Pith. sign in

REVIEW 3 major objections 5 minor 74 references

Near the big bang, a non-stiff fluid's velocity in inhomogeneous cosmologies oscillates between tilted and orthogonal states, generating matter-density ripples.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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

T0 review

2026-08-03 16:52 UTC pith:J4YPL2GJ

load-bearing objection Solid, novel numerical evidence for tilt transitions in T2-symmetric non-stiff fluid cosmologies, but the untested path-dependence of the shock-capturing scheme and the over-strong causal claim in the abstract need attention before this is more than conditional. the 3 major comments →

arxiv 2512.11375 v3 pith:J4YPL2GJ submitted 2025-12-12 gr-qc

Mixmaster Fluids Near the Big Bang

Elliot Marshall This is my paper
classification gr-qc
keywords tilt transitionsmixmaster oscillationsBKL conjectureT^2 symmetrynon-stiff perfect fluidcosmological singularitydensity inhomogeneitiesnumerical relativity
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 paper reports numerical simulations of T^2-symmetric cosmological spacetimes containing a non-stiff perfect fluid (pressure p=Kρ with 0≤K<1) as the big-bang singularity is approached. It claims to show, for the first time, that the fluid velocity develops mixmaster-like oscillations — repeated transitions between orthogonal and extremely tilted states — driven by the gravitational Kasner oscillations. These tilt transitions are shown to follow simple trigger conditions based on the Kasner exponents and the sound speed K. The paper further finds that these oscillations generate local inhomogeneities in the matter density, scattered chaotically across space, in a way that supports the generalised BKL picture for fluid-filled inhomogeneous cosmologies. A sympathetic reader would care because it extends the oscillatory big-bang scenario to realistic non-stiff matter and links it to primordial density fluctuations.

Core claim

The central claim is that the approach to the singularity in inhomogeneous T^2-symmetric spacetimes with a non-stiff perfect fluid is local, oscillatory, and governed by the vacuum mixmaster dynamics with the fluid acting as a test field. The new observation is that the fluid velocity undergoes repeated tilt transitions, oscillating between orthogonal (|ν|=0) and extremely tilted (|ν|→1) states, in response to the Kasner oscillations of the gravitational field. The paper derives linearised trigger conditions — a velocity component grows when the corresponding Kasner exponent exceeds the sound-speed square K — and shows numerically that these conditions correctly predict the growth, decay, an

What carries the argument

The analysis is carried out in the β-normalised orthonormal-frame formulation of the Einstein-Euler equations for T^2-symmetric spacetimes in a timelike areal gauge. The key numerical tool is a path-conservative finite-volume scheme that evolves the primitive fluid variables (log T^00 and the three-velocity ν^A) directly, avoiding the primitive-recovery failures that occur when conserved variables decay; it also enforces the sub-luminal constraint |ν|<1 by rescaling. The physical mechanism is the tilt transition: at a given point, the fluid alternates between orthogonal and extremely tilted Kasner states, with the transition triggered by the sign of PA−K, where PA are the Kasner exponents an

Load-bearing premise

The numerical results assume that the periodic boundary conditions used for all runs do not alter the qualitative late-time behaviour — the paper itself notes that periodicity can suppress spatial-curvature growth and possibly change the initial inhomogeneous phase, and it does not compare with excision-type boundaries.

What would settle it

Run the same simulations with an excision-type or 'zooming' boundary (e.g., as used in some G2 cosmological codes) and compare the tilt-transition times and the spatial distribution of density-gradient spikes at late times; a qualitative mismatch with the periodic-boundary runs would falsify the claim that the observed oscillations are boundary-independent.

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

Share X Bluesky LinkedIn Reddit HN

If this is right

  • If the claim holds, the generalised BKL conjecture extends to inhomogeneous cosmologies with non-stiff perfect fluids: the approach to the singularity is local, oscillatory, and the fluid is a test field undergoing tilt transitions.
  • Matter density inhomogeneities can be generated throughout the spatial domain by tilt-transition dynamics, providing a possible mechanism for primordial density fluctuations and structure formation.
  • The tilt-transition triggers depend on the sign of PA−K, so the sound speed K controls the statistical behaviour: low-K fluids tend to extreme tilt, high-K (stiff) fluids to orthogonality, and intermediate K to sustained oscillation.
  • The numerical method — evolving primitive fluid variables with a path-conservative finite-volume scheme — proves stable through many tilt transitions, where previous simulations crashed.

