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REVIEW 3 major objections 7 minor 292 references

Detailed lattice schemes can evolve perfect and viscous fluids alone or coupled to scalars and gauge fields on an expanding universe, with self-consistent expansion and machine-precision gauge invariance in some formulations.

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2026-07-30 11:17 UTC pith:JA7RLU2E

load-bearing objection Solid methods monograph that delivers usable multi-scheme lattice kernels for relativistic scalar-gauge-fluid dynamics in FLRW; the algebra is careful, the main practical caveat is missing stability/shock analysis for the centered fluid fluxes. the 3 major comments →

arxiv 2607.27141 v1 pith:JA7RLU2E submitted 2026-07-29 astro-ph.CO

The art of simulating the early Universe. Part III: Scalar-Gauge-Fluid Dynamics

Daniel G. Figueroa , Kenneth Marschall , Antonino S. Midiri , Alberto Roper Pol This is my paper
classification astro-ph.CO PACS 98.80.Cq47.11.-j04.30.Db
keywords lattice cosmologyearly universerelativistic fluid dynamicsviscous fluidsmagnetohydrodynamicsscalar-fluid couplinggravitational wavesFLRW expansion
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This monograph lays out how to put early-universe fluid dynamics on a lattice. It starts from continuum fluid equations in flat space and in an expanding FLRW background, covering perfect fluids, first-order viscous fluids, and fluids coupled to Abelian gauge fields or singlet scalars, in both conservation and non-conservation forms and in fully relativistic or subrelativistic bulk motion. It then gives explicit discretization schemes—collocated, semi-collocated, and staggered—so that the discrete equations can be stepped in time while all scalar, gauge, and fluid sectors source the expansion. Some schemes keep lattice gauge invariance to machine precision. The same framework includes initial-condition methods and gravitational-wave evolution sourced by every sector. The point is practical: these recipes are meant to make controlled numerical studies of acoustic and MHD turbulence, phase-transition bubble–plasma dynamics, and the gravitational waves they produce possible in a fully expanding background.

Core claim

The paper establishes a complete set of lattice evolution algorithms for isolated perfect and first-order imperfect fluids, and for fluids coupled to Abelian gauge fields or singlet scalars, on a spatially flat expanding background. The schemes handle conservation and non-conservation forms, relativistic and subrelativistic bulk motion, self-consistent expansion from all sectors, and—in selected formulations—preservation of the Gauss constraint to machine precision, together with lattice initial conditions and gravitational-wave sourcing by fluid, scalar, and gauge degrees of freedom.

What carries the argument

Collocated, semi-collocated, and staggered finite-difference discretizations of the fluid equations in conservation and non-conservation form (with matching gauge and scalar kernels, program-variable rescalings, and low-storage Runge–Kutta time stepping), including reconstruction of primitive variables from stress-energy components via the z(r²) relation when the equation of state is constant.

Load-bearing premise

Dissipative effects can be captured by first-order Navier–Stokes viscosity restricted to subrelativistic bulk velocities and small bulk viscosity relative to the Hubble time.

What would settle it

Run the claimed gauge-invariant staggered or semi-collocated schemes on a charged fluid plus U(1) field and check whether the discrete Gauss constraint stays at round-off for many Hubble times; or recover the known continuum Alfvén speed (with and without Boris correction) and pure-radiation conformal conservation laws to the design order in lattice spacing and time step.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Fully relativistic bulk fluid motion in expanding FLRW can be simulated together with scalar and gauge sectors rather than only in flat space or Higgsless setups.
  • Acoustic and MHD turbulence and first-order phase-transition bubble–plasma dynamics can source gravitational waves with all sectors backreacting on the expansion.
  • Ideal and resistive MHD, including large-conductivity induction equations and bounded Alfvén speeds via Boris correction, become available on the same lattice footing as the fluid.
  • Initial fluid configurations (vortical and compressional) and multi-sector GW extraction can be set consistently with the same discrete operators.
  • Public implementation of these schemes would let independent groups cross-check nonlinear early-universe fluid predictions against shared algorithms.

