Literature verification report — Einstein–Vlasov PBH brief (2026-09-23)

Ledger #

A. Einstein–Vlasov numerics #

1. Rein, Rendall & Schaeffer 1998 — VERIFIED #

Source: https://arxiv.org/abs/gr-qc/9804040 (PDF fetched and converted), Phys. Rev. D 58, 044007.

Formulation. "We use Schwarzschild coordinates (i.e. polar slicing and area radius)." Metric: ds² = −e^{2μ(t,r)}dt² + e^{2λ(t,r)}dr² + r²(dθ²+sin²θdφ²), with λ(t,0)=0 and λ,μ→0 at infinity. Field eqs (2.2)–(2.3), (2.6): e^{−2λ}(2rλ′−1)+1 = 8πr²ρ, e^{−2λ}(2rμ′+1)−1 = 8πr²p, λ̇ = −4πr e^{λ+μ} j. Vlasov eq. in variables (r, w = x·v/r, L = |x×v|²), eq. (2.9). No-trapped-surface condition (2.10): ∫_{|x|≤r}√(1+v²) f dv dx < r/2.

Method (Sec. 3). Particle method: cells in (r,u,α) with weights f⁰_{ijk} = f̊(r_i,u_j,α_k)·4πr_i²Δr·2πu_j²Δu·sinα_k Δα; characteristics Dr = e^{μ−λ}w/√(1+u²), Dw = −wλ̇ − e^{μ−λ}√(1+u²)μ′ + e^{μ−λ}L/(r³√(1+u²)), DL = 0, "simple Euler time stepping"; volume-element update (Δt)⁻¹(f¹−f⁰) = −f⁰(λ̇ + wλ′e^{μ−λ}(1+u²)^{−1/2}). Steady-state test with φ(E)=1 for E<E₀=0.9: "mass 3.36·10⁻² and support contained in 0 ≤ r ≤ 0.36", 2550 particles, errors table (Δt = 1/16000: 2.1% in m, 1.3% in μ at t=10).

Benchmark run (Sec. 4), verbatim: f̊ = A f₀ with "f₀(x,v) = [50,000(2.2 − r)(r − 2)(10.2 − u)(u − 10)(3.1 − α)(α − 2.9)]² for 2 < r < 2.2, 10 < u < 10.2, 2.9 < α < 3.1 and f₀(x,v) = 0 otherwise. Thus the mass is initially concentrated between r = 2 and 2.2 and is moving inward rapidly. In most of the following simulations the support of f₀ is divided into 40 by 20 by 20 cells (40 in r) resulting in 16000 particles and the time step is 0.005." A = 0.69 disperses; A = 0.75 forms a hole ("abrupt transition in μ occurs at r = 0.235"). "Thus it seems that the critical value of A for this choice of f₀ is A* ≈ 0.70." "lim_{A→A⁺} M(A) = ½ lim r(A) ≈ 0.11, which indicates type I behaviour … Since Δr = 0.005, the discontinuity in M(A) at A is significant." Two further families: f₀ = 0.1(1−r²)²(1−u²)² (r<1,u<1; Δt=0.00125; threshold near A≈1.6) and f₀ = 0.1(3−r)²(2−r)²(1−r)²(1−u²)² (1<r<3, u<1; threshold near A≈0.76).

Detection: black hole identified by the step in the lapse and the radius where λ = −½ ln(1−2m/r) is maximal; M(A) = r(A)/2; cross-checked with radial null geodesics ("Those starting after a certain time T₁ remain within a finite radius"). "in all the cases which were computed the mass of the black hole is more than 90% of the total ADM mass".

2. Olabarrieta & Choptuik 2002 — VERIFIED-WITH-CORRECTION #

Source: https://arxiv.org/abs/gr-qc/0107076 (PDF converted), Phys. Rev. D 65, 024007.

