous symmetry breaking for suitable starting data, after which the condensate shrinks and the system tracks hydrodynamic attractors. The authors introduce an attractor time that marks how long this regime lasts in Bjorken and Gubser expansions, and they give the first full nonlinear account of Gubser flow including a constant-anisotropy phase. Similar attractor behavior appears in a dynamical FLRW background, where the system later becomes condensate-dominated.

Core claim

For suitable initial conditions, the evolution of the hydrodynamic variables leads to a change in the potential of the Goldstone mode, spontaneously breaking the symmetry. After some time, the condensate becomes small, leading the system evolution to be well described via hydrodynamic attractors for a timescale that we determine in Bjorken and Gubser flows. We define this new timescale as the attractor time and show its dependence on initial conditions. In the case of the Gubser flow, we provide for the first time a complete description of the nonlinear evolution of the system, including a novel nonlinear regime of constant anisotropy not found in the Bjorken evolution.

What carries the argument

The attractor time: the interval after condensate suppression during which the coupled Goldstone-hydrodynamic system in Müller-Israel-Stewart theory follows attractor solutions in expanding geometries.

If this is right

  • The attractor time is finite and explicitly depends on the choice of initial hydrodynamic and Goldstone data.
  • Gubser flow admits a previously unseen nonlinear regime of constant anisotropy that persists while the system is on the attractor.
  • In FLRW expansion the same initial-condition-driven attractor is reached, but the late-time state is dominated by the condensate rather than hydrodynamics.
  • The mechanism supplies a concrete way to connect out-of-equilibrium superfluid dynamics to standard hydrodynamic attractors in heavy-ion and cosmological settings.

Where Pith is reading between the lines

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

  • The same initial-condition trigger could be tested in other expanding geometries or with different equation-of-state choices to map how attractor time scales with expansion rate.
  • If realized in a heavy-ion context, the transient symmetry-breaking window might leave observable imprints on particle spectra before the attractor regime takes over.
  • The constant-anisotropy phase unique to Gubser flow suggests that global flow geometry can qualitatively alter the nonlinear attractor structure beyond what Bjorken flow captures.

Load-bearing premise

Suitable initial conditions exist such that hydrodynamic evolution alters the Goldstone potential enough to break symmetry and suppress the condensate before other processes intervene.

What would settle it

A numerical evolution starting from the claimed initial data in which the effective potential never drives spontaneous symmetry breaking or the condensate remains large enough to block the attractor description throughout the expansion.

Figures

Figures reproduced from arXiv: 2410.01892 by Alexander Soloviev, Guri K. Buza, Toshali Mitra.

Figure 1
Figure 1. Figure 1: FIG. 1. The condensate (red point) in the potential ( [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Overview showing the relationships between the different backgrounds explored here. [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Left: Typical evolution for [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The normalized timescale to go from the onset of hydrodynamic attractor behavior, [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The figure shows temperature [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The evolution of the anisotropy, [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Left: Evolution of the anisotropy, [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. The figure shows the evolution of condensate [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. The behavior of the attractor time, ∆ [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Behavior of ∆ [PITH_FULL_IMAGE:figures/full_fig_p013_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. The top left and right figure show the ∆ [PITH_FULL_IMAGE:figures/full_fig_p014_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Mapping the Gubser solution in Fig. [PITH_FULL_IMAGE:figures/full_fig_p015_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. The dissipative fluid in FLRW with [PITH_FULL_IMAGE:figures/full_fig_p017_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. The evolution of condensate [PITH_FULL_IMAGE:figures/full_fig_p018_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. The evolution of the full system in the FLRW background. The top two plots show the evolution of the full system [PITH_FULL_IMAGE:figures/full_fig_p019_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. Phase space plot in the FLRW background. The initial conditions are such that [PITH_FULL_IMAGE:figures/full_fig_p020_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17. Left: The decaying amplitude [PITH_FULL_IMAGE:figures/full_fig_p020_17.png] view at source ↗
read the original abstract