Where Pith is reading between the lines

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

  • Extending the viscous sector from first-order Navier–Stokes to Maxwell–Cattaneo or Israel–Stewart form is the natural next methods step the text itself flags as future work.
  • The same operator toolkit likely ports to multi-component or non-Abelian-charged fluids once the corresponding continuum forces are written in conservation or non-conservation form.
  • Controlled continuum-limit tests of GW spectra from fluid-only versus fluid-plus-gauge runs would quantify how much prior flat-space or subrelativistic approximations biased predicted stochastic backgrounds.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 7 minor

Summary. This monograph (third in the "Art of Simulating the Early Universe" series) develops the continuum theory and lattice discretization of relativistic fluid dynamics on an FLRW background, coupled to Abelian gauge fields and singlet scalars. It derives, from a kinetic-theory/variational starting point, conservation and non-conservation forms of the fluid equations for perfect and first-order (Navier-Stokes) imperfect fluids, including Lorentz-force and scalar-damping couplings, and presents explicit discrete kernels in collocated, semi-collocated, and staggered formulations with arbitrary even-order spatial accuracy, low-storage Runge-Kutta time integration, self-consistent Friedmann expansion sourced by all sectors, and gravitational-wave sourcing. It is intended as the theoretical basis for the scalar-gauge-fluid module of CosmoLattice v3.0. The continuum derivations are careful and are cross-checked against known limits (conformal flatness for ω=1/3, the subtle subrelativistic limit retaining ∂0γ, entropy production inequalities). The discrete operators are written explicitly with order-p continuum-recovery statements. The manuscript contains no numerical tests, no stability analysis, and no discussion of aliasing, shock capturing, or time-step restrictions.

Significance. If the kernels perform as derived, this is a substantial and useful contribution. It is, to my knowledge, the first systematic lattice-cosmology treatment of relativistic fluids — perfect and first-order viscous — coupled to Abelian gauge fields and singlet scalars on an FLRW background, with self-consistent expansion sourced by all sectors. Notable strengths: the derivations are parameter-free in the sense that all discrete operators are given to explicit arbitrary even order with stated continuum limits; the continuum theory is cross-checked against independent consistency requirements (conformal flatness for radiation in conformal time, the non-trivial subrelativistic reduction requiring retention of ∂0 ln γ, entropy production inequalities in Eqs. 63/91/126); and the staggered gauge schemes inherit exact discrete Gauss-constraint preservation from Parts I/II, giving falsifiable, machine-precision-level diagnostics. The work directly underpins a public code release (CosmoLattice v3.0), making the results reproducible in principle. Its significance is conditional on the stability/applicability questions in the major comments being addressed, since the headline applications (FOPTs