Coordinates (correction): "we … have chosen maximal-areal coordinates", metric (4): ds² = −(α² − a²β²)dt² + 2a²β dtdr + a²dr² + r²dΩ²; constraints (5)–(6), slicing condition (7), β = αrK^θ_θ (8); "fully constrained evolution". Mass aspect (9): M = (r/2)(1 + β²/α² − 1/a²). Method: particle-mesh (Shapiro–Teukolsky approach); particles are "infinitesimally thin shells", each with "net angular momentum zero, l⃗ = 0, but |l⃗|² ≡ l² ≠ 0"; geodesics integrated with lsoda, O(Δt); N = 10⁵ particles (Fig. 4); statistical error requires N ≈ 1/h⁴. Initial data (56)–(58): R(r) = r² e^{−(r−r₀)²/Δr²}Θ(r), P(p̄_r) = e^{−(p̄_r−p̄_{r0})²/Δp̄_r²}, L(l) = e^{−(l−l₀)²/Δl²}Θ(l), with r₀=5, Δr=1, p̄_{r0}=0, Δp̄_r=2, l₀=12, Δl=2 ("almost time symmetric", family (a)); families (b)–(e): p̄_{r0}=−4, l₀=3,5,7,12; family (f): tanh profiles (63)–(65) with l₀=7. Tuning parameter = total rest mass M₀ (55); "we generally determined M₀* to a relative precision of about 4 × 10⁻¹¹"; "The critical parameter for this family is M₀* ≈ 1.3". Results: Type I; static critical geometry (β→0, ȧ→0); lifetime law (59) τ ~ −σ ln|M₀ − M₀*|; Table I: σ = 5.1±0.2, 5.3±0.2, 5.2±0.2 (a); 5.7 (b); 5.5 (c); 5.0, 5.0 (d); 4.9 (e); 5.9±0.2 (f). "The maximum value of 2M(r)/r is about 0.76" in the critical solution. Apparent horizon: "when 2M(t,r)/r = 1 − 1/a² + β²/α² becomes equal to 1, a marginally trapped surface has been formed".

3. Akbarian & Choptuik 2014 — VERIFIED-WITH-CORRECTION #

Source: https://arxiv.org/abs/1409.5176 (PDF converted), Phys. Rev. D 90, 104023. (The brief's 1409.5847 is "Extremisers for the trace theorem on the sphere".)

Coordinates: "We adopt polar-areal coordinates (t, r)", metric (8) ds² = −α²dt² + a²dr² + r²dΩ²; Hamiltonian constraint (9), momentum constraint (10), polar slicing (11); a(t,0)=1 "which follows from the demand of elementary flatness at the origin"; α(t,r_max) = 1/a(t,r_max). Not horizon-penetrating. Method: direct finite-volume integration of the Vlasov PDE in (r, p_r, l²), conservation form (27); Roe solver; O(h²) Runge–Kutta; 2D mode with fixed l² or 3D with l-distribution. Centre: constraints "from r = 0 outward"; I found no further statement on a special treatment of f at r=0. L=0 limit: never used — "apart from the obvious fact that the particles do have angular momentum in all of our computations"; all families carry δ(l−l₀), Gaussian or step distributions in l (Table I). Angular momentum "does not have a significant impact on the features of the critical solution". Initial data: f = S(r,p_r)F(l) with S Gaussian A exp(−(r−r_c)²/Δr² − (p_r−p_c)²/Δp²) or bi-quadratic; families G1–G10. Near-static: Φ(E,l) = C(1−E/E₀)^b Θ(E₀−E) δ(l−l₀) (70), perturbed via (75)–(77). Results: massless generic: "σ = 1.4 ± 0.1" (69), Γ ≡ max_r 2m/r at criticality "0.79 ≲ Γ ≲ 0.81" (67); near-static massless: "σ = 1.43 ± 0.07" (78), "0.80 ≲ Γ ≲ 0.89" (72); massive (Table IV): σ ≈ 1.32–1.54, "the observed variation in σ is significant"; a massive near-static solution "with E_b negative, but relatively close to 0 … had an associated scaling exponent σ = 3.0 ± 0.1 … clearly distinct". "For differing families of initial data we find distinct critical solutions, so there is no universality of the critical configuration itself."

4. Andréasson & Rein 2006 — VERIFIED-WITH-CORRECTION #

Source: https://arxiv.org/abs/gr-qc/0601112 (ar5iv), CQG 23, 3659 (2006). (gr-qc/0510105 is an unrelated DGP-gravity paper.)

Maximal-areal coordinates ds² = −(α²+a²β²)dt² + 2a²β dtdr + a²dr² + r²dΩ²; PIC scheme (≈14 000 particles typical, up to 10⁶); Euler steps. Steady states f = Φ(E,L) = (E₀ − E)₊^k (L − L₀)₊^l, cases (k,l) = (0,0), (0,½), (1,½), (0,3/2), all with L₀ = 0.1 (shell-like). Binding energy E_b = (M₀ − M)/M₀, central redshift Z_c = 1/√α(0) − 1; binding-energy maximum near Z_c ≈ 0.40–0.43, E_b,max ≈ 0.039–0.041 (Tables 1–4). "the conjecture by Novikov and Zeldovich … is true in a numerical sense also for states that depend on the angular momentum L." Type I: "for type I matter there is thus a mass gap". Non-universality: "Since there are infinitely many unstable static solutions, and any one will do as critical solution, universality is contradicted". BH detection: 2m/r > 1 ⇔ a² > α²/β², plus null-geodesic capture checks. King models are not in this paper (they are in item 5).

