{"id":"26f2c3b2-96fe-4faa-9b3d-eb7f2c691565","arxiv_id":"2610.35405","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"For Eckart-like viscous dark matter with a generic pressure function of the matter density, viable histories must start with no viscosity and end at the Lambda-CDM de Sitter attractor, so the background mimics Lambda-CDM.","lead":"This paper uses dynamical systems methods to analyze a broad class of Eckart-type viscous dark matter models that include the radiation era. It concludes that viable cosmic histories must track standard Lambda-CDM at the background level, while viscosity can still suppress small-scale structure growth through a scale-dependent drag.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek/deepseek-v4-flash","headline":"The general VM stability statement in Sec. III.C is miscomputed: the Ωm eigenvalue is -3(1-Ωm)(1+ΩΠ'), not 0, and admissible ΩΠ can make VM a stable physical attractor, so the abstract's 'only Λ' claim is false in the stated class.","rationale":"The reader's weakest_assumption correctly identified VM as the weak point, but the concern is stronger than a non-hyperbolic gap. The general Jacobian at VM is not generically singular: the correct Ωm eigenvalue is -3(1-Ωm)(1+ΩΠ'), which is nonzero for generic ΩΠ. The paper's own power-law case with s<1 already yields a stable VM, which is inconsistent with its general claim of a zero eigenvalue at VM; the explicit smooth function ΩΠ=-(16/3)Ωm²(1-Ωm) satisfies ΩΠ(0)=0, ΩΠ≤0 on the physical interval, and even ΩΠ'(0)=0, yet has a stable physical VM at Ωm=3/4 alongside the Λ sink. This function is within the stated general class, so the abstract's 'only physically admissible late-time attractor' statement fails for the full class. The power-law analysis and the numerical illustrations remain useful if the claims are narrowed, but the central general conclusion needs correction, not just a conditionally acceptable gap.","tokens_in":25408,"tokens_out":28860,"duration_ms":213163,"concrete_test":"Compute the general Jacobian at VM from Eqs. (29)-(32) and check λ_m=-3(1-Ωm)(1+ΩΠ'), rather than the stated λ2=0. Then integrate the autonomous system with ΩΠ(Ωm)=-(16/3)Ωm²(1-Ωm), starting from Ωr=1-ε, Ωm=ε, ΩΛ=0 for small ε, and record the late-time endpoint; if the trajectory terminates at Ωm=3/4, Ωr=0 instead of at Λ, the abstract's unique-attractor claim is settled false.","verdict_should_be":"REJECT","load_bearing_attack":"At the fixed point VM (Ωr=0, Ωm=-ΩΠ, q=-1), the Jacobian from Eqs. (29)-(32) is upper triangular with eigenvalues -4 and λ_m = -3(1-Ωm)(1+dΩΠ/dΩm). The text's report of 'λ2=0' in Sec. III.C is therefore not the general result: it holds only when dΩΠ/dΩm=-1 at VM, a condition never imposed. For any admissible ΩΠ with dΩΠ/dΩm>-1 at a positive VM, VM is a hyperbolic stable node inside the physical phase space, contrary to the abstract sentence b3. The power-law determinant of Sec. IV.B, det(J)=12(1-s)(1-Ωm), is just the special case p'=-s at VM, not the general Jacobian. A concrete function within every stated hypothesis, smooth, with ΩΠ(0)=0 and ΩΠ≤0 on Ωm∈[0,1] (indeed with ΩΠ'(0)=0), is ΩΠ(Ωm)=-(16/3)Ωm²(1-Ωm). It has VM points at Ωm=1/4 and Ωm=3/4; at Ωm=3/4, ΩΠ(Ωm)=-Ωm and dΩΠ/dΩm=1>-1, so λ_m=-3/2. At R the eigenvalues are (4,1), at M they are (-1,3), and at Λ they are (-3,-4). Thus R is a source, M is a saddle, and both Λ and the physically admissible VM at Ωm=3/4 are sinks. The power-law restrictions of Sec. IV do not cover this element of the general class, so the claimed general conclusion is not merely unproved; it is false as stated. The valid conclusion must be restricted to the power-law family (or to additional assumptions not present in the paper).","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek/deepseek-v4-flash","summary":"The paper analyzes FLRW cosmologies containing radiation, Eckart bulk-viscous