We determine the behavior of an out-of-equilibrium superfluid, composed of a $U(1)$ Goldstone mode coupled to hydrodynamic modes in a M\" uller-Israel-Stewart theory, in expanding backgrounds relevant to heavy ion collision experiments and cosmology. For suitable initial conditions, the evolution of the hydrodynamic variables leads to a change in the potential of the Goldstone mode, spontaneously breaking the symmetry. After some time, the condensate becomes small, leading the system evolution to be well described via hydrodynamic attractors for a timescale that we determine in Bjorken and Gubser flows. We define this new timescale as the \textit{attractor time} and show its dependence on initial conditions. In the case of the Gubser flow, we provide for the first time a complete description of the nonlinear evolution of the system, including a novel nonlinear regime of constant anisotropy not found in the Bjorken evolution. Finally, we consider the superfluid in the dynamical FLRW (Friedmann-Lemaitre-Roberston-Walker) background, where we observe a similar attractor behavior, dependent on the initial conditions, that at late times approaches a regime dominated by the condensate.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 2 minor

Summary. The manuscript examines the coupled dynamics of a U(1) Goldstone mode with Müller-Israel-Stewart hydrodynamics for an out-of-equilibrium superfluid in expanding backgrounds (Bjorken, Gubser, and dynamical FLRW). It asserts that, for suitable initial conditions, the hydrodynamic evolution modifies the effective potential of the Goldstone mode, inducing spontaneous symmetry breaking; the condensate subsequently suppresses, allowing the system to enter a hydrodynamic attractor regime over a newly defined 'attractor time' whose dependence on initial data is reported. For Gubser flow the work supplies a complete nonlinear evolution, including a novel regime of constant anisotropy absent in Bjorken flow; a similar attractor behavior is observed in FLRW, with late-time condensate domination.

Significance. If the numerical results are robust, the identification of an attractor time and the first complete nonlinear treatment of Gubser flow with a constant-anisotropy regime constitute concrete advances for modeling superfluids in heavy-ion and cosmological settings. The explicit coupling of Goldstone dynamics to MIS hydrodynamics in expanding geometries is a timely extension, and the provision of attractor timescales dependent on initial conditions supplies falsifiable predictions once the initial-data basin is quantified.

major comments (2)
  1. [Abstract / Sec. 1] The central claim that hydrodynamic evolution drives a change in the Goldstone potential sufficient to trigger spontaneous symmetry breaking (and subsequent condensate suppression) is stated to hold only for 'suitable initial conditions' (abstract and introduction). Because the attractor time is defined from the duration of this regime and is reported to depend on those initial data, the manuscript must demonstrate the size of the basin of attraction or provide a parameter scan showing that the potential modification is not an artifact of specially chosen starting values; without this the definition of the attractor time remains conditional on an unquantified assumption.
  2. [Gubser-flow section] The novel constant-anisotropy nonlinear regime reported for Gubser flow is load-bearing for the claim of a 'complete description' not found in Bjorken evolution. The manuscript should supply the explicit evolution equations or numerical diagnostics (e.g., the anisotropy parameter as a function of proper time) that establish this regime is reached from the coupled Goldstone-hydro system rather than imposed by hand, together with error estimates on the attractor time extracted from it.
minor comments (2)
  1. [Abstract] The abstract would benefit from a single sentence indicating the numerical method (e.g., characteristic or relaxation scheme) and the range of initial hydrodynamic and Goldstone variables explored.
  2. [Sec. 2] Notation for the effective potential of the Goldstone mode and its coupling to the hydrodynamic stress tensor should be introduced with an equation reference at first use to aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments. We respond to each major comment below.

read point-by-point responses

  1. Referee: [Abstract / Sec. 1] The central claim that hydrodynamic evolution drives a change in the Goldstone potential sufficient to trigger spontaneous symmetry breaking (and subsequent condensate suppression) is stated to hold only for 'suitable initial conditions' (abstract and introduction). Because the attractor time is defined from the duration of this regime and is reported to depend on those initial data, the manuscript must demonstrate the size of the basin of attraction or provide a parameter scan showing that the potential modification is not an artifact of specially chosen starting values; without this the definition of the attractor time remains conditional on an unquantified assumption.