major comments (3)
  1. [§4.1.1 and §4.2.1, Eqs. (227)-(230), (234)-(244), (256)-(257)] The fluid-sector kernels discretize all nonlinear fluxes with neutral (centered) stencils of order p=2m and no dealiasing, skew-symmetric splitting, upwinding, or filtering. In conservation form the flux is pointwise-quadratic in the dynamical variables (Eq. 229: z/(z+ω)·T̄^{0i}T̄^{0j}/T̄^{00}); in non-conservation form the advection u_j∂_j u_i is discretized in plain (non-skew-symmetric) form (term M.3 of Eqs. 257 and 267d). For the under-resolved turbulence applications advertised in §1 (acoustic and MHD turbulence as GW sources), such schemes are known to alias energy from above ~2/3 k_UV back into the resolved band and to admit nonlinear (aliasing) instability unless the physical viscosity resolves the grid scale. The manuscript nowhere states a resolution criterion of the form Re_grid ≡ u_rms·δx/ν̃ ≲ O(1), nor performs any von Neumann or energy-method stability analysis of the discr
  2. [§4 and §5.2; application context §1 and §2.4.2] First-order phase-transition bubble walls — an explicitly advertised target (§1 and §2.4.2) — involve shocks/detonations/deflagrations. By Godunov's theorem, the linear centered schemes of Eqs. (195)/(198) with p>1 cannot be monotone near discontinuities, so wall profiles thinner than a few grid spacings will develop Gibbs oscillations unless the physical Navier-Stokes viscosity resolves the wall width (roughly ν̃ ≳ u_wall·δx) or artificial viscosity/limiters are added. The manuscript does not state the viscous-resolution condition for walls, does not discuss monotonicity, and provides no shock-tube-type test. Given that the scalar-fluid sector (§5.2) is designed specifically for FOPT simulations, the authors should either quantify the regime in which the presented schemes handle wall discontinuities (and say what to do outside it) or demonstrate with a canonical benchmark that the physi
  3. [§§4-7 (no numerical tests); §3.4 (time-step criteria)] For a methods monograph of this length, the complete absence of numerical verification is a significant gap. The abstract asserts that gauge invariance is preserved 'to machine precision in some cases,' and §4–§5 present the kernels as ready-to-implement algorithms, but no convergence test at the claimed order p, no constraint-evolution plot (Gauss/Hubble constraints of Eqs. 233, 262, 298–299), no energy-conservation check, and no linear-wave test is shown anywhere in the document. The gauge-sector constraint results have precedent in Parts I/II, which could simply be cited; but the fluid and fluid-coupling kernels are new here and are not demonstrated at all. I request at least minimal evidence — e.g., propagation of a linear sound wave at the nominal convergence order, a Gauss-constraint and Hubble-constraint drift plot for the staggered/semi-collocated schemes, and one nonlinear test
minor comments (7)
  1. [§4.2.1] Eq. (267e): the derivative in term (M.4) is written ∇̄^{(+,0)}_i; the superscript should presumably be (+,p) for consistency with the rest of the staggered formulation.
  2. [§3.3] Eq. (212b) is typographically garbled ('9f(n+ 9f(n) + ˆı)−f(n+ 2ˆı)−f(n−ˆı)/16'); please restore the intended semi-sum expression.
  3. [§2.4.2, Eqs. (138)-(144)] The temperature inversion for the scalar-fluid conservation form reduces to an eighth-order polynomial (Eq. 139) that must in general be solved numerically at every lattice site and every Runge-Kutta substep. A brief comment on the computational cost and on practical strategies (tabulation, warm-start from the previous substep, the analytic C1=0 and toy-model cases of Eqs. 140-144) would be valuable to implementers.
  4. [§2.3.2, Eq. (74)] For the imperfect-fluid conservation form, the Cardano roots in Eq. (74) require selecting 'on a case-by-case basis' the real root with z≥1. In a time evolution one additionally needs the selected root to be continuous in time as r² evolves; a sentence on root tracking (and on the ω_A < -1 corner where uniqueness fails) would help.
  5. [Abstract / §1] The abstract states gauge invariance is preserved to machine precision 'in some cases'; please specify in the abstract or introduction which discretizations (staggered vs. semi-collocated vs. all-collocated) have this property and where the evidence is given.
  6. [§3.2 / §4] The discussion of alternatives to centered differencing would benefit from pointers to the relativistic-hydrodynamics literature on high-resolution shock-capturing methods (e.g., reviews by Font, and Porth et al. on higher-order schemes), both to situate the present choices and to guide users whose applications fall outside the validated regime.
  7. [Conventions and notation] The conventions note that δx is used loosely for both lattice spacing and time step; given that the stability discussion requested above involves both, it would be cleaner to keep δx and δη distinct throughout §§4-5.

Circularity Check

0 steps flagged

Methods monograph: discretizations are constructive translations of continuum EOMs; no fitted-as-prediction or definitional circularity.

full rationale

This paper is a lattice-methods monograph. Continuum fluid, MHD, and scalar-fluid equations are standard kinetic/thermodynamic derivations (Secs. 2.1–2.4); lattice kernels (Secs. 4–5) are finite-difference transcriptions of those EOMs in conservation and non-conservation form, with collocated/staggered placements. Nothing is fitted to data and then relabeled a prediction. Self-citations to Art I/II and CosmoLattice supply shared lattice infrastructure (derivatives, Runge–Kutta, program variables) and do not close a load-bearing uniqueness or ansatz loop that forces the fluid results. Gauge-invariance and energy/Hubble constraints are external consistency checks on the schemes, not tautologies. Numerical-stability concerns about centered fluxes (aliasing, shocks) are correctness risks, not circularity. Score 0; steps empty.

Axiom & Free-Parameter Ledger

5 free parameters · 5 axioms · 0 invented entities

The central claim is a set of discretization algorithms. It rests on standard relativistic hydrodynamics, FLRW geometry, and the modeling choice that first-order viscous hydro plus ideal/resistive MHD and a phenomenological scalar damping term adequately capture the target physics. No parameters are fitted to observational data; free choices are conventional numerical or effective-theory inputs.