dark matter with dimensionless viscous pressure ΩΠ = ΩΠ(Ωm), and a cosmological constant. Using a two-dimensional dynamical system, it derives the fixed points (R, R1, M, VM, Λ), gives a general stability classification, and concludes (abstract sentence b3) that a viable radiation era requires the complete absence of viscosity, that the matter era requires small viscosity, that the only physically admissible late-time attractor is the Λ-dominated point, and hence that the background must closely track ΛCDM. The second half of the paper studies the power-law ansatz ΩΠ = −ξ̂0 Ωm^s: for s ≥ 1 and ξ̂0 ≪ 1 the background mimics ΛCDM, while for s < 1 backward integration terminates at a finite e-fold Ncrash where Ωm becomes negative and ΩΠ complex. A final section derives a scale-dependent bulk-viscous drag on linear matter perturbations, leading to a k²-suppression of the small-scale power spectrum. The power-law results, including Eq. (43) and the closed-form Ncrash of Eq. (53), are derived cleanly and corroborated numerically; the problem is the extension of those results to the general class.","tokens_in":25965,"tokens_out":18877,"duration_ms":465590,"significance":"If the general-class claim were correct, it would be a strong no-go statement: Eckart-like viscous dark matter could not resolve the H0 tension and would be observationally indistinguishable from ΛCDM at the background level. The paper has real strengths: the radiation-repeller determinant, Eq. (35), is a clean general result; the power-law stability classification and the explicit Ncrash boundary are falsifiable, concrete predictions; the perturbative section transparently states its sub-horizon and quasi-static assumptions and honestly flags the breakdown of linear theory at large k; and the power-law parameters are scanned rather than fitted to produce the conclusions, with the perturbation estimate importing ξ̂0 = 10^-6 from the published value in Ref. [62]. However, the central general-class conclusion does not follow from the analysis, and a concrete admissible counterexample shows it is false as stated. The paper's substantive and defensible contribution is the power-law case study plus the general bound on the radiation point; that contribution is significant but narrower than what the abstract claims.","major_comments":[{"comment":"The eigenvalue λ2 = 0 reported for the viscous matter point (VM) in Sec. III.C is not the general result. At VM one has Ωr = 0, so J21 in Eq. (31) vanishes and the Jacobian is upper triangular, with J22 = −4 and, using q = −1 and Ωm = −ΩΠ in Eqs. (29) and (33), J11 = −3(1−Ωm)(1 + dΩΠ/dΩm). The reported zero eigenvalue therefore requires dΩΠ/dΩm = −1 at the fixed point, a condition that is neither imposed nor implied by the stated assumptions. For a generic admissible ΩΠ the point VM is hyperbolic with eigenvalues (−4, −3(1−Ωm)(1+dΩΠ/dΩm)), so the assertion that the linear theory is inconclusive applies only to the measure-zero case dΩΠ/dΩm = −1. The power-law determinant in Eq. (43), det(J) = 12(1−s)(1−Ωm), is the special case ΩΠ′ = −s at VM and cannot support stability conclusions for the general class.","section":"Sec. III.C, VM bullet; Eqs. (29)-(33)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The central claim that 'the only physically admissible late-time attractor is the cosmological constant-dominated fixed point' is false within the stated general class. The smooth function ΩΠ(Ωm) = −(16/3)Ωm²(1−Ωm) satisfies every hypothesis of Sec. III.A: it vanishes at Ωm = 0 and is non-positive on Ωm ∈ [0,1], so it lies in the physically allowed Region (IV). The VM condition Ωm = −ΩΠ gives Ωm(1−Ωm) = 3/16, with roots Ωm = 1/4 and Ωm = 3/4; at Ωm = 3/4 one has dΩΠ/dΩm = 1 > −1, so the corrected eigenvalues are −4 and −3/2, making VM a hyperbolic stable node inside the physical phase space. The radiation point has eigenvalues (4,1), the