5. Günther et al. 2021 — VERIFIED-WITH-CORRECTION #

Source: https://arxiv.org/abs/2009.08163 (PDF converted), CQG 38, 035003.

Abstract (verbatim): "We investigate stability issues for steady states of the spherically symmetric Einstein-Vlasov system numerically in Schwarzschild, maximal areal, and Eddington-Finkelstein coordinates. Across all coordinate systems we confirm the conjecture that the first binding energy maximum along a one-parameter family of steady states signals the onset of instability. Beyond this maximum perturbed solutions either collapse to a black hole, form heteroclinic orbits, or eventually fully disperse. Contrary to earlier research, we find that a negative binding energy does not necessarily correspond to fully dispersing solutions. We also comment on the so-called turning point principle …" PIC with "at least 15 million numerical particles", Δr, Δt ~ 10⁻⁴, dynamically accessible perturbations; polytropes (1−E/E₀)₊^k(L−L₀)₊^l and King f₀ = (e^{1−E/E₀} − 1)₊; E_b = (N−M)/N. BH criteria: 2m/r→1 (Schwarzschild), 1/a − rκ < 0 (maximal areal), a < 0 (EF). Code availability: none stated; only "The movie corresponding to these snapshots is available at [31]" with [31] = http://www.diffgleichg.uni-bayreuth.de/en/research/einstein-vlasov-numerics/.

6. Ames, Andréasson & Rinne — VERIFIED #

• 2021: https://arxiv.org/abs/2010.15771, CQG 38, 105003. Abstract: "(2+1)+1 formulation … particle-in-cell and finite difference code … Solutions are launched from non-stationary initial data and exhibit type I critical behaviour … lifetime scaling … support that the critical solutions are stationary … We prove that complete dispersal of the solution implies that it has nonpositive binding energy."
• 2023 hoop/WCC: https://arxiv.org/abs/2305.04360, PRD 108, 064054. "We find formation of an apparent horizon in all cases we consider, which provides support for the weak cosmic censorship conjecture." Highly prolate data become only mildly prolate by horizon formation; polar circumference "𝒞_{H,p}/(4πM_H) < 1.12 and 𝒞_{H,e}/(4πM_H) < 1, at the time when the horizon forms" (κ_p ∈ [1.00,1.12]). Key argument: dust-like (Lin–Mestel–Shu) data are inappropriate for Vlasov because velocity dispersion regularizes ("for a sufficiently small amplitude … the global existence results … can be applied"). Only total angular momentum zero implemented; rotating case left open. Relevance to anisotropy: for collisionless matter in GR, prolate elongation did not prevent horizon formation in any run — anisotropy did not help avoid a horizon.
• No 2024–2026 dynamical follow-up found; Ames–Andréasson 2024 CQG review (arXiv 2310.00776) covers stationary solutions only.

B. Rigorous results for a causal-bound argument #

7. Andréasson, Kunze & Rein 2011 — VERIFIED #

Source: https://arxiv.org/abs/0706.3787 (PDF converted), Math. Ann. 350, 683.

Set-up (Schwarzschild coordinates, metric (1.6)–(1.9)): given 0<r₀<r₁, M = r₁/2, 0<M_out<M with (2.6) 2(M−M_out)/r₀ < 8/9; case (i) R₁−r₁ < (r₁−r₀)/6 (2.7); case (ii) (2.8); R₀ = (r₁+R₁)/2; all matter outside r₀ initially in [R₀,R₁] with mass M_out (2.9); inner mass M−M_out inside r₀ (2.10).

Theorem 2.1 (verbatim): "Let r₀, r₁, M, and M_out be given as above, and let R₁ satisfy (2.7). Then there exists a set I₁ of regular initial data for the spherically symmetric Einstein-Vlasov system such that if f̊ ∈ I₁, then (2.9) and (2.10) hold, the corresponding solution exists on D, and lim_{s→∞} γ⁺(s) < ∞, lim_{s→∞} ∫_{γ⁺(s)}^∞ 4πr²ρ(s,r)dr > 0, where γ⁺ satisfies (1.10)." (Theorem 2.2: same with (2.8), κ=6.) D := {(t,r) : r ≥ γ⁺(t)} is the region outside the outgoing radial null geodesic from r₀ (1.11).

Theorem 2.4 (verbatim, parts): "(a) There exist constants α, β > 0 … such that if t ≥ 0 and r ≥ 2M + αe^{−βt} then f(t,r,·,·) = 0, i.e., we have vacuum, and the metric equals the Schwarzschild metric … (b) … lim_{t