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Instability of Slowly Expanding FLRW Spacetimes

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Pith's one-line read Under Gowdy symmetry, small perturbations of decelerated FLRW spacetimes form shocks in finite time for every linear equation of state $0\le K\le 1$.

desk verdict A solid numerical study of shock formation in coupled Gowdy Einstein-Euler FLRW models; the K > 0 evidence is credible, but the 'all K / arbitrarily small perturbations' framing outstrips what the simulations actually show. read the letter →

arxiv 2502.08095 v2 pith:KDIHFXZM submitted 2025-02-12 gr-qc

Elliot Marshall This is my paper · ORCID
classification gr-qc
keywords shockformationFLRWspacetimesEinstein-EulerequationsGowdysymmetryrelativisticfluidsdeceleratedexpansionnumericalrelativityshock-capturingscheme
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper numerically studies nonlinear perturbations of the decelerated FLRW fluid solutions to the coupled Einstein-Euler equations, assuming Gowdy symmetry and a linear equation of state $p=K\rho$ with $0\le K\le 1$. The central claim is that for every $K$, sufficiently small perturbations develop shocks in finite time, and for $K>0$ the shock-formation time scales as $t_*\sim A\exp(B/\|\mathrm{I.D.}\|_2^2 + C/\|\mathrm{I.D.}\|_2)$. This matters because it contrasts sharply with the accelerated-expansion regime, where known results show that expansion suppresses shock formation, and because for dust ($K=0$) it suggests that coupling to gravity destroys stability that holds on fixed decelerating backgrounds. The evidence is obtained with a high-resolution shock-capturing scheme that can evolve past the shock, overcoming a limitation of earlier fixed-background studies.

What carries the argument

The argument is carried by a first-order flux-conservative formulation of the Gowdy-symmetric Einstein-Euler equations in areal coordinates, which reduces the problem to a $(1+1)$-dimensional periodic system of balance laws for the metric variables $\alpha$, $U$, $A$ and the fluid variables $\mu$, $v$. The fluid equations are evolved with a second-order shock-capturing central-upwind scheme, which allows the simulation to continue past shock formation, and shocks are flagged when the norm ratio $\|(v,\mu)\|_3/\|(v,\mu)\|_0$ exceeds $10^6$. The initial data are a one-parameter family of sinusoidal perturbations of the exact FLRW solution, with convergence checked against higher resolutions and against the exact FLRW solution.

What would settle it

Evolve the same Gowdy-symmetric system from a different small perturbation, for example a sum of several sinusoidal modes with comparable total $L^2$ norm, and check whether the norm ratio $\|(v,\mu)\|_3/\|(v,\mu)\|_0$ stays bounded for all time for some $K\in(0,1)$; a bounded ratio would disprove universal shock formation.

Watch

Extended reading notes

Core claim

On its own terms, the discovery is that the decelerated FLRW solution is future-unstable in the fluid sector: perturbed Gowdy-symmetric solutions of the Einstein-Euler system form shocks in finite time for every $K\in[0,1]$. For $K\in(0,1]$, the fluid variables become highly oscillatory and develop shocks while the metric variables remain continuous but acquire kinks at the shock location; for $K=0$, the velocity forms a single stationary shock and the gravitational variables develop sharp features, though the dust simulations suffer numerical issues. The paper also finds for $K>0$ that the shock time decreases exponentially with both the perturbation size and $K$, fitting $t_*\sim A\exp(B/\|\mathrm{I.D.}\|_2^2 + C/\|\mathrm{I.D.}\|_2)$ for fixed $K$. These results are presented as strong numerical evidence, not a proof, and are stated to contrast starkly with the suppression of shocks known for accelerated expansion.

Load-bearing premise

The conclusion that all sufficiently small perturbations form shocks rests on the assumption that the one-parameter sinusoidal initial-data family with $a=b=c=d=0.05$ and $b$ varying down to $0.02$ is representative of arbitrary small perturbations of FLRW.