Where Pith is reading between the lines

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

  • Because the paper uses periodic boundary conditions and flags that periodicity can suppress spatial-curvature growth, the chaotic spatial pattern of density inhomogeneities may depend on boundary conditions; testing with excision-type boundaries would clarify this.
  • The tilt-transition mechanism is essentially a test-field response to a known vacuum attractor, so it likely persists for other matter such as magnetic fields or multiple fluids; the trigger condition PA−K might act as a general selection rule.
  • A quantitative test of the density-fluctuation claim would be to compare the statistics of ∂xρ/ρ spikes against a simple Markov map for the Kasner-to-Kasner transition; the paper does not provide such a statistical model.
  • In 3+1 models removing the T^2 symmetry, tilt transitions would likely occur in all velocity components, potentially making the density-fluctuation mechanism generic rather than symmetry-induced.

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 / 5 minor

Summary. This paper presents numerical simulations of T^2-symmetric cosmological spacetimes with a non-stiff perfect fluid obeying p=Kρ for K∈[0,1). The author uses β-normalized orthonormal-frame variables, evolves the primitive fluid variables directly, and employs a path-conservative finite-volume scheme to handle shocks. The central claims are: (i) the fluid velocity in inhomogeneous cosmologies undergoes mixmaster-like tilt transitions, consistent with the generalized BKL conjecture of Uggla et al.; (ii) these tilt transitions generate local matter-density inhomogeneities of the Rendall type. The paper also derives linearized trigger conditions for tilt transitions from spatially homogeneous Kasner states, and reports good agreement between those conditions and the observed growth/decay of the velocity components. The numerical setup is described in detail, with convergence tests in the pre-mixmaster phase and constraint monitoring in the chaotic phase.

Significance. If the central claims are correct, this is the first numerical evidence of repeated tilt transitions in inhomogeneous non-stiff fluid cosmologies near the big bang, providing concrete support for the generalized BKL picture beyond vacuum mixmaster models. The connection between tilt transitions and the Rendall instability is also physically interesting, suggesting a mechanism for primordial density inhomogeneities. The paper has several genuine strengths: the trigger conditions in Section 3 are derived from linearized homogeneous equations rather than fitted to the numerical output, so the agreement in Section 5.1 is a validation rather than a tautology; the primitive-variable evolution is well motivated by the failure of conservative variable recovery; and the monitoring of constraint residuals in the chaotic regime is a useful sanity check. However, the headline claims currently rest on a path-conservative scheme whose path dependence is acknowledged but untested, and on pointwise comparisons in a chaotic regime where formal numerical convergence is absent.

major comments (3)
  1. [§4.1, Eqs. (4.4)-(4.5), Figs. 3 and 10] The generalized Rankine-Hugoniot condition (4.5) depends on the DLM path, and the paper acknowledges that the path-selection problem is not addressed and convergence to the path-defined solution is not guaranteed. This is load-bearing because Fig. 3 shows that shocks in ν^1 form during the initial inhomogeneous phase, at t≈-0.25 to -1.0, before the ODE-dominated mixmaster regime begins. During this phase the non-conservative products are non-negligible, so different paths will in general produce different shock speeds and strengths. The resulting state at the onset of the mixmaster regime is therefore path-dependent, and since the subsequent Kasner/tilt sequence is chaotic, the pointwise tilt transitions (Fig. 5) and the spatial pattern of density gradients (Fig. 10) may depend on the arbitrary choice of the segment path. The argument that the path plays a small role because the mixmaste
  2. [§4.4 and §5.1, Figs. 5-10] The convergence tests establish formal second-order behavior only before the mixmaster regime; the paper states that in the chaotic mixmaster regime convergence is 'notably worse'. Yet the central evidence—tilt transitions in Fig. 5, the trigger-condition comparisons in Figs. 6-8, and the density inhomogeneity in Fig. 10—is presented at single resolutions (N=500 or N=1000) without a resolution study of the qualitative features. Because chaotic systems do not converge pointwise, the meaningful objects should be statistical or coarse-grained (e.g., distributions of tilt-transition times, spatial density-gradient statistics, or counts of transitions) and shown to be stable under resolution refinements and under the path choice. Monitoring constraint residuals alone does not establish that the pointwise tilt sequence is physical. This is a load-bearing gap for the paper's specific claims.
  3. [Remark 4.1] The paper explicitly acknowledges that periodic boundary conditions may suppress the growth of spatial curvature and influence the initial inhomogeneous phase, which in turn may alter the state when mixmaster dynamics begin. This is exactly the phase that sets the initial data for the chaotic regime, so the concern is not merely technical. No test with alternative boundary treatments (e.g., excision-type or 'zooming' boundaries) or with different domain sizes is provided. Given that the abstract claims generic inhomogeneous behavior and attributes density inhomogeneities to the oscillations, the possible sensitivity to periodic boundaries should be either addressed numerically or reflected in a significantly more cautious phrasing of the conclusions.
minor comments (5)
  1. [Fig. 2] The caption refers to the constraint (CM)_1, but the constraint equations are labeled (CM)_2 and (CM)_3 in Eqs. (2.9)-(2.10). This labeling should be corrected for consistency.
  2. [§4.1, footnote 20] The footnote states that λ±_{j+1/2} in the numerical scheme are not the fluid characteristic speeds λ± defined in (2.20), but the notation is confusing because the same symbol is used. Please use distinct notation for the numerical dissipation speeds.
  3. [§3 and Table 1] The derivation text writes the growth condition as K−P_A<0 while Table 1 states P_A−K>0; these are equivalent but the sign reversal may confuse readers. A brief sentence aligning the two notations would help.
  4. [Figs. 5 and 6] Different resolutions are used for the principal tilt-transition plots (N=500 in Fig. 5, N=1000 in Fig. 6) without explanation. It would be clearer to either use the same resolution or justify the difference.
  5. [General] No code or data are shipped, and the paper does not state whether the author is willing to provide them upon request. Given the numerical nature of the claims and the acknowledged subtleties of path-conservative schemes, a reproducibility statement would be valuable.