free parameters (5)
  • EOS parameter ω (or cs²)
    Taken as constant for pure fluids; for scalar-fluid systems it becomes T- and ϕ-dependent through Veff. Not fitted here, but must be supplied by the user.
  • Shear and bulk viscosities ν, ξ (or comoving ν̃, ξ̃)
    Transport coefficients fixed by microphysics or set by hand; paper gives order-of-magnitude early-Universe estimates but treats them as input.
  • Electrical conductivity σf (or diffusivity ηdiff)
    Controls Ohmic dissipation and the ideal-MHD limit; user-supplied.
  • Scalar damping coefficient ηϕ (or Aϕ)
    Phenomenological out-of-equilibrium coupling of the fluid to the scalar; taken from prior literature or set constant.
  • Lattice order p, time-stepper coefficients, α-time choice
    Purely numerical choices that affect truncation error but not the continuum target.
axioms (5)
  • domain assumption Matter content admits a hydrodynamic description with λdB ≪ lmfp ≪ lcont ≪ L, so that a stress-energy tensor and constitutive relations close the system.
    Stated at the opening of Sec. 2; standard kinetic-theory premise for early-Universe plasma.
  • domain assumption Deviations from LTE are small (Knudsen number ≪ 1) and may be truncated at first-order Navier-Stokes in the Landau frame.
    Sec. 2.3.2; paper explicitly notes acausality and defers higher-order causal hydro.
  • domain assumption Background metric is flat FLRW; back-reaction appears only through the averaged Friedmann equations.
    Sec. 2.2; metric perturbations are not dynamical except for the passive GW sector of Sec. 7.
  • domain assumption Abelian Ohm’s law with conductivity σf closes the fluid-gauge coupling; non-Abelian fluids are not treated.
    Sec. 2.4.1.
  • standard math Standard continuum vector calculus, Einstein summation in the continuum, and periodic lattice Fourier analysis.
    Used throughout Secs. 2–3.

pith-pipeline@v1.2.0-grok45-kimik3 · 82689 in / 3034 out tokens · 58170 ms · 2026-07-30T11:17:36.804037+00:00 · methodology

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We discuss lattice methods for the simulation of fluid dynamics in the early Universe. This review represents a third entry in the monographic series on lattice cosmology techniques~\cite{Figueroa:2020rrl,Baeza-Ballesteros:2025tme}, which previously covered canonical and non-canonical field theory dynamics. Here, we first review the continuum theory of fluid dynamics in flat spacetime, and then in an FLRW background. We consider conservation and non-conservation forms of the equations of motion for fluids in isolation or coupled to scalar and/or gauge fields, and either fully relativistic or subrelativistic regimes of fluid bulk motion. After reviewing basic lattice concepts, we introduce detailed discretization schemes for fluid dynamics in expanding backgrounds for: $i)$ isolated perfect fluids, $ii)$ isolated imperfect (viscous) fluids, $iii)$ fluids coupled to gauge fields, and $iv)$ fluids coupled to scalar fields. Our evolution algorithms accommodate self-consistent expansion sourced by all scalar, gauge, and fluid sectors, preserving gauge invariance to machine precision in some cases. We also review lattice methods to set up the initial conditions for fluids, and the implementation of gravitational wave dynamics sourced by all scalar, gauge, and fluid degrees of freedom. This document represents the theoretical basis for the scalar-gauge-fluid module that will be publicly released as part of ${\mathcal C}{\tt osmo}{\mathcal L}{\tt attice}~{\tt v3.0}$ after publication of this monograph, check http://www.cosmolattice.com for updates.

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Reference graph

Works this paper leans on

292 extracted references · 10 canonical work pages

  1. [1]

    D. G. Figueroa, A. Florio, F. Torrenti and W. Valken- burg, JCAP04, 035 (2021), [2006.15122], 10.1088/1475- 7516/2021/04/035

    Pith/arXiv arXiv 2021

  2. [2]

    Baeza-Ballesteroset al., JCAP06, 087 (2026), [2512.15627], 10.1088/1475-7516/2026/06/087

    J. Baeza-Ballesteroset al., JCAP06, 087 (2026), [2512.15627], 10.1088/1475-7516/2026/06/087

    arXiv 2026

  3. [5]

    Albrecht and P

    A. Albrecht and P. J. Steinhardt, Phys. Rev. Lett.48, 1220 (1982), 10.1103/PhysRevLett.48.1220

    doi:10.1103/physrevlett.48.1220 1982

  4. [6]

    Brout, F

    R. Brout, F. Englert and E. Gunzig, Annals Phys.115, 78 (1978), 10.1016/0003-4916(78)90176-8

    work page doi:10.1016/0003-4916(78)90176-8 1978

  5. [8]

    Kazanas, Astrophys

    D. Kazanas, Astrophys. J. Lett.241, L59 (1980), 10.1086/183361

    doi:10.1086/183361 1980

  6. [9]

    Sato, Mon

    K. Sato, Mon. Not. Roy. Astron. Soc.195, 467 (1981)

    1981

  7. [10]