Editorial extensions

If this is right

  • If the central claim is correct, decelerated FLRW spacetimes with $T^3$ spatial topology are not future-stable for any linear equation of state $0\le K\le 1$; arbitrarily small Gowdy-symmetric perturbations shock in finite time.
  • The shock-formation time for $K>0$ grows roughly like $A\exp(B/\|\mathrm{I.D.}\|_2^2)$, so very small data shock at exponentially large times, consistent with a slow instability rather than an immediate blow-up.
  • For dust, the results imply that gravitational coupling can overturn stability that holds on a fixed decelerating background, since shocks form even though dust on fixed backgrounds with $\sigma>1/2$ is stable.
  • The absence of the Rendall instability for $K>1/3$ in the decelerated regime indicates that the late-time dynamics is not homogenized, in contrast to the accelerated case.
  • If the claim holds, the stability boundary for the coupled system is empty for the whole $K$ range, so the transition observed in fixed-background studies is effectively erased by gravitational coupling.

Reading between the lines

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

  • The universality claim rests on a single family of sinusoidal initial data; testing other profiles—multi-mode, compact support, or random phases—could reveal whether shock formation is generic or tied to this particular profile.
  • Gowdy symmetry restricts perturbations to depend on one spatial direction; a full three-dimensional evolution could change the threshold for shock formation, since wave focusing and shock formation are dimension-dependent.
  • The $K=0$ runs show numerical fragility, so the dust shock claim is the least secure part of the paper; a higher-order scheme or adaptive mesh would provide a sharper test of whether the apparent metric discontinuities are genuine.
  • The exponential scaling with $\|\mathrm{I.D.}\|_2^{-2}$ resembles the unstable regime of the fixed-background problem, suggesting that a proof might be built by treating the gravitational coupling as a perturbation that pushes the fluid into the unstable parameter region.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. The paper numerically studies Gowdy-symmetric nonlinear perturbations of decelerated FLRW solutions to the Einstein-Euler system with linear equations of state p=Kρ, 0≤K≤1, using a shock-capturing Kurganov-Tadmor scheme. The author reports that perturbations develop shocks in finite time for all K, with an exponential scaling of shock formation time for K>0 and a stationary shock for K=0. The manuscript includes code validation against the exact FLRW solution, convergence tests, constraint monitoring, and a scaling analysis of the shock time.

Significance. If the K>0 results are correct, they provide numerical evidence for instability of decelerated FLRW spacetimes, complementing prior fixed-background studies and contrasting with the well-known suppression of shocks under accelerated expansion. The use of shock-capturing methods to evolve past shock formation is a genuine technical advance over earlier work, and the detailed convergence tests, constraint monitoring, and FLRW validation are commendable strengths. However, the universal 'for all K' and 'arbitrarily small perturbations' claims are not supported by the evidence as presented: the K=0 branch is explicitly qualified in §4.4.3, the K=1 branch is compromised by code crashes, and the scaling extrapolation in §4.4.2 never approaches the FLRW limit.

major comments (3)
  1. [Abstract, §4.4.3, §5] The abstract and the discussion in §5 claim that perturbations of the FLRW solution develop shocks in finite time for all values of K, but §4.4.3 explicitly states that for K=0 'further simulations of the dust case with an alternative numerical scheme would be needed to confidently assert the presence of shocks,' and §4.4.1 reports that at K=1 the code crashes before unambiguous shock formation can be confirmed. Because the headline result is a universal statement over K, these acknowledged caveats are load-bearing and the claim as written is not supported for the K=0 and K=1 branches.
  2. [§4.4.2, §5] The scaling law (4.19) is fitted from simulations in which b is varied from 0.02 to 0.35 while a=c=d=0.05 are held fixed, so reducing b does not drive the initial data toward the FLRW solution (which is a=b=c=d=0). The L2 norm (4.20) therefore remains bounded below by the fixed a, c, d components, and the statement in §5 that 'arbitrarily small perturbations' form shocks is an extrapolation outside the simulated range, not a tested result.
  3. [§4.1.1, §4.4] All shock-formation claims are based on the single-parameter family of sinusoidal initial data (4.9) with equal amplitudes a=b=c=d=0.05 (except in the scaling sweep), and the abstract and §5 generalize to 'perturbations of the FLRW solution' without qualification. No evidence is provided that other perturbation profiles or relative mode amplitudes behave in the same way, so the universality of the claim across arbitrary perturbations is not established by the simulations presented.
minor comments (3)
  1. [Section 3 heading] The heading 'FLR W Solutions' contains a spacing typo and should read 'FLRW Solutions'.
  2. [§4.4.2] The sentence 'the shock time decreases exponentially as both the size of our perturbations and K are increased' is contradicted by Figure 12 for K≥0.9, where the shock time increases again; the text later acknowledges this, but the statement should be qualified at first mention.
  3. [Figure 13 caption] The caption states that the Kurganov-Tadmor scheme was used to evolve A0, A1, U0, and U1 for this simulation, which is an exception to the scheme described in §4.1; the reason for this modification and its potential effect on the K=0 results should be stated in the text.