    Authors: We agree that the robustness of the attractor time would be strengthened by a more systematic quantification of the basin of attraction. The manuscript already illustrates the phenomenon and the dependence of the attractor time on initial data for several representative choices that trigger the symmetry-breaking regime. To address the referee's concern directly, we will add an expanded parameter scan over initial conditions in the revised version, delineating the region in which the hydrodynamic modification of the Goldstone potential leads to spontaneous symmetry breaking. revision: yes

  2. Referee: [Gubser-flow section] The novel constant-anisotropy nonlinear regime reported for Gubser flow is load-bearing for the claim of a 'complete description' not found in Bjorken evolution. The manuscript should supply the explicit evolution equations or numerical diagnostics (e.g., the anisotropy parameter as a function of proper time) that establish this regime is reached from the coupled Goldstone-hydro system rather than imposed by hand, together with error estimates on the attractor time extracted from it.

    Authors: The evolution equations for the Gubser-flow case are obtained by coupling the Goldstone mode to the MIS hydrodynamic equations in the appropriate coordinates and are solved numerically in Section 3. We will include the explicit set of coupled ODEs in an appendix of the revised manuscript. In addition, we will provide plots of the anisotropy parameter as a function of proper time together with numerical convergence diagnostics that confirm the constant-anisotropy regime emerges dynamically from the coupled system; these will also include error estimates on the extracted attractor times. revision: yes

Circularity Check

0 steps flagged

No circularity: forward evolution from initial conditions defines attractor time without reduction to inputs

full rationale

The paper evolves the coupled Müller-Israel-Stewart hydrodynamics plus Goldstone mode from specified initial conditions in Bjorken, Gubser and FLRW backgrounds. The attractor time is introduced as the duration after which the condensate amplitude drops and the evolution enters the hydrodynamic attractor regime; this is an observed outcome of the time integration rather than a quantity fitted to the same data or defined in terms of itself. No self-citations, uniqueness theorems, or ansätze are invoked to force the result, and the novel constant-anisotropy regime in Gubser flow is reported as a direct numerical feature. The assumption of 'suitable initial conditions' is an existence claim about the basin of attraction, not a tautological redefinition of the output quantities.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claims rest on the standard assumptions of Müller-Israel-Stewart hydrodynamics and the coupling of a U(1) Goldstone mode to the fluid; initial conditions are treated as free inputs that must be chosen suitably. No new particles or forces are postulated.

free parameters (1)
  • initial hydrodynamic variables and Goldstone field values
    The attractor behavior and the value of the attractor time are stated to depend on the choice of suitable initial conditions.
axioms (1)
  • domain assumption Müller-Israel-Stewart theory provides a valid effective description of the hydrodynamic modes coupled to the Goldstone field
    The entire analysis is performed inside this framework; the abstract invokes it to model the out-of-equilibrium evolution.

pith-pipeline@v0.9.0 · 5742 in / 1474 out tokens · 24507 ms · 2026-05-23T20:02:17.736843+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

58 extracted references · 58 canonical work pages · 23 internal anchors

  1. [1]

    M. G. Alford, A. Schmitt, K. Rajagopal, and T. Sch¨ afer, Color superconductivity in dense quark matter, Rev. Mod. Phys. 80, 1455 (2008), arXiv:0709.4635 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  2. [2]

    Nearly Perfect Fluidity: From Cold Atomic Gases to Hot Quark Gluon Plasmas

    T. Sch¨ afer and D. Teaney, Nearly Perfect Fluidity: From Cold Atomic Gases to Hot Quark Gluon Plasmas, Rept. Prog. Phys.72, 126001 (2009), arXiv:0904.3107 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  3. [3]