    V. F. Mukhanov and G. V. Chibisov, JETP Lett.33, 532 (1981)

    1981

  8. [11]

    A. H. Guth and S. Y. Pi, Phys. Rev. Lett.49, 1110 (1982), 10.1103/PhysRevLett.49.1110

    doi:10.1103/physrevlett.49.1110 1982

  9. [12]

    A. A. Starobinsky, Phys. Lett. B117, 175 (1982), 10.1016/0370-2693(82)90541-X

    doi:10.1016/0370-2693(82)90541-x 1982

  10. [14]

    J. M. Bardeen, P. J. Steinhardt and M. S. Turner, Phys. Rev. D28, 679 (1983), 10.1103/PhysRevD.28.679

    doi:10.1103/physrevd.28.679 1983

  11. [15]

    Aghanimet al., Astron

    Planck, N. Aghanimet al., Astron. Astrophys.641, A1 (2020), [1807.06205], 10.1051/0004-6361/201833880

    Pith/arXiv arXiv 2020

  12. [16]

    Aghanimet al., Astron

    Planck, N. Aghanimet al., Astron. Astrophys.641, A6 (2020), [1807.06209], 10.1051/0004-6361/201833910

    Pith/arXiv arXiv 2020

  13. [17]

    Akramiet al., Astron

    Planck, Y. Akramiet al., Astron. Astrophys.641, A10 (2020), [1807.06211], 10.1051/0004-6361/201833887

    Pith/arXiv arXiv 2020

  14. [18]

    J. H. Traschen and R. H. Brandenberger, Phys. Rev. D42, 2491 (1990), 10.1103/PhysRevD.42.2491

    doi:10.1103/physrevd.42.2491 1990

  15. [19]

    Kofman, A

    L. Kofman, A. D. Linde and A. A. Starobinsky, Phys. Rev. Lett.73, 3195 (1994), [hep-th/9405187], 10.1103/Phys- RevLett.73.3195

    Pith/arXiv arXiv 1994

  16. [20]

    Shtanov, J

    Y. Shtanov, J. H. Traschen and R. H. Brandenberger, Phys. Rev. D51, 5438 (1995), [hep-ph/9407247], 10.1103/Phys- RevD.51.5438

    Pith/arXiv arXiv 1995

  17. [21]

    D. I. Kaiser, Phys. Rev. D53, 1776 (1996), [astro- ph/9507108], 10.1103/PhysRevD.53.1776

    arXiv 1996

  18. [22]

    Kofman, A

    L. Kofman, A. D. Linde and A. A. Starobinsky, Phys. Rev. D56, 3258 (1997), [hep-ph/9704452], 10.1103/Phys- RevD.56.3258

    Pith/arXiv arXiv 1997

  19. [23]

    P. B. Greene, L. Kofman, A. D. Linde and A. A. Starobin- sky, Phys. Rev. D56, 6175 (1997), [hep-ph/9705347], 10.1103/PhysRevD.56.6175

    Pith/arXiv arXiv 1997

  20. [24]

    D. I. Kaiser, Phys. Rev. D56, 706 (1997), [hep-ph/9702244], 10.1103/PhysRevD.56.706

    Pith/arXiv arXiv 1997

  21. [25]

    D. I. Kaiser, Phys. Rev. D57, 702 (1998), [hep-ph/9707516], 10.1103/PhysRevD.57.702

    Pith/arXiv arXiv 1998

  22. [26]

    P. B. Greene and L. Kofman, Phys. Lett. B448, 6 (1999), [hep-ph/9807339], 10.1016/S0370-2693(99)00020-9

    Pith/arXiv arXiv 1999

  23. [27]

    P. B. Greene and L. Kofman, Phys. Rev. D62, 123516 (2000), [hep-ph/0003018], 10.1103/PhysRevD.62.123516

    Pith/arXiv arXiv 2000

  24. [28]

    Peloso and L

    M. Peloso and L. Sorbo, JHEP05, 016 (2000), [hep- ph/0003045], 10.1088/1126-6708/2000/05/016

    arXiv 2000

  25. [29]

    Berges, D

    J. Berges, D. Gelfand and J. Pruschke, Phys. Rev. Lett.107, 061301 (2011), [1012.4632], 10.1103/PhysRevLett.107.061301

    Pith/arXiv arXiv 2011

  26. [30]

    Enqvist, D

    K. Enqvist, D. G. Figueroa and R. N. Lerner, JCAP01, 040 (2013), [1211.5028], 10.1088/1475-7516/2013/01/040

    Pith/arXiv arXiv 2013

  27. [31]