Circularity Check

0 steps flagged · score 1.0 of 10

Numerical shock-formation claims are directly simulated; empirical scaling fit is labeled as such and not circular. Minor self-citations are contextual, not load-bearing.

full rationale

The paper's central claims rest on direct numerical evolution of the Gowdy-symmetric Einstein-Euler system, not on a derivation that reduces to its inputs. Shock formation is identified by monitoring the resolution-dependent norm ratio (4.18) and by visual inspection of the evolved fluid variables, and the shocks are observed in the simulations themselves (e.g., Figures 7, 11). The scaling law (4.19) is explicitly described as an empirical fit ('we find ... t* scales approximately as') with fitted constants A, B, C, and it is not used to define or predict the shock-formation time independently of the simulations; it is a summary of observed data, so it is not a fitted input renamed as a prediction. The initial-data family (4.9) is a perturbation of FLRW in the sense that setting a=b=c=d=0 recovers FLRW, and the claim about small perturbations is an extrapolation from finite-amplitude runs; this is a question of evidential support, not circularity. The paper cites previous work by Fajman et al. for context and comparison, but the load-bearing instability evidence is generated by the present numerical scheme, and no uniqueness theorem or ansatz is imported from self-citations to force the conclusion. The K=0 caveat in Section 4.4.3 ('further simulations of the dust case with an alternative numerical scheme would be needed to confidently assert the presence of shocks') weakens the universal claim in the abstract, but this is an overstatement relative to the evidence, not a circular reduction. Overall, the derivation chain is self-contained against the numerical experiments, and any circularity is negligible.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claim rests on hand-chosen initial data amplitudes, a hand-chosen shock threshold, and fitted constants in the scaling law, plus standard assumptions about the Gowdy foliation and the numerical scheme. No new physical entities are introduced.

free parameters (3)
  • A, B, C (scaling law constants) = not reported in text
    Fit to shock times in Figure 10 using Eq. (4.19); used to extrapolate shock formation to arbitrarily small perturbations.
  • Initial data amplitudes a, b, c, d = a=b=c=d=0.05 in most runs; b varied 0.02-0.35 in scaling runs
    Hand-chosen amplitudes of the sinusoidal perturbations; define what 'small perturbation' means in the experiments.
  • Shock detection threshold = 10^6 for ratio ||(v,mu)||_3 / ||(v,mu)||_0
    Chosen by hand following [18] to define the shock formation time t*; affects all reported t* values.
assumptions (4)
  • standard math The Gowdy areal coordinate foliation covers the maximal globally hyperbolic development of the Einstein-Euler system (LeFloch-Rendall [33]).
    Used in Section 2.1 to justify evolving to t=infinity and dropping the Hamiltonian constraint; if the foliation broke down first, shocks might be coordinate artifacts.
  • domain assumption The Kurganov-Tadmor finite volume scheme converges to the entropy weak solution of the balance-law system (2.51).
    The paper relies on the scheme's shock-capturing fidelity; Section 4.2 provides convergence tests but no rigorous convergence proof for this coupled system.
  • ad hoc to paper The sinusoidal initial data (4.9) with equal amplitudes represent small perturbations of the FLRW solution.
    Only this family is evolved; the universal conclusion is inferred from it.
  • domain assumption The linear equation of state p=K rho with 0<=K<=1 is an adequate fluid model.
    Standard modeling assumption throughout the paper, stated in Section 1.

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Pith. "Pith review of Instability of Slowly Expanding FLRW Spacetimes." pith.science (2026). https://pith.science/paper/KDIHFXZM

@misc{pith2026250208095,
  author       = {Pith},
  title        = {Pith review of: Instability of Slowly Expanding FLRW Spacetimes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KDIHFXZM}},
  note         = {Machine review of arXiv:2502.08095}
}
abstract

We numerically study, under a Gowdy symmetry assumption, nonlinear perturbations of the decelerated FLRW fluid solutions to the Einstein-Euler system toward the future for linear equations of state $p=K\rho$ with $0\leq K\leq 1$. This article builds on the work of Fajman et al. (2024 arXiv:2405.03431) in which perturbations of the homogeneous fluid solution on a fixed, decelerating FLRW background were studied. Our numerical results show that for all values of $K$, perturbations of the FLRW solution develop shocks in finite time. This behaviour contrasts known results for spacetimes with accelerated expansion in which shock formation is suppressed.