matter point has (−1,3), and Λ has (−3,−4), so R → M → VM is a complete chronology ending at a viscous attractor. Even within the power-law family, the paper's own Eq. (40) places VM at Ωm = ξ̂0^{1/(1−s)} ∈ (0,1) with VM the stable attractor for 0 < s < 1 (Sec. IV.B); the exclusion of that case relies on the separate backward-integration criterion Ncrash of Eq. (53), which concerns past-time extendibility, not the late-time stability of the fixed point. The abstract and conclusions must be restricted to the power-law family with s ≥ 1 (or to additional assumptions that explicitly exclude functions such as the one above).","section":"Abstract (sentence b3); Sec. V, 'Late-Time Attractors' bullet","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The inference stated in Sec. V that 'a rigorous assessment reveals that this point lies outside the physical phase space of the theory' is unsupported for the general class. In Sec. III.C the general VM is declared non-hyperbolic and deferred to 'other techniques,' which are never carried out in the general section; the only rigorous assessment present is the power-law computation of Sec. IV.B, which moves VM outside [0,1] for s > 1 through Eq. (40) and Fig. 2. The gap between the general non-hyperbolicity statement of Sec. III.C and the Sec. V exclusion statement is precisely where the counterexample in the previous comment lives. The conclusion is therefore not merely unproved for the general class; it is contradicted by an element of the class the paper claims to have constrained.","section":"Sec. III.C vs. Sec. V transition","authors_acknowledge":null,"authors_acknowledge_p":null}],"minor_comments":[{"comment":"The particular solution of Eq. (61) with right-hand side −R̃v(k) is ε = −(R̃v/9)a³, which yields ΔP/PΛ = −2R̃v/9 consistent with Eq. (64); the coefficient displayed in Eq. (63) is inconsistent with Eq. (64) and should be corrected.","section":"Sec. IV.E, Eqs. (63)-(64)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The eigenvalue expressions in Eqs. (41) and (44) are evaluated in the limit Ωm → 0 rather than at the fixed point itself. For s > 1 the exact eigenvalues at R and Λ are (4,1) and (−3,−4); for 0 < s < 1 the derivative ΩΠ′ diverges at Ωm = 0, so the linearization at the fixed point is ill-defined rather than possessing a diverging eigenvalue. The limiting procedure should be stated explicitly.","section":"Sec. IV.B, Eqs. (41) and (44)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The text alternates between s > 1 and s ≥ 1 (e.g., the abstract's 's ≥ 1' versus the stability discussion of Sec. IV.B). For s = 1 the VM condition Ωm = ξ̂0Ωm collapses VM onto Λ for ξ̂0 ≠ 1, so the stability classification should treat s = 1 separately rather than grouping it with s > 1.","section":"Abstract and Sec. IV.B/C, s=1 boundary","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"There are several typographical errors: 'organized ain the following way' (end of Sec. I), 'Gibbs–Duheim' (Sec. II), and 'yield to two mutually exclusive branches' (Sec. III.B). In addition, the statement in Sec. IV.D that the background deviation stays 'strictly under 1%' would benefit from specifying the range of s and ξ̂0 over which the bound holds.","section":"General presentation","authors_acknowledge":null,"authors_acknowledge_p":null}],"recommendation":"major_revision","confidential_remarks":"The paper overstates its main result relative to what is actually proved. The power-law case study of Sec. IV and the general radiation-point bound of Sec. III.C are sound, and the Ncrash formula (Eq. (53)) is a genuinely useful contribution. However, the VM eigenvalue bullet in Sec. III.C is inconsistent with the paper's own Eqs. (29)-(33), and this inconsistency propagates directly into the abstract's general-class conclusion. A simple consistency check of the Jacobian at VM would have caught the problem; as it stands, the abstract promises more than the analysis delivers. If the authors restrict the claims to the power-law family and correct the VM eigenvalue computation, I think the paper would be a solid contribution to the viscous-cosmology literature. The paper fits the journal's scope well; the main issue is scope control of the conclusions, not the underlying technical work. I also recommend fixing the coefficient inconsistency between Eqs. (63) and (64) before resubmission."