Circularity Check

0 steps flagged

No significant circularity: the tilt-trigger conditions are derived independently and then validated against the numerics, not fitted or renamed as predictions.

full rationale

I traced the paper's derivation chain: the Einstein-Euler equations (Section 2) are reduced to β-normalized form; Section 3 linearizes the spatially homogeneous fluid equations about the orthogonal and extremely tilted Kasner equilibrium points to derive the trigger conditions in Table 1 (e.g., P_A - K > 0 for growth of ν^A). These conditions are obtained analytically from the stated equations and are not fitted to the numerical output. The subsequent comparison in Section 5.1 (e.g., 'changes in the growth and decay of the velocity components coincide with the changes in sign and size of the quantities P_A - K') is a validation against the simulation, not a tautology. The paper's self-citations [9,10,33,34,58] are used for context—gauge conventions, asymptotic stability results in other settings, and the known Rendall-instability mechanism—but they are not load-bearing for the central claim that the fluid develops mixmaster-like tilt transitions in inhomogeneous T^2-symmetric spacetimes. The acknowledged caveats about periodic boundaries (Remark 4.1) and the path-dependence of the path-conservative scheme (Section 4.1) are numerical limitations and physical uncertainties, not circular reasoning: the path choice affects the shock evolution, but the tilt-trigger prediction is not derived from that path choice. The generalized-BKL framework of Uggla et al. is used as an interpretive lens, not as an input fitted to produce the observed oscillations. Overall, the central derivation is self-contained against its own assumptions, and no prediction reduces by construction to an input or to a self-citation chain.

Axiom & Free-Parameter Ledger

4 free parameters · 5 axioms · 0 invented entities

The central claim relies on symmetry reduction, BKL locality, periodic boundary conditions, the non-conservative path choice, and the validity of homogeneous trigger conditions in an inhomogeneous setting. No new particles, forces, or conserved quantities are introduced. The listed free parameters are initial-data, physical-scan, and numerical-scheme choices rather than parameters fitted to force the observed result.

free parameters (4)
  • Initial data amplitudes a,b,c,d = a=b=-1, c=0.01, d=0.1
    Freely specifiable parameters in (4.13) chosen as a nonlinear perturbation of FLRW; not fitted to the target result, but the generality of the claim across initial data is not established.
  • Sound speed K = 0.1, 0.5, 0.9 in scans
    Physical parameter scanned to test the K-dependence of tilt behaviour; the oscillatory claim is demonstrated for these sampled values rather than the whole interval K in [0,1).
  • Velocity rescaling epsilon = 10^-13
    Ad hoc enforcement of |nu|<1 via (4.12); frequent clipping near extreme tilt could affect the fine dynamics, though the author reports it is robust.
  • CFL constant C = 0.01
    Numerical stability parameter; not physical, but affects the discrete evolution and thus the observed chaotic trajectory.
axioms (5)
  • domain assumption T2-symmetric reduction with timelike areal gauge and beta-normalized frame equations faithfully captures the relevant near-singularity dynamics.
    Used throughout Section 2; restricts results to 1+1 dimensional models with Abelian G2 symmetry and excludes Class B Bianchi cosmologies.
  • domain assumption BKL locality: as E1_1 decays, spatial-derivative terms become negligible so pointwise dynamics is ODE-dominated.
    Section 3; justifies comparing pointwise numerics to spatially homogeneous trigger conditions; supported indirectly by decay of E1_1 and T_ab in Figure 4.
  • domain assumption Periodic boundary conditions do not qualitatively change the late-time mixmaster/tilt-transition behaviour.
    Remark 4.1 acknowledges periodicity can suppress curvature growth and influence the initial inhomogeneous phase; this is not tested against excision-type boundaries.
  • domain assumption The segment path in the path-conservative scheme yields the physically relevant non-conservative shock solution.
    Section 4.1: DLM weak solutions depend on the chosen path; the author uses a linear segment and cites non-convergence caveats [1,17], arguing the path plays a small role once the ODE-dominated regime is reached.
  • domain assumption Spatially homogeneous linearized trigger conditions (Table 1) remain valid pointwise in the inhomogeneous setting.
    Section 3 derives them in the homogeneous truncation; Section 5.1 compares them to inhomogeneous simulations and finds agreement, but no rigorous justification is given for the pointwise transfer.