    D. G. Figueroa, J. Garcia-Bellido and F. Torrenti, Phys. Rev. D92, 083511 (2015), [1504.04600], 10.1103/Phys- RevD.92.083511

    Pith/arXiv arXiv 2015

  28. [32]

    B. A. Bassett, D. I. Kaiser and R. Maartens, Phys. Lett. B455, 84 (1999), [hep-ph/9808404], 10.1016/S0370- 2693(99)00478-5

    Pith/arXiv arXiv 1999

  29. [33]

    B. A. Bassett, F. Tamburini, D. I. Kaiser and R. Maartens, Nucl. Phys. B561, 188 (1999), [hep-ph/9901319], 10.1016/S0550-3213(99)00495-2

    Pith/arXiv arXiv 1999

  30. [34]

    B. A. Bassett, C. Gordon, R. Maartens and D. I. Kaiser, Phys. Rev. D61, 061302 (2000), [hep-ph/9909482], 10.1103/Phys- RevD.61.061302

    Pith/arXiv arXiv 2000

  31. [35]

    Finelli and R

    F. Finelli and R. H. Brandenberger, Phys. Rev. D62, 083502 (2000), [hep-ph/0003172], 10.1103/PhysRevD.62.083502

    Pith/arXiv arXiv 2000

  32. [36]

    Chambers and A

    A. Chambers and A. Rajantie, Phys. Rev. Lett.100, 041302 (2008), [0710.4133], 10.1103/PhysRevLett.100.041302

    Pith/arXiv arXiv 2008

  33. [37]

    J. R. Bond, A. V. Frolov, Z. Huang and L. Kofman, Phys. Rev. Lett.103, 071301 (2009), [0903.3407], 10.1103/Phys- RevLett.103.071301

    Pith/arXiv arXiv 2009

  34. [38]

    Linde, S

    A. Linde, S. Mooij and E. Pajer, Phys. Rev. D87, 103506 (2013), [1212.1693], 10.1103/PhysRevD.87.103506

    Pith/arXiv arXiv 2013

  35. [39]

    S. V. Imrith, D. J. Mulryne and A. Rajantie, Phys. Rev. D100, 043543 (2019), [1903.07487], 10.1103/Phys- RevD.100.043543

    Pith/arXiv arXiv 2019

  36. [40]

    Musoke, S

    N. Musoke, S. Hotchkiss and R. Easther, Phys. Rev. Lett.124, 061301 (2020), [1909.11678], 10.1103/Phys- RevLett.124.061301

    Pith/arXiv arXiv 2020

  37. [41]

    Martin, T

    J. Martin, T. Papanikolaou, L. Pinol and V. Vennin, JCAP05, 003 (2020), [2002.01820], 10.1088/1475-7516/2020/05/003

    Pith/arXiv arXiv 2020

  38. [42]

    Adshead, J

    P. Adshead, J. T. Giblin, R. Grutkoski and Z. J. Weiner, JCAP03, 017 (2024), [2311.01504], 10.1088/1475- 7516/2024/03/017

    Pith/arXiv arXiv 2024

  39. [43]

    Cotner, A

    E. Cotner, A. Kusenko, M. Sasaki and V. Takhistov, JCAP10, 077 (2019), [1907.10613], 10.1088/1475-7516/2019/10/077

    Pith/arXiv arXiv 2019

  40. [44]

    Martin, T

    J. Martin, T. Papanikolaou and V. Vennin, JCAP01, 024 (2020), [1907.04236], 10.1088/1475-7516/2020/01/024

    Pith/arXiv arXiv 2020

  41. [45]

    Garcia-Bellido, A

    J. Garcia-Bellido, A. D. Linde and D. Wands, Phys. Rev. D54, 6040 (1996), [astro-ph/9605094], 10.1103/PhysRevD.54.6040

    Pith/arXiv arXiv 1996

  42. [46]

    A. M. Green and K. A. Malik, Phys. Rev. D64, 021301 (2001), [hep-ph/0008113], 10.1103/PhysRevD.64.021301

    Pith/arXiv arXiv 2001

  43. [47]

    Cotner, A

    E. Cotner, A. Kusenko and V. Takhistov, Phys. Rev. D98, 083513 (2018), [1801.03321], 10.1103/PhysRevD.98.083513

    Pith/arXiv arXiv 2018

  44. [48]

    Brandenburg, K

    A. Brandenburg, K. Enqvist and P. Olesen, Phys. Rev. D54, 1291 (1996), [astro-ph/9602031], 10.1103/PhysRevD.54.1291

    Pith/arXiv arXiv 1996

  45. [49]