},"author_rebuttal":null,"desk_editor":{"model":"deepseek/deepseek-v4-flash","letter":"Two things to know up front. The power-law analysis—radiation era included, the s≥1 versus s<1 dichotomy, the analytic N_crash boundary—is solid and genuinely useful. But the abstract's headline claim is false for the stated general class. The error sits in Sec. III.C: at the VM fixed point the paper reports eigenvalues (−4, 0) and calls the point non-hyperbolic. The general Jacobian from Eqs. (29)–(33), evaluated at VM, is upper triangular with eigenvalues −4 and −3(1−Ωm)(1+dΩΠ/dΩm). The zero eigenvalue appears only in the special case dΩΠ/dΩm = −1, which the paper never imposes. I checked this against the equations in the text; the stress-test note is right.\n\nThis is not a cosmetic mismatch. The function ΩΠ(Ωm) = −(16/3)Ωm²(1−Ωm) satisfies every stated hypothesis—smooth, ΩΠ(0)=0, ΩΠ ≤ 0 on [0,1]—and has a hyperbolic stable VM at Ωm = 3/4 with ΩΛ = 1/4, alongside the R source, M saddle, and Λ sink. So the general class does admit a physically admissible dissipative de Sitter attractor, and the background need not track ΛCDM. What the paper actually proves is the power-law statement: s ≥ 1 pushes VM outside the physical phase space, and s < 1 fails on the past-time boundary. The early-universe and matter-era claims are general and correct; the late-time claim is not. The reader's conditional verdict is directionally right, but this is a false statement, not merely an unsupported one.\n\nWhat the paper does well: the radiation-era exclusion argument (the q = 1 fixed point requires ΩΠ = Ωm/3 ≥ 0, contradicting the Eckart sign for ξ > 0, so the only admissible source is the zero-viscosity R) is clean and survives contact with the counterexample. The Bernoulli integration and N_crash formula are checkable and new. The authors are honest about scope: the perturbative suppression is explicitly not new, they flag that the δΠ ≈ ξδ′ closure inherits the Eckart gradient-instability problem, and they concede linear theory breaks down where the suppression is large. Citation practice looks fine.\n\nMinor arithmetic: the −23% suppression quoted for k = 600H0 does not follow from their own Eq. (64); the stated inputs give about −9%. The k = 300H0 value is also slightly high (−2.2%). Easy fix.\n\nWho it is for: people working in viscous and causal cosmology, and anyone using phase-space methods on dark-sector models. It deserves a serious referee, with the explicit instruction that the abstract and conclusions must be narrowed to the power-law family (or complemented with assumptions that exclude stable VM) and the Sec. III.C eigenvalues corrected. That is a conditional accept with required revision, not a reject.","headline":"Solid power-law case study with a clean past-time boundary result, but the abstract's general-class late-time claim is false: the Sec. III.C VM eigenvalue is miscomputed, and a stable viscous attractor exists inside the stated general class.","tokens_in":26532,"tokens_out":14362,"would_cite":true,"duration_ms":98466,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"pith_extraction":{"msc":[],"pacs":["98.80.-k","95.35.