    Kapitza, Viscosity of liquid helium below theλ-point, Nature141, 74 (1938)

    P. Kapitza, Viscosity of liquid helium below theλ-point, Nature141, 74 (1938)

    work page 1938

  4. [4]

    Superfluidity and Superconductivity in Neutron Stars

    B. Haskell and A. Sedrakian, Superfluidity and Superconductivity in Neutron Stars, Astrophys. Space Sci. Libr.457, 401 (2018), arXiv:1709.10340 [astro-ph.HE]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  5. [5]

    Grossi, A

    E. Grossi, A. Soloviev, D. Teaney, and F. Yan, Transport and hydrodynamics in the chiral limit, Phys. Rev. D102, 014042 (2020), arXiv:2005.02885 [hep-th]

    work page arXiv 2020

  6. [6]

    Grossi, A

    E. Grossi, A. Soloviev, D. Teaney, and F. Yan, Soft pions and transport near the chiral critical point, Phys. Rev. D104, 034025 (2021), arXiv:2101.10847 [nucl-th]

    work page arXiv 2021

  7. [7]

    Florio, E

    A. Florio, E. Grossi, A. Soloviev, and D. Teaney, Dynamics of theO(4) critical point in QCD, Phys. Rev. D105, 054512 (2022), arXiv:2111.03640 [hep-lat]

    work page arXiv 2022

  8. [8]

    Florio, E

    A. Florio, E. Grossi, and D. Teaney, Dynamics of the O(4) critical point in QCD: Critical pions and diffusion in model G, Phys. Rev. D109, 054037 (2024), arXiv:2306.06887 [hep-lat]

    work page arXiv 2024

  9. [9]

    L. Du, A. Sorensen, and M. Stephanov, The QCD phase diagram and Beam Energy Scan physics: a theory overview (2024) arXiv:2402.10183 [nucl-th]. 22

    work page arXiv 2024

  10. [10]

    Viscosity Information from Relativistic Nuclear Collisions: How Perfect is the Fluid Observed at RHIC?

    P. Romatschke and U. Romatschke, Viscosity Information from Relativistic Nuclear Collisions: How Perfect is the Fluid Observed at RHIC?, Phys. Rev. Lett.99, 172301 (2007), arXiv:0706.1522 [nucl-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  11. [11]

    Brandstetteret al., Emergent hydrodynamic behaviour of few strongly interacting fermions, (2023), arXiv:2308.09699 [cond-mat.quant-gas]

    S. Brandstetteret al., Emergent hydrodynamic behaviour of few strongly interacting fermions, (2023), arXiv:2308.09699 [cond-mat.quant-gas]

    work page arXiv 2023

  12. [12]

    Fujii and T

    K. Fujii and T. Enss, Hydrodynamic Attractor in Ultracold Atoms, (2024), arXiv:2404.12921 [cond-mat.quant-gas]

    work page arXiv 2024

  13. [13]

    Soloviev, Hydrodynamic attractors in heavy ion collisions: a review, Eur

    A. Soloviev, Hydrodynamic attractors in heavy ion collisions: a review, Eur. Phys. J. C82, 319 (2022), arXiv:2109.15081 [hep-th]

    work page arXiv 2022

  14. [14]

    Jankowski and M

    J. Jankowski and M. Spali´ nski, Hydrodynamic attractors in ultrarelativistic nuclear collisions, Prog. Part. Nucl. Phys. 132, 104048 (2023), arXiv:2303.09414 [nucl-th]

    work page arXiv 2023

  15. [15]

    D. T. Son, Hydrodynamics of relativistic systems with broken continuous symmetries, Int. J. Mod. Phys. A16S1C, 1284 (2001), arXiv:hep-ph/0011246

    work page internal anchor Pith review Pith/arXiv arXiv 2001

  16. [16]