reviewed 2026-08-03 · how reviews work

0 comments
Cite this review

Pith. "Pith review of Mixmaster Fluids Near the Big Bang." pith.science (2026). https://pith.science/paper/J4YPL2GJ

@misc{pith2026251211375,
  author       = {Pith},
  title        = {Pith review of: Mixmaster Fluids Near the Big Bang},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/J4YPL2GJ}},
  note         = {Machine review of arXiv:2512.11375}
}
Share X Bluesky LinkedIn Reddit HN
read the original abstract

We numerically study the approach to the singularity in $\mathbb{T}^{2}$-symmetric cosmological spacetimes containing a non-stiff perfect fluid satisfying a linear equation of state $p=K\rho$, $K \in [0,1)$. Near the singularity, the dynamics are found to be local and oscillatory. In particular, our results show, for the first time, that the fluid velocity in inhomogeneous cosmologies develops mixmaster-esque oscillations consistent with the generalised BKL conjecture of Uggla et al. Moreover, we find these fluid oscillations are responsible for the development of local inhomogeneities in the matter density of the early universe.

Figures

Figures reproduced from arXiv: 2512.11375 by Elliot Marshall.

Figure 1
Figure 1. Figure 1: Convergence plots of Σ− and ν 1 at t ≈ −20, K = 0.5. As the chaotic mixmaster regime is reached, however, the convergence is notably worse. This is to be expected; numerical simulations of chaotic systems using standard techniques will not, in general, recover convergence to a reference solution. Nonetheless, as in [36], we observe that the constraint violation remains small during this regime, shown for t… view at source ↗
Figure 2
Figure 2. Figure 2: Convergence plots of log2 (∥(CM)1∥2), K = 0.5. 0 1 2 3 4 5 6 x −0.010 −0.005 0.000 0.005 0.010 ν 1 (a) t = 0.0 0 1 2 3 4 5 6 x −0.3 −0.2 −0.1 0.0 0.1 0.2 0.3 ν 1 (b) t = −0.25 0 1 2 3 4 5 6 x −1.0 −0.5 0.0 0.5 1.0 ν 1 (c) t = −0.5 0 1 2 3 4 5 6 x −1.0 −0.5 0.0 0.5 1.0 ν 1 (d) t = −1.0 [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Shock formation in ν 1 . N = 4000, K = 0.5. 5.1. Tilt Transitions. Since the stress-energy components are negligible, the evolution of the gravitational variables is indistinguishable from previous studies in vacuum, see for example [3,7,36]. Thus, for the remainder of this article we will focus on the behaviour of the fluid. In [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Maximum values of log(∣T˜00∣) and log(∣E 1 1 ∣) over time. N = 4000, K = 0.5. −300 −250 −200 −150 −100 −50 0 t 0.0 0.2 0.4 0.6 0.8 1.0 |ν| [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Tilt transitions in ∣ν∣ at single point in space. N = 500, K = 0.5. the components ∣ν A∣ with the quantities PA − K at a single point. The value of the norm ∣ν∣ at the same point is shown in [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Comparison of ν A and PA − K at the 520th cell. Observe that changes in the growth and decay of the velocity components coincide with the changes in sign and size of the quantities PA − K, following the trigger rules in [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The value of the norm ∣ν∣ at the 520th cell. N = 1000, K = 0.5. We also expect the size of the sound speed parameter K to affect the frequency of tilt transitions. To see this, recall that the Kasner exponents must satisfy the conditions (1.2), 3 ∑ A=1 PA = 1, 3 ∑ A=1 P 2 A = 1. In particular, the second condition implies P1 ≤ 1 and, hence, P1 − K ≤ 1 − K Thus, as K ↗ 1, we expect that P1 − K will tend to … view at source ↗
Figure 8
Figure 8. Figure 8: Comparison of ∣ν 1 ∣ and P1 − K at a single point in space. N = 500, K = 0.5. in between these two extremes, the fluid will consistently oscillate between the orthogonal and extremely tilted states. Indeed, this heuristic argument is supported by our numerical results. In [PITH_FULL_IMAGE:figures/full_fig_p018_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: , we compare the evolution of ∣ν∣ for K = 0.1, 0.5, and 0.9. As predicted, the K = 0.1 and K = 0.9 solutions tend to extremely tilted and orthogonal states, respectively, while the K = 0.5 solution oscillates between the two states. −250 −200 −150 −100 −50 0 t 0.0 0.2 0.4 0.6 0.8 1.0 |ν| K = 0.1 K = 0.5 K = 0.9 [PITH_FULL_IMAGE:figures/full_fig_p018_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The density gradient of the fluid at t ≈ −290, N = 500. either an orthogonal or extremely tilted state but cannot not oscillate between the two. Typically, this means that density inhomogeneities are only generated around a few points in space. However, the oscillatory nature of the fluid near the big bang in T 2 -symmetry means that these density inhomogeneities can be generated all over the spatial doma… view at source ↗

discussion (0)