+d","98.80.Jk"],"model":"deepseek/deepseek-v4-flash","headline":"Eckart-like viscous dark matter is forced by phase-space geometry to track $\\Lambda$CDM at the background level, leaving scale-dependent structure-growth suppression as its remaining channel.","keywords":["bulk viscosity","Eckart formalism","viscous dark matter","dynamical system analysis","phase-space constraints","ΛCDM background","Hubble tension","structure growth suppression"],"falsifier":"Choose an admissible function $\\Omega_\\Pi(\\Omega_m)$ that vanishes at $\\Omega_m = 0$, is non-positive in the physical region, and behaves linearly near $\\Omega_m = 0$ (for example $\\Omega_\\Pi = -\\hat{\\xi}_0 \\Omega_m$ plus a sub-leading correction), then numerically integrate the two-dimensional system forward from the radiation source $(\\Omega_r, \\Omega_m) = (1, 0)$. If trajectories for such a function end at the dissipative de Sitter-like point ($\\Omega_m = -\\Omega_\\Pi$, $w_{\\rm eff} = -1$) rather than at the $\\Lambda$ node, the paper's general claim that the cosmological-constant point is the only admissible attractor is refuted. A complementary check: any viable model must support backward integration to at least the BBN epoch ($z \\sim 10^{10}$); the paper's own $s < 1$ case already crashes at $z \\sim 5 \\times 10^8$, so a crash boundary shallower than BBN for any proposed viscous model would settle that the model is observationally dead.","tokens_in":26925,"feed_emoji":"🌌","tokens_out":28176,"duration_ms":185689,"temperature":0.7,"pith_summary":"This paper asks whether dark matter endowed with a bulk viscous pressure, treated in the first-order Eckart formalism, can sustain a complete cosmic history from radiation domination through matter domination to late-time acceleration without contradicting itself. The authors try to establish that for the entire class of models in which the dimensionless viscous pressure $\\Omega_\\Pi$ is an arbitrary function of the matter density parameter $\\Omega_m$, vanishing at $\\Omega_m = 0$, the phase-space geometry forbids viscosity during the radiation era, demands an extremely small viscous pressure during matter domination, and leaves the cosmological-constant point as the only physically admissible late-time attractor. If the argument is right, the background expansion of these models is nearly indistinguishable from $\\Lambda$CDM, so they cannot resolve the $H_0$ tension. The constructive side of the result is that the same viscosity acts on sub-horizon density fluctuations as a scale-dependent dissipative drag, a mechanism the paper connects to proposals for remedying the $S_8$ tension.","feed_headline":"Viscous dark matter is forced to mimic ΛCDM","feed_subtitle":"Eckart bulk-viscous models cannot fix the Hubble tension; their only signature is small-scale growth suppression.","key_machinery":"The load-bearing mechanism is a two-dimensional autonomous dynamical system in the plane $(\\Omega_r, \\Omega_m)$, with the cosmological-constant density parameter eliminated through $\\Omega_r + \\Omega_m + \\Omega_\\Lambda = 1$. The dynamics are set by the dimensionless bulk viscous pressure $\\Omega_\\Pi \\equiv (8\\pi G/3H^2)\\Pi$, assumed to be a general function $\\Omega_\\Pi(\\Omega_m)$ with $\\Omega_\\Pi(0) = 0$; by the Eckart closure $\\Pi = -\\xi\\Theta$, which makes $\\Omega_\\Pi$ negative in expansion; and by the deceleration parameter $q = -1 + 2\\Omega_r + \\tfrac{3}{2}(\\Omega_m + \\Omega_\\Pi)$, whose value $q = 1$ selects the radiation branch. Three structural facts do the work: the radiation-era equilibrium needs $\\Omega_\\Pi = \\Omega_m/3 > 0$, placing it in the thermodynamically forbidden half-plane; the matter point remains a usable saddle only when $|\\Omega_\\Pi| \\ll 1$; and the stability eigenvalues at the radiation source and the de Sitter sink force parameter regimes in which the dissipative de Sitter-like point leaves the physical phase space. On the perturbative side, the mechanism is the closure $\\delta\\Pi \\approx \\xi \\delta'_m$ (from $\\delta\\Theta = \\theta_m/a$ under the sub-horizon and quasi-static approximations), which adds