    Jedamzik, V

    K. Jedamzik, V. Katalinic and A. V. Olinto, Phys. Rev. D57, 3264 (1998), [astro-ph/9606080], 10.1103/PhysRevD.57.3264

    Pith/arXiv arXiv 1998

  46. [50]

    Subramanian and J

    K. Subramanian and J. D. Barrow, Phys. Rev. D58, 083502 (1998), [astro-ph/9712083], 10.1103/PhysRevD.58.083502

    Pith/arXiv arXiv 1998

  47. [51]

    Christensson, M

    M. Christensson, M. Hindmarsh and A. Brandenburg, Phys. Rev. E64, 056405 (2001), [astro-ph/0011321], 10.1103/Phys- RevE.64.056405

    Pith/arXiv arXiv 2001

  48. [52]

    Kosowsky, A

    A. Kosowsky, A. Mack and T. Kahniashvili, Phys. Rev. D66, 024030 (2002), [astro-ph/0111483], 10.1103/Phys- RevD.66.024030

    Pith/arXiv arXiv 2002

  49. [53]

    Caprini and R

    C. Caprini and R. Durrer, Phys. Rev. D74, 063521 (2006), [astro-ph/0603476], 10.1103/PhysRevD.74.063521

    Pith/arXiv arXiv 2006

  50. [54]

    Gogoberidze, T

    G. Gogoberidze, T. Kahniashvili and A. Kosowsky, Phys. Rev. D76, 083002 (2007), [0705.1733], 10.1103/Phys- RevD.76.083002

    Pith/arXiv arXiv 2007

  51. [55]

    Caprini, R

    C. Caprini, R. Durrer and G. Servant, JCAP12, 024 (2009), [0909.0622], 10.1088/1475-7516/2009/12/024

    Pith/arXiv arXiv 2009

  52. [56]

    Kahniashvili, A

    T. Kahniashvili, A. Brandenburg, A. G. Tevzadze and B. Ratra, Phys. Rev. D81, 123002 (2010), [1004.3084], 10.1103/PhysRevD.81.123002. 83

    Pith/arXiv arXiv 2010

  53. [57]

    Brandenburg, T

    A. Brandenburg, T. Kahniashvili and A. G. Tevzadze, Phys. Rev. Lett.114, 075001 (2015), [1404.2238], 10.1103/Phys- RevLett.114.075001

    Pith/arXiv arXiv 2015

  54. [58]

    Brandenburget al., Phys

    A. Brandenburget al., Phys. Rev. D96, 123528 (2017), [1711.03804], 10.1103/PhysRevD.96.123528

    Pith/arXiv arXiv 2017

  55. [59]

    Brandenburget al., Phys

    A. Brandenburget al., Phys. Rev. Fluids.4, 024608 (2019), [1710.01628], 10.1103/PhysRevFluids.4.024608

    Pith/arXiv arXiv 2019

  56. [60]

    Roper Pol and A

    A. Roper Pol and A. S. Midiri, JCAP06, 070 (2026), [2501.05732], 10.1088/1475-7516/2026/06/070

    Pith/arXiv arXiv 2026

  57. [61]

    Gleiser, Phys

    M. Gleiser, Phys. Rev. D49, 2978 (1994), [hep-ph/9308279], 10.1103/PhysRevD.49.2978

    Pith/arXiv arXiv 1994

  58. [62]

    E. J. Copeland, M. Gleiser and H. R. Muller, Phys. Rev. D52, 1920 (1995), [hep-ph/9503217], 10.1103/PhysRevD.52.1920

    Pith/arXiv arXiv 1920

  59. [63]

    M. A. Amin, R. Easther and H. Finkel, JCAP12, 001 (2010), [1009.2505], 10.1088/1475-7516/2010/12/001

    Pith/arXiv arXiv 2010

  60. [64]

    M. A. Amin, R. Easther, H. Finkel, R. Flauger and M. P. Hertzberg, Phys. Rev. Lett.108, 241302 (2012), [1106.3335], 10.1103/PhysRevLett.108.241302

    Pith/arXiv arXiv 2012

  61. [65]

    Gleiser, N

    M. Gleiser, N. Graham and N. Stamatopoulos, Phys. Rev. D 83, 096010 (2011), [1103.1911], 10.1103/PhysRevD.83.096010

    Pith/arXiv arXiv 2011

  62. [66]