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

Reference graph

Works this paper leans on

74 extracted references · 1 canonical work pages

  1. [1]

    Abgrall and S

    R. Abgrall and S. Karni,A Comment on the Computation of Non-Conservative Pproducts, Journal of Compu- tational Physics229(2010), no. 8, 2759–2763

    2010

  2. [2]

    Andersson and A.D

    L. Andersson and A.D. Rendall,Quiescent Cosmological Singularities, Commun. Math. Phys.218(2001), no. 3, 479–511

    2001

  3. [3]

    Andersson, H

    L. Andersson, H. van Elst, W.C. Lim, and C. Uggla,Asymptotic Silence of Generic Cosmological Singularities, Phys. Rev. Lett.94(2005), 051101

    2005

  4. [4]

    Aurrekoetxea, K

    J.C. Aurrekoetxea, K. Clough, and E.A. Lim,CTTK: A New Method to Solve the Initial Data Constraints in Numerical Relativity, Classical and Quantum Gravity40(March 2023), no. 7, 075003. 20

    2023

  5. [5]

    Belinskii, I.M

    V.A. Belinskii, I.M. Khalatnikov, and E.M. Lifshitz,Oscillatory Approach to a Singular Point in Relativistic Cosmology, Adv. Phys.19(1970), no. 80, 525–573

    1970

  6. [6]

    Berger and D

    B.K. Berger and D. Garfinkle,Phenomenology of the Gowdy universe onT 3×R, Physical Review D57(1998), no. 8, 4767–4777. DOI: 10.1103/PhysRevD.57.4767

    work page doi:10.1103/physrevd.57.4767 1998

  7. [7]

    Berger, J

    B.K. Berger, J. Isenberg, and M. Weaver,Oscillatory Approach to the Singularity in Vacuum Spacetimes with T 2 Isometry, Phys. Rev. D64(2001), no. 8

    2001

  8. [8]

    1, 64–84

    M.J.BergerandP.Colella,Local Adaptive Mesh Refinement for Shock Hydrodynamics,JournalofComputational Physics82(1989), no. 1, 64–84

    1989

  9. [9]

    Beyer, E

    F. Beyer, E. Marshall, and T.A. Oliynyk,Future Instability of FLRW Fluid Solutions for Linear Equations of Statep=Kρwith1/3<K<1, Phys. Rev. D107(2023), 104030

    2023

  10. [10]

    ,Past Instability of FLRW Solutions of the Einstein-Euler-Scalar Field Equations for Linear Equations of Statep=Kρwith0≤K<1/3, Physical Review D110(2024), no. 4

    2024

  11. [11]

    Beyer and T.A

    F. Beyer and T.A. Oliynyk,Localized Big Bang Stability for the Einstein-Scalar Field Equations, Archive for Rational Mechanics and Analysis248(December 2023), no. 1

    2023

  12. [12]

    2, 515–637

    ,Past Stability of FLRW Solutions to the Einstein-Euler-Scalar Field Equations and their Big Bang Singularities, Beijing Journal of Pure and Applied Mathematics1(2024), no. 2, 515–637

    2024

  13. [13]

    6, 1701–1794

    ,Relativistic Perfect Fluids Near Kasner Singularities, Communications in Analysis and Geometry32 (2024), no. 6, 1701–1794

    2024

  14. [14]

    Beyer, T.A

    F. Beyer, T.A. Oliynyk, and W. Zheng,Localized Past Stability of the Subcritical Kasner-Scalar Field Spacetimes (2025), available at2502.09210

    Pith/arXiv arXiv 2025

  15. [15]

    Bernhard Brehm,Bianchi VIII and IX Vacuum Cosmologies: Almost Every Solution Forms Particle Horizons and Converges to the Mixmaster Attractor(2016), available at1606.08058

    Pith/arXiv arXiv 2016

  16. [16]

    Castro, U.S

    M.J. Castro, U.S. Fjordholm, S. Mishra, and C. Parés,Entropy Conservative and Entropy Stable Schemes for Nonconservative Hyperbolic Systems, SIAM Journal on Numerical Analysis51(2013), no. 3, 1371–1391

    2013

  17. [17]

    Castro, P.G

    M.J. Castro, P.G. LeFloch, M.L. Muñoz-Ruiz, and C. Parés,Why Many Theories of Shock Waves are Necessary: Convergence Error in Formally Path-Consistent Schemes, Journal of Computational Physics227(2008), no. 17, 8107–8129