a scale-dependent friction $R_v \\propto k^2(\\xi/\\bar{\\rho}_m)(H/a)$ to the matter-density contrast equation and converts viscosity into a growth suppressor on small scales.","core_discovery":"Restricting to a flat FLRW universe with radiation, viscous dark matter, and a cosmological constant, the paper reduces the dynamics to a two-dimensional autonomous system in $(\\Omega_r, \\Omega_m)$ with $\\Omega_\\Lambda$ fixed by the Friedmann constraint, and classifies its fixed points. On the radiation branch, the only equilibrium that carries viscous matter would sit at $\\Omega_m = 3\\Omega_\\Pi$ (equivalently $\\Pi = \\rho_m/3$), which requires a positive viscous pressure; but the Eckart closure $\\Pi = -\\xi\\Theta$ together with the second law $\\xi > 0$ makes $\\Pi$ negative in an expanding universe. The paper's central claim is that no trajectory can therefore start from a viscous radiation state, so a viable radiation era forces the complete absence of viscosity. At late times the fixed points are the matter point (a saddle whenever $|\\Omega_\\Pi| \\ll 1$), a dissipative de Sitter-like matter point, and the cosmological-constant point (a stable node); for the general class the dissipative point is non-hyperbolic, and the decisive exclusion of it as an admissible attractor is carried by the power-law case study. There, $\\Omega_\\Pi = -\\hat{\\xi}_0 \\Omega_m^s$ with $s \\geq 1$ and $\\hat{\\xi}_0 \\ll 1$ reproduces the $\\Lambda$CDM background to better than 1 percent while pushing the dissipative point outside the physical phase space, whereas $s < 1$ creates a past-time boundary $N_{\\rm crash}$ beyond which $\\Omega_m$ turns negative and $\\Omega_\\Pi$ becomes complex. The paper concludes that the background is therefore locked to $\\Lambda$CDM, making these models unlikely to resolve the $H_0$ tension, and that their remaining physical channel is a scale-dependent dissipative drag that suppresses the growth of sub-horizon matter fluctuations.","pith_inferences":["Inference: the general-class claim that the $\\Lambda$ point is the only admissible attractor is stronger than the general proof alone supports, because at the dissipative de Sitter-like point the Jacobian has a zero eigenvalue and linear stability is inconclusive there; the exclusion is fully demonstrated only in the power-law subclass, where $s \\geq 1$ pushes the point outside the physical phase ","Inference: the scale-dependent drag $R_v \\propto k^2$ is a known remnant of Eckart gradient instabilities, so the structure-suppression prediction should be re-derived in a causal second-order setting before being used as an $S_8$-tension remedy; if it survives with a relaxation-time cutoff, the model predicts a characteristic $k^2$ tilt in the small-scale matter power spectrum that existing galax","Inference: the paper's closing conjecture that interacting dark-sector models with energy exchange $Q \\propto \\rho^s$ may suffer analogous past-time crashes is directly testable with the same two-dimensional phase-space machinery, and the paper's Bernoulli-type solution gives a template for locating the crash boundary.","Inference: because the background is degenerate with $\\Lambda$CDM, the sharpest observational discriminator for the surviving regime is the growth sector, namely a suppression that steepens with wavenumber, which distinguishes it from scale-independent screening mechanisms in modified gravity."],"forward_implications":["If the central claim holds, this Eckart class of viscous dark matter cannot move the background expansion away from $\\Lambda$CDM: the matter-density deviation stays below 1 percent from $z = 10$ to today, so there is no route to the $H_0$ tension.","A viable early universe requires $\\Omega_\\Pi = 0$ during radiation domination, so viscosity is confined to late times and any viscous modification of BBN-era expansion is excluded within