    Antusch and S

    S. Antusch and S. Orani, JCAP03, 026 (2016), [1511.02336], 10.1088/1475-7516/2016/03/026

    Pith/arXiv arXiv 2016

  63. [67]

    K. D. Lozanov and M. A. Amin, Phys. Rev. D97, 023533 (2018), [1710.06851], 10.1103/PhysRevD.97.023533

    Pith/arXiv arXiv 2018

  64. [68]

    Hasegawa and J.-P

    F. Hasegawa and J.-P. Hong, Phys. Rev. D97, 083514 (2018), [1710.07487], 10.1103/PhysRevD.97.083514

    Pith/arXiv arXiv 2018

  65. [69]

    M. A. Aminet al., Phys. Rev. D98, 024040 (2018), [1803.08047], 10.1103/PhysRevD.98.024040

    Pith/arXiv arXiv 2018

  66. [70]

    Kitajima, J

    N. Kitajima, J. Soda and Y. Urakawa, JCAP10, 008 (2018), [1807.07037], 10.1088/1475-7516/2018/10/008

    Pith/arXiv arXiv 2018

  67. [71]

    Antusch, F

    S. Antusch, F. Cefal` a and F. Torrent ´ ı, JCAP10, 002 (2019), [1907.00611], 10.1088/1475-7516/2019/10/002

    Pith/arXiv arXiv 2019

  68. [72]

    M. Ibe, M. Kawasaki, W. Nakano and E. Sonomoto, Phys. Rev. D100, 125021 (2019), [1908.11103], 10.1103/Phys- RevD.100.125021

    Pith/arXiv arXiv 2019

  69. [73]

    Sang and Q.-G

    Y. Sang and Q.-G. Huang, Phys. Rev. D100, 063516 (2019), [1905.00371], 10.1103/PhysRevD.100.063516

    Pith/arXiv arXiv 2019

  70. [74]

    X.-X. Kou, C. Tian and S.-Y. Zhou, Class. Quant. Grav.38, 045005 (2021), [1912.09658], 10.1088/1361-6382/abd09f

    Pith/arXiv arXiv 2021

  71. [75]

    Nazari, M

    Z. Nazari, M. Cicoli, K. Clough and F. Muia, JCAP05, 027 (2021), [2010.05933], 10.1088/1475-7516/2021/05/027

    Pith/arXiv arXiv 2021

  72. [76]

    Sang and Q.-G

    Y. Sang and Q.-G. Huang, Phys. Lett. B823, 136781 (2021), [2012.14697], 10.1016/j.physletb.2021.136781

    Pith/arXiv arXiv 2021

  73. [77]

    J. C. Aurrekoetxea, K. Clough and F. Muia, Phys. Rev. D108, 023501 (2023), [2304.01673], 10.1103/PhysRevD.108.023501

    Pith/arXiv arXiv 2023

  74. [78]

    Mahbub and S

    R. Mahbub and S. S. Mishra, Phys. Rev. D108, 063524 (2023), [2303.07503], 10.1103/PhysRevD.108.063524

    Pith/arXiv arXiv 2023

  75. [79]

    Piani and J

    M. Piani and J. Rubio, JCAP12, 002 (2023), [2304.13056], 10.1088/1475-7516/2023/12/002

    Pith/arXiv arXiv 2023

  76. [80]

    Shafi, E

    M. Shafi, E. J. Copeland, R. Mahbub, S. S. Mishra and S. Basak, JCAP10, 082 (2024), [2406.00108], 10.1088/1475- 7516/2024/10/082

    Pith/arXiv arXiv 2024

  77. [81]

    Drees and C

    M. Drees and C. Wang, JCAP04, 078 (2025), [2501.13811], 10.1088/1475-7516/2025/04/078

    Pith/arXiv arXiv 2025

  78. [82]

    Piani, J

    M. Piani, J. Rubio and F. Torrenti, JCAP06, 024 (2025), [2501.14869], 10.1088/1475-7516/2025/06/024

    Pith/arXiv arXiv 2025

  79. [83]

    A. H. Guth and E. J. Weinberg, Nucl. Phys. B212, 321 (1983), 10.1016/0550-3213(83)90307-3

    doi:10.1016/0550-3213(83)90307-3 1983

  80. [85]

    Kosowsky, M

    A. Kosowsky, M. S. Turner and R. Watkins, Phys. Rev. D45, 4514 (1992), 10.1103/PhysRevD.45.4514

    doi:10.1103/physrevd.45.4514 1992

Showing first 80 references.