    2008

  18. [18]

    Chiodaroli,A Counterexample to Well-Posedness of Entropy Solutions to the Compressible Euler System, Journal of Hyperbolic Differential Equations11(September 2014), no

    E. Chiodaroli,A Counterexample to Well-Posedness of Entropy Solutions to the Compressible Euler System, Journal of Hyperbolic Differential Equations11(September 2014), no. 03, 493–519

    2014

  19. [19]

    Chiodaroli, C

    E. Chiodaroli, C. De Lellis, and O. Kreml,Global Ill-Posedness of the Isentropic System of Gas Dynamics, Communications on Pure and Applied Mathematics68(August 2014), no. 7, 1157–1190

    2014

  20. [20]

    Coley and S

    A. Coley and S. Hervik,A Dynamical Systems Approach to the Tilted Bianchi Models of Solvable Type, Classical and Quantum Gravity22(2005), no. 3, 579

    2005

  21. [21]

    ,Bianchi Models with Vorticity: The Type III Bifurcation, arXiv preprint arXiv:0802.3629 (2008)

    Pith/arXiv arXiv 2008

  22. [22]

    Coley and S

    A.A. Coley and S. Hervik,Bianchi Cosmologies: A Tale of Two Tilted Fluids, Classical and Quantum Gravity 21(2004), no. 17, 4193

    2004

  23. [23]

    Coley, S

    A.A. Coley, S. Hervik, and W.C. Lim,Fluid Observers and Tilting Cosmology, Classical and Quantum Gravity 23(2006), no. 10, 3573

    2006

  24. [24]

    4, 310–313

    ,Tilt and Phantom Cosmology, Physics Letters B638(2006), no. 4, 310–313

    2006

  25. [25]

    Coley and W.C

    A.A. Coley and W.C. Lim,Generating Matter Inhomogeneities in General Relativity, Phys. Rev. Lett.108 (2012), 191101

    2012

  26. [26]

    1, 015009

    ,Spikes and Matter Inhomogeneities in Massless Scalar Field Models, Classical and Quantum Gravity 33(2016), no. 1, 015009

    2016

  27. [27]

    1, 015009

    ,Periodic Boundary Conditions and G2 Cosmology, Classical and Quantum Gravity41(2023), no. 1, 015009

    2023

  28. [28]

    Curtis and D

    J. Curtis and D. Garfinkle,Numerical Simulations of Stiff Fluid Gravitational Singularities, Physical Review D 72(2005), no. 6, 064003

    2005

  29. [29]

    Dal Maso, P.G

    G. Dal Maso, P.G. LeFloch, and F. Murat,Definition and Weak Stability of Nonconservative Products, Journal de mathématiques pures et appliquées74(1995), no. 6, 483–548

    1995

  30. [30]

    M.J.C. Diaz, A. Kurganov, and T.M. de Luna,Path-Conservative Central-Upwind Schemes for Nonconservative Hyperbolic Systems, ESAIM: Mathematical Modelling and Numerical Analysis53(2019), no. 3, 959–985

    2019

  31. [31]

    Ellis,Dynamics of Pressure-Free Matter in General Relativity, Journal of Mathematical Physics8 (1967), no

    G.F.R. Ellis,Dynamics of Pressure-Free Matter in General Relativity, Journal of Mathematical Physics8 (1967), no. 5, 1171–1194

    1967

  32. [32]

    Ellis and M.A.H

    G.F.R. Ellis and M.A.H. MacCallum,A Class of Homogeneous Cosmological Models, Communications in Mathematical Physics12(1969), no. 2, 108 –141

    1969

  33. [33]

    Fournodavlos, E

    G. Fournodavlos, E. Marshall, and T.A. Oliynyk,Future Stability of Perfect Fluids with Extreme Tilt and Linear Equation of Statep=c 2 sρ for the Einstein-Euler System with Positive Cosmological Constant: The Range 1 3 <c 2 s < 3 7 (2024), available at2404.06789

    Pith/arXiv arXiv 2024

  34. [34]

    ,Future Stability of Tilted Two-Fluid Bianchi I Spacetimes(2025), available at2508.15155

    Pith/arXiv arXiv 2025

  35. [35]

    Fournodavlos, I

    G. Fournodavlos, I. Rodnianski, and J. Speck,Stable Big Bang Formation for Einstein’s Equations: The Complete Sub-Critical Regime, J. Amer. Math. Soc.36(July 2023), 827–916

    2023

  36. [36]

    Garfinkle,Numerical Simulations of Generic Singularities, Physical Review Letters93(2004), no

    D. Garfinkle,Numerical Simulations of Generic Singularities, Physical Review Letters93(2004), no. 16, 6

    2004

  37. [37]

    12, S295–S306

    ,Numerical simulations of general gravitational singularities, Classical and Quantum Gravity24(2007), no. 12, S295–S306. 21