this class.","The viscous pressure must stay small ($\\hat{\\xi}_0 \\ll 1$): increasing it to $\\hat{\\xi}_0 = 0.01$ with $s = 1.2$ shortens the matter era from roughly 8 to 4 e-folds, cutting the linear growth factor from about $3 \\times 10^3$ to about 55 and failing to form observed structures.","In the surviving regime the model's imprint is scale-dependent: with $\\hat{\\xi}_0 = 10^{-6}$, the matter power spectrum is suppressed by about 2.5 percent at $k \\approx 300 H_0$, with the suppression growing as $k^2$, and with almost no effect on large-scale CMB observables.","For $s < 1$ the model only exists up to a past-time boundary ($N_{\\rm crash} \\approx -20$ to $-37$, i.e. redshifts $z \\lesssim 5 \\times 10^8$), which is too late to support the radiation era that BBN requires; only the $s \\geq 1$, small-$\\hat{\\xi}_0$ regime survives."],"supporting_citations":[{"why":"The first-order relativistic fluid theory; supplies the closure $\\Pi = -\\xi\\Theta$ whose negative sign in expansion, set by the second law, drives the exclusion of the viscous radiation era.","marker":"[22]"},{"why":"The bulk-viscous cosmology formalism adopted in Section II; supplies the energy-momentum tensor and viscous-pressure equations that define the model class.","marker":"[70]"},{"why":"The causal second-order thermodynamics that frames the paper's scope; also supplies the truncated regime ($H\\tau_r \\ll 1$) used in Appendix A to suggest the results extend beyond the first-order theory.","marker":"[24, 25]"},{"why":"The dynamical-systems review; provides the eigenvalue classification of sources, saddles, and sinks, and the treatment of non-hyperbolic fixed points that the stability analysis relies on.","marker":"[72]"},{"why":"Origin of the power-law ansatz $\\xi = \\hat{\\xi}_0/(8\\pi G H) \\, \\Omega_m^s$ used as the case study for testing the general phase-space constraints.","marker":"[57]"},{"why":"Prior structure-formation studies of viscous dark matter; established the dissipative-drag mechanism whose chronological boundaries the paper re-derives and constrains.","marker":"[14, 50, 66]"},{"why":"Supplies the benchmark value $\\hat{\\xi}_0 = 10^{-6}$ and the large-scale-structure context used for the power-spectrum suppression estimate.","marker":"[62]"},{"why":"Source of the sub-horizon and quasi-static approximation framework used to close the perturbative equations in Appendix B.","marker":"[86]"},{"why":"The generic-instability analysis of first-order dissipative fluid theories; supports the paper's caveat that the $k^2$ dissipative drag is a remnant of gradient instabilities.","marker":"[23]"}],"fun_headline_variants":["Eckart viscous dark matter forced to track ΛCDM","Bulk viscosity fails to ease Hubble tension","Viscous dark matter's only signal: growth suppression","Radiation era rules out Eckart bulk viscosity","Viscous cosmology corners itself at ΛCDM"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The argument rests on assuming that the power-law conclusions hold for every admissible function $\\Omega_\\Pi(\\Omega_m)$: the viscosity-driven de Sitter-like point must never act as a reachable late-time attractor, even though at that point the general linear-stability test is inconclusive because one eigenvalue vanishes, and the point is explicitly excluded only in the power-law subclass, where the condition $s \\geq 1$ moves it outside the physical phase space.","fun_headline_variants_meta":{"raw":{"variants":["Eckart viscous dark matter forced to track ΛCDM","Bulk viscosity fails to ease Hubble tension","Viscous dark matter's only signal: growth suppression","Radiation era rules out Eckart bulk viscosity","Viscous cosmology corners itself at