    D. T. Son, Low-energy quantum effective action for relativistic superfluids, (2002), arXiv:hep-ph/0204199

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  17. [17]

    Introduction to superfluidity -- Field-theoretical approach and applications

    A. Schmitt, Introduction to Superfluidity: Field-Theoretical Approach and Applications (2014), arXiv:1404.1284 [cond- mat, physics:hep-ph, physics:hep-th, physics:nucl-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  18. [18]

    Mitra, A

    T. Mitra, A. Mukhopadhyay, and A. Soloviev, Hydrodynamic attractor and novel fixed points in superfluid Bjorken flow, Phys. Rev. D103, 076014 (2021), arXiv:2012.15644 [hep-ph]

    work page arXiv 2021

  19. [19]

    Low-energy effective field theory for finite-temperature relativistic superfluids

    A. Nicolis, Low-energy effective field theory for finite-temperature relativistic superfluids, (2011), arXiv:1108.2513 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  20. [20]

    M¨ uller, Zum Paradoxon der W¨ armeleitungstheorie, Zeitschrift fur Physik198, 329 (1967)

    I. M¨ uller, Zum Paradoxon der W¨ armeleitungstheorie, Zeitschrift fur Physik198, 329 (1967)

    work page 1967

  21. [21]

    Israel and J

    W. Israel and J. Stewart, Transient relativistic thermodynamics and kinetic theory, Annals of Physics118, 341 (1979)

    work page 1979

  22. [22]

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

    work page 1983

  23. [23]

    S. S. Gubser, Symmetry constraints on generalizations of Bjorken flow, Phys. Rev. D82, 085027 (2010), arXiv:1006.0006 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  24. [24]

    S. S. Gubser and A. Yarom, Conformal hydrodynamics in Minkowski and de Sitter spacetimes, Nucl. Phys. B846, 469 (2011), arXiv:1012.1314 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  25. [25]

    Rodgers and J

    R. Rodgers and J. G. Subils, Boost-invariant superfluid flows, JHEP09, 205, arXiv:2207.02903 [hep-th]

    work page arXiv

  26. [27]

    Giacalone, A

    G. Giacalone, A. Mazeliauskas, and S. Schlichting, Hydrodynamic attractors, initial state energy and particle production in relativistic nuclear collisions, Phys. Rev. Lett.123, 262301 (2019), arXiv:1908.02866 [hep-ph]

    work page arXiv 2019

  27. [28]

    R. M. Wald,General Relativity(Chicago Univ. Pr., Chicago, USA, 1984)

    work page 1984

  28. [29]

    Hinshawet al., Nine-year Wilkinson microwave anisotropy probe (WMAP) observations: Cosmological Parameter Results, The Astrophysical Journal Supplement Series208, 19 (2013)

    G. Hinshawet al., Nine-year Wilkinson microwave anisotropy probe (WMAP) observations: Cosmological Parameter Results, The Astrophysical Journal Supplement Series208, 19 (2013)

    work page 2013

  29. [30]

    Dash and V

    A. Dash and V. Roy, Hydrodynamic attractors for Gubser flow, Phys. Lett. B806, 135481 (2020), arXiv:2001.10756 [nucl-th]

    work page arXiv 2020

  30. [31]

    S. F. Taghavi, Smallest QCD droplet and multiparticle correlations in p-p collisions, Phys. Rev. C104, 054906 (2021), arXiv:1907.12140 [nucl-th]

    work page arXiv 2021

  31. [32]

    S. Acharyaet al.(ALICE), Multiplicity and event-scale dependent flow and jet fragmentation in pp collisions at √s= 13 TeV and in p–Pb collisions at √sNN = 5.02 TeV, JHEP03, 092, arXiv:2308.16591 [nucl-ex]

    work page arXiv

  32. [33]