    2007

  38. [38]

    Garfinkle, A

    D. Garfinkle, A. Ijjas, and P.J. Steinhardt,Initial Conditions Problem in Cosmological Inflation Revisited, Physics Letters B843(2023), 138028

    2023

  39. [39]

    Garfinkle and F

    D. Garfinkle and F. Pretorius,Spike Behavior in the Approach to Spacetime Singularities, Physical Review D 102(December 2020), no. 12

    2020

  40. [40]

    Groeniger, O

    H.O. Groeniger, O. Petersen, and H. Ringström,Formation of Quiescent Big Bang Singularities(2023), available at2309.11370

    arXiv 2023

  41. [41]

    Hawking and G.F.R

    S.W. Hawking and G.F.R. Ellis,The Large Scale Structure of Space-Time, First, Cambridge University Press, 1973

    1973

  42. [42]

    Uggla,A New Proof of the Bianchi Type IX Attractor Theorem, Class

    J.M Heinzle and C. Uggla,A New Proof of the Bianchi Type IX Attractor Theorem, Class. Quantum Grav.26 (2009), no. 7, 075015

    2009

  43. [43]

    Hervik, W.C

    S. Hervik, W.C. Lim, P. Sandin, and C. Uggla,Future Asymptotics of Tilted Bianchi Type II Cosmologies, Class. Quantum Gravity27(2010), no. 18, 185006

    2010

  44. [44]

    Hervik, R.J

    S. Hervik, R.J. Van Den Hoogen, W.C. Lim, and A.A. Coley,The Futures of Bianchi Type VII0 Cosmologies with Vorticity, Classical and Quantum Gravity23(2006), no. 3, 845

    2006

  45. [45]

    Hervik, R.J

    S. Hervik, R.J. van den Hoogen, W.C. Lim, and A.A. Coley,Late-Time Behaviour of the Tilted Bianchi Type V I−1/9 Models, Classical and Quantum Gravity25(2007), no. 1, 015002

    2007

  46. [46]

    Hervik, R.J

    S. Hervik, R.J. Van den Hoogen, W.C. Lim, and A.A. Coley,Late-Time Behaviour of the Tilted Bianchi Type V Ih Models, Classical and Quantum Gravity24(2007), no. 15, 3859

    2007

  47. [47]

    Hewitt, R

    C.G. Hewitt, R. Bridson, and J. Wainwright,The Asymptotic Regimes of Tilted Bianchi II Cosmologies, General Relativity and Gravitation33(January 2001), no. 1, 65–94

    2001

  48. [48]

    King and G.F.R

    A.R. King and G.F.R. Ellis,Tilted Homogeneous Cosmological Models, Communications in Mathematical Physics31(1973/09/01), no. 3, 209–242

    1973

  49. [49]

    Kurganov and E

    A. Kurganov and E. Tadmor,New High-Resolution Central Schemes for Non-Linear Conservation Laws and Convection-Diffusion Equations, Journal of computational physics160(2000), no. 1, 241–282

    2000

  50. [50]

    LeFloch and A.D

    P.G. LeFloch and A.D. Rendall,A Global Foliation of Einstein-Euler Spacetimes with Gowdy-Symmetry on T 3, Arch. Rat. Mech.201(2011), no. 3, 841–870

    2011

  51. [51]

    LeVeque,Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics, Cambridge University Press, 2002

    R.J. LeVeque,Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics, Cambridge University Press, 2002

    2002

  52. [52]

    Li,BKL Bounces Outside Homogeneity: Einstein-Maxwell-Scalar Field in Surface Symmetry(2024), available at2408.12434

    W. Li,BKL Bounces Outside Homogeneity: Einstein-Maxwell-Scalar Field in Surface Symmetry(2024), available at2408.12434

    Pith/arXiv arXiv 2024

  53. [53]

    ,BKL Bounces Outside Homogeneity: Gowdy Symmetric Spacetimes(2024), available at2408.12427

    Pith/arXiv arXiv 2024

  54. [54]

    Lim,The Dynamics of Inhomogeneous Cosmologies, Ph.D

    W.C. Lim,The Dynamics of Inhomogeneous Cosmologies, Ph.D. Thesis, 2004

    2004

  55. [55]

    W.C. Lim, L. Andersson, D. Garfinkle, and F. Pretorius,Spikes in the Mixmaster Regime ofG2 Cosmologies, Physical Review D79(2009), no. 12, 123526

    2009

  56. [56]

    Lim and A.A

    W.C. Lim and A.A. Coley,General Relativistic Density Perturbations, Classical and Quantum Gravity31 (November 2013), no. 1, 015020

    2013

  57. [57]

    MacCallum,Cosmological Models from a Geometric Point of View, Cargese Lect

    M.A.H. MacCallum,Cosmological Models from a Geometric Point of View, Cargese Lect. Phys.6(1973), 61–174, available at2001.11387