Figures

Figures reproduced from arXiv: 2502.08095 by the authors.

Figure 1
Figure 1. Plot of the (σ, K) parameter space for fluids on a power-law back￾ground established in [18]. The green and red shaded regions denote the stable and unstable regimes, respectively. The critical line, Kcrit = 1 − 2 3σ is coloured black while the blue line denotes the FLRW solutions to the Einstein-Euler system with zero cosmological constant. A key limitation of the numerical study in [18] was that their scheme could… view at source ↗
Figure 2
Figure 2. Convergence plots of v before and after shocks have formed. There are several other tests we can perform to check the accuracy of our solutions. First, we can monitor the violation of the constraints (2.52)-(2.53). Clearly, when CA1 = CU1 = 0 (4.10) the constraints are identically satisfied. The quantity log2 (∥C∥2) can therefore be understood as the violation error of the constraint as a function of time. We observ… view at source ↗
Figure 3
Figure 3. Convergence plot of log2 (∥CA1 ∥2 + ∥CU1 ∥2), K = 0.1 [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: Convergence of ∫D (τ˜(t, x) − τ˜(0, x) − ∫ t 0 G(T)dT)dx, K = 0.1. Let us now consider the integral form of a general conservation law on a domain D without boundaries, d dt ∫ D udx + ∫ D ∂xf = 0. (4.11) By applying the divergence theorem, the integral of the flux disa…
Figure 5
Figure 5. Figure 5: Convergence plot of L2-norm of α−αexact for several different values of the timestep ∆t. All evolutions used N = 200 and K = 0.25. 4.3. Testing for Shock Formation. Our numerical scheme, unlike that of [18], uses shock capturing methods which allows us to evolve beyond…
Figure 6
Figure 6. Figure 6: Values of the norm ratio (4.18) over time for various resolutions, K = 0.1. Shocks form just after t = 10. 4.4.1. Shock Formation for K > 0. We now discuss the behaviour of numerical solutions generated from initial data of the form (4.9) for K ∈ (0, 1]. As discussed i…
Figure 7
Figure 7. Figure 7: Fluid velocity v at various times. N = 6400, K = 0.1. Two shocks are clearly present in [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]
Figure 8
Figure 8. Figure 8: Fluid velocity v (top), α (middle), and ∂xα (bottom) at t = 7.41. The derivative of α was estimated using second order finite differences. N = 6400, K = 0.3. 3.8 4.0 4.2 4.4 4.6 x 0.100 0.105 0.110 0.115 0.120 ∂x α [PITH_FULL_IMAGE:figures/full_fig_p016_8.png]
Figure 9
Figure 9. Figure 9: Close up of kink in ∂xα at t = 7.41. N = 6400, K = 0.3. oscillations are seen in the metric variables, however as mentioned before shocks do not form. This behaviour is consistent with previous analytical work in the linear and decelerated regimes [17, 19, 20]. However…
Figure 10
Figure 10. Figure 10: Plot of the estimated shock formation time against the size of initial data, ∥I.D.∥2 for various values of K. The dashed lines are curves of best fit following the formula (4.19). Values of b were taken between 0.02 and 0.35. N = 6400, a = c = d = 0.05. 0 5 10 t 10−2 …
Figure 11
Figure 11. Figure 11: Values of the norm ratio (4.18) over time for various values of K. N = 6400 and a = b = c = d = 0.05. We wish to emphasise that our numerical scheme becomes significantly more unstable when K = 0. We find sharp, seemingly discontinuous, features form in certain metric…
Figure 12
Figure 12. Figure 12: Estimated shock formation time for different values of K. N = 6400 and a = b = c = d = 0.05. [4, 5, 11, 15, 23, 24, 51]) which our code is not sufficiently resolving. This behaviour is shown for U1, along with a shock in the fluid velocity, in [PITH_FULL_IMAGE:figure…
Figure 13
Figure 13. Figure 13: v and U1 at t = 6.59. N = 1000, K = 0. The Kurganov-Tadmor scheme was used to evolve A0, A1, U0, and U1 for this simulation. 5. Discussion We have numerically simulated Gowdy-symmetric nonlinear perturbations of FLRW solutions to the Einstein-Euler system for the full…

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