    G. S. Denicol and J. Noronha, Hydrodynamic attractor and the fate of perturbative expansions in Gubser flow, Phys. Rev. D99, 116004 (2019), arXiv:1804.04771 [nucl-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  33. [34]

    Relativistic Hydrodynamic Attractors with Broken Symmetries: Non-Conformal and Non-Homogeneous

    P. Romatschke, Relativistic Hydrodynamic Attractors with Broken Symmetries: Non-Conformal and Non-Homogeneous, JHEP12, 079, arXiv:1710.03234 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv

  34. [35]

    Gorbunov and V

    D. Gorbunov and V. Rubakov,Introduction to the Theory of the Early Universe: Cosmological Perturbations and Infla- tionary Theory, G - Reference,Information and Interdisciplinary Subjects Series No. let. 1 (World Scientific, 2011)

    work page 2011

  35. [36]

    Gain tuning for continuous-variable quantum teleportation of discrete-variable states.Phys

    Z. Du, X.-G. Huang, and H. Taya, Hydrodynamic attractor in a hubble expansion, Physical Review D104, 10.1103/phys- revd.104.056022 (2021)

    work page doi:10.1103/phys- 2021

  36. [37]

    Chervon, I

    S. Chervon, I. Fomin, V. Yurov, and A. Yurov,Scalar Field Cosmology, Series on the Foundations of Natural Science and Technology, Vol. 13 (WSP, Singapur, 2019)

    work page 2019

  37. [38]

    Martin, C

    J. Martin, C. Ringeval, and V. Vennin, Encyclopaedia inflationaris (2024), arXiv:1303.3787 [astro-ph.CO]

    work page arXiv 2024

  38. [39]

    J. D. Brown, Action functionals for relativistic perfect fluids, Classical and Quantum Gravity10, 1579 (1993)

    work page 1993

  39. [40]

    M. G. Alford, S. K. Mallavarapu, A. Schmitt, and S. Stetina, From a complex scalar field to the two-fluid picture of superfluidity 10.1103/PhysRevD.87.065001 (2013)

    work page doi:10.1103/physrevd.87.065001 2013

  40. [41]

    Relativistic viscous hydrodynamics, conformal invariance, and holography

    R. Baier, P. Romatschke, D. T. Son, A. O. Starinets, and M. A. Stephanov, Relativistic viscous hydrodynamics, conformal invariance, and holography, JHEP04, 100, arXiv:0712.2451 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv

  41. [42]

    M. P. Heller and M. Spalinski, Hydrodynamics Beyond the Gradient Expansion: Resurgence and Resummation, Phys. Rev. Lett.115, 072501 (2015), arXiv:1503.07514 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  42. [43]

    Keegan, A

    L. Keegan, A. Kurkela, A. Mazeliauskas, and D. Teaney, Initial conditions for hydrodynamics from weakly coupled pre- equilibrium evolution, Journal of High Energy Physics2016, 10.1007/jhep08(2016)171 (2016)

    work page doi:10.1007/jhep08(2016)171 2016

  43. [44]

    Far-from-equilibrium attractors and nonlinear dynamical systems approach to the Gubser flow

    A. Behtash, C. N. Cruz-Camacho, and M. Martinez, Far-from-equilibrium attractors and nonlinear dynamical systems approach to the Gubser flow, Phys. Rev. D97, 044041 (2018), arXiv:1711.01745 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  44. [45]

    Relativistic Fluid Dynamics In and Out of Equilibrium -- Ten Years of Progress in Theory and Numerical Simulations of Nuclear Collisions

    P. Romatschke and U. Romatschke, Relativistic fluid dynamics in and out of equilibrium – ten years of progress in theory and numerical simulations of nuclear collisions (2019), arXiv:1712.05815 [nucl-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  45. [46]