ΛCDM"]},"model":"deepseek/deepseek-v4-flash","effort":"low","cost_usd":0.000137,"raw_usage":{"cost":0.0001366638,"is_byok":false,"cost_details":{"upstream_inference_cost":0.0001366638,"upstream_inference_prompt_cost":3.83838e-05,"upstream_inference_completions_cost":9.828e-05},"total_tokens":1857,"prompt_tokens":1317,"completion_tokens":540,"prompt_tokens_details":{"audio_tokens":0,"video_tokens":0,"cached_tokens":1119,"cache_write_tokens":0},"completion_tokens_details":{"audio_tokens":0,"image_tokens":0,"reasoning_tokens":476}},"tokens_in":1317,"tokens_out":540,"cost_source":"openrouter_usage","duration_ms":6179,"image_pages":[],"temperature":1.0,"billed_attempts":1,"reasoning_tokens":476,"transport_provider":"openrouter","cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-10-27T10:13:26.647464+00:00","model_set":{"reader":"deepseek/deepseek-v4-flash"},"falsifier":"Choose an admissible function $\\Omega_\\Pi(\\Omega_m)$ that vanishes at $\\Omega_m = 0$, is non-positive in the physical region, and behaves linearly near $\\Omega_m = 0$ (for example $\\Omega_\\Pi = -\\hat{\\xi}_0 \\Omega_m$ plus a sub-leading correction), then numerically integrate the two-dimensional system forward from the radiation source $(\\Omega_r, \\Omega_m) = (1, 0)$. If trajectories for such a function end at the dissipative de Sitter-like point ($\\Omega_m = -\\Omega_\\Pi$, $w_{\\rm eff} = -1$) rather than at the $\\Lambda$ node, the paper's general claim that the cosmological-constant point is the only admissible attractor is refuted. A complementary check: any viable model must support backward integration to at least the BBN epoch ($z \\sim 10^{10}$); the paper's own $s < 1$ case already crashes at $z \\sim 5 \\times 10^8$, so a crash boundary shallower than BBN for any proposed viscous model would settle that the model is observationally dead.","supporting_citations":[{"cited_title":"Viscous universes, Phys. Lett. A120, 433","cited_arxiv_id":null,"evidence_quote":"The first-order relativistic fluid theory; supplies the closure $\\Pi = -\\xi\\Theta$ whose negative sign in expansion, set by the second law, drives the exclusion of the viscous radiation era."},{"cited_title":"Dark Matter Self-interactions and Small Scale Structure","cited_arxiv_id":"1705.02358","evidence_quote":"The bulk-viscous cosmology formalism adopted in Section II; supplies the energy-momentum tensor and viscous-pressure equations that define the model class."},{"cited_title":"On the feasibility of truncated Israel-Stewart model in the context of late acceleration","cited_arxiv_id":"2010.15355","evidence_quote":"The dynamical-systems review; provides the eigenvalue classification of sources, saddles, and sinks, and the treatment of non-hyperbolic fixed points that the stability analysis relies on."},{"cited_title":"A hybrid model of viscous and Chaplygin gas to tackle the Universe acceleration","cited_arxiv_id":"2103.16733","evidence_quote":"Origin of the power-law ansatz $\\xi = \\hat{\\xi}_0/(8\\pi G H) \\, \\Omega_m^s$ used as the case study for testing the general phase-space constraints."},{"cited_title":"Dissipative Cosmology and the Nature of Dark Energy: Insights from Bulk Viscosity with DESI DR2 observations","cited_arxiv_id":"2606.05539","evidence_quote":"Supplies the benchmark value $\\hat{\\xi}_0 = 10^{-6}$ and the large-scale-structure context used for the power-spectrum suppression estimate."},{"cited_title":"Dark matter and dark energy interactions: theoretical challenges, cosmological implications and observational signatures","cited_arxiv_id":"1603.08299","evidence_quote":"Source of the sub-horizon and quasi-static approximation framework used to close the perturbative equations in Appendix B."},{"cited_title":"The Thermodynamics of irreversible pro- cesses. 3.. Relativistic theory of the simple fluid, Phys. Rev.58, 919","cited_arxiv_id":null,"evidence_quote":"The generic-instability analysis of first-order dissipative fluid theories; supports the paper's caveat that the $k^2$ dissipative drag is a remnant of gradient instabilities."}],"review_version":1}