    Pith/arXiv arXiv 1973

  58. [58]

    Marshall,Instability of Slowly Expanding FLRW Spacetimes, Classical and Quantum Gravity42(2025), no

    E. Marshall,Instability of Slowly Expanding FLRW Spacetimes, Classical and Quantum Gravity42(2025), no. 10, 105005

    2025

  59. [59]

    Oliynyk,On the Fractional Density Gradient Blow-Up Conjecture of Rendall, Communications in Mathematical Physics405(2024), no

    T.A. Oliynyk,On the Fractional Density Gradient Blow-Up Conjecture of Rendall, Communications in Mathematical Physics405(2024), no. 8, 197

    2024

  60. [60]

    Parés,Numerical Methods for Nonconservative Hyperbolic Systems: A Theoretical Framework., SIAM Journal on Numerical Analysis44(2006), no

    C. Parés,Numerical Methods for Nonconservative Hyperbolic Systems: A Theoretical Framework., SIAM Journal on Numerical Analysis44(2006), no. 1, 300–321

    2006

  61. [61]

    Rendall,Asymptotics of Solutions of the Einstein Equations with Positive Cosmological Constant, Ann

    A.D. Rendall,Asymptotics of Solutions of the Einstein Equations with Positive Cosmological Constant, Ann. Henri Poincaré5(2004), no. 6, 1041–1064

    2004

  62. [62]

    Ringström,The Bianchi IX attractor, Ann

    H. Ringström,The Bianchi IX attractor, Ann. Henri Poincaré2(2001), 405–500

    2001

  63. [63]

    Rodnianski and J

    I. Rodnianski and J. Speck,A Regime of Linear Stability for the Einstein-Scalar Field System with Applications to Non-Linear Big Bang Formation, Ann. Math.187(2018), no. 1, 65–156

    2018

  64. [64]

    ,Stable Big Bang Formation in Near-FLRW Solutions to the Einstein-Scalar Field and Einstein-Stiff Fluid Systems, Sel. Math. New Ser.24(2018), no. 5, 4293–4459

    2018

  65. [65]

    G. Rácz, I. Szapudi, I. Csabai, and L. Dobos,The Anisotropy of the Power Spectrum in Periodic Cosmological Simulations, Monthly Notices of the Royal Astronomical Society503(March 2021), no. 4, 5638–5645

    2021

  66. [66]

    Uggla, H

    C. Uggla, H. van Elst, J. Wainwright, and G.F.R. Ellis,Past Attractor in Inhomogeneous Cosmology, Physical Review D68(November 2003), no. 10

    2003

  67. [67]

    van Elst and C

    H. van Elst and C. Uggla,General Relativistic Orthonormal Frame Approach, Classical and Quantum Gravity 14(September 1997), no. 9, 2673–2695

    1997

  68. [68]

    van Elst, C

    H. van Elst, C. Uggla, and J. Wainwright,Dynamical Systems Approach toG2 Cosmology, Classical and Quantum Gravity19(December 2001), no. 1, 51–82

    2001

  69. [69]

    Wainwright and G.F.R

    J. Wainwright and G.F.R. Ellis,Dynamical Systems in Cosmology, Cambridge University Press, 1997

    1997

  70. [70]

    Wainwright and L

    J. Wainwright and L. Hsu,A Dynamical Systems Approach to Bianchi Cosmologies: Orthogonal Models of Class A, Classical and Quantum Gravity6(1989oct), no. 10, 1409

  71. [71]

    Weaver, J

    M. Weaver, J. Isenberg, and B.K. Berger,Mixmaster Behavior in Inhomogeneous Cosmological Spacetimes, Physical Review Letters80(April 1998), no. 14, 2984–2987. 22

    1998

  72. [72]

    Wu,Design of Provably Physical-Constraint-Preserving Methods for General Relativistic Hydrodynamics, Physical Review D95(2017), no

    K. Wu,Design of Provably Physical-Constraint-Preserving Methods for General Relativistic Hydrodynamics, Physical Review D95(2017), no. 10, 103001

    2017

  73. [73]

    Zhang and C

    X. Zhang and C. Shu,On Positivity-Preserving High Order Discontinuous Galerkin Schemes for Compressible Euler Equations on Rectangular Meshes, Journal of Computational Physics229(2010), no. 23, 8918–8934

    2010

  74. [74]

    4, 1238–1248

    ,Positivity-Preserving High Order Discontinuous Galerkin Schemes for Compressible Euler Equations with Source Terms, Journal of Computational Physics230(2011), no. 4, 1238–1248. School of Mathematics, 9 Rainforest W alk, Monash University, VIC 3800, Australia Email address:elliot.marshall@monash.edu

    2011

This paper was first reviewed by deepseek-v4-flash on August 3, 2026.