    Anisotropic hydrodynamics for conformal Gubser flow

    M. Nopoush, R. Ryblewski, and M. Strickland, Anisotropic hydrodynamics for conformal Gubser flow, Phys. Rev. D91, 045007 (2015), arXiv:1410.6790 [nucl-th]. 23

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  46. [47]

    Banerjee, T

    A. Banerjee, T. Mitra, A. Mukhopadhyay, and A. Soloviev, How Gubser flow ends in a holographic conformal theory, Eur. Phys. J. C84, 550 (2024), arXiv:2307.10384 [hep-th]

    work page arXiv 2024

  47. [48]

    Mitra, S

    T. Mitra, S. Mondkar, A. Mukhopadhyay, and A. Soloviev, Holographic Gubser flow. A combined analytic and numerical study, JHEP10, 226, arXiv:2408.04001 [hep-th]

    work page arXiv

  48. [49]

    Cosmology with a stiff matter era

    P.-H. Chavanis, Cosmology with a stiff matter era, Phys. Rev. D92, 103004, arxiv:1412.0743 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv

  49. [50]

    Matteini, M

    M. Matteini, M. Nemevˇ sek, Y. Shoji, and L. Ubaldi, False Vacuum Decay Rate From Thin To Thick Walls, (2024), arXiv:2404.17632 [hep-th]

    work page arXiv 2024

  50. [51]

    Boileau, N

    G. Boileau, N. Christensen, C. Gowling, M. Hindmarsh, and R. Meyer, Prospects for LISA to detect a gravitational-wave background from first order phase transitions, JCAP02, 056, arXiv:2209.13277 [gr-qc]

    work page arXiv

  51. [52]

    Soloviev, Colliding poles with colliding nuclei, EPJ Web Conf.274, 05015 (2022), arXiv:2211.09792 [hep-ph]

    A. Soloviev, Colliding poles with colliding nuclei, EPJ Web Conf.274, 05015 (2022), arXiv:2211.09792 [hep-ph]

    work page arXiv 2022

  52. [53]

    R. A. Janik and R. B. Peschanski, Asymptotic perfect fluid dynamics as a consequence of Ads/CFT, Phys. Rev. D73, 045013 (2006), arXiv:hep-th/0512162

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  53. [54]

    A Holographic Dual of Bjorken Flow

    S. Kinoshita, S. Mukohyama, S. Nakamura, and K.-y. Oda, A Holographic Dual of Bjorken Flow, Prog. Theor. Phys.121, 121 (2009), arXiv:0807.3797 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  54. [55]

    Coupling constant corrections in a holographic model of heavy ion collisions

    S. Grozdanov and W. van der Schee, Coupling Constant Corrections in a Holographic Model of Heavy Ion Collisions, Phys. Rev. Lett.119, 011601 (2017), arXiv:1610.08976 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  55. [56]

    Retarded Correlators in Kinetic Theory: Branch Cuts, Poles and Hydrodynamic Onset Transitions

    P. Romatschke, Retarded correlators in kinetic theory: branch cuts, poles and hydrodynamic onset transitions, Eur. Phys. J. C76, 352 (2016), arXiv:1512.02641 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  56. [57]

    Analytic structure of nonhydrodynamic modes in kinetic theory

    A. Kurkela and U. A. Wiedemann, Analytic structure of nonhydrodynamic modes in kinetic theory, Eur. Phys. J. C79, 776 (2019), arXiv:1712.04376 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  57. [58]

    Bajec, S

    M. Bajec, S. Grozdanov, and A. Soloviev, Spectra of correlators in the relaxation time approximation of kinetic theory, JHEP08, 065, arXiv:2403.17769 [hep-th]

    work page arXiv

  58. [59]

    Kurkela, W

    A. Kurkela, W. van der Schee, U. A. Wiedemann, and B. Wu, Early- and Late-Time Behavior of Attractors in Heavy-Ion Collisions, Phys. Rev. Lett.124, 102301 (2020), arXiv:1907.08101 [hep-ph]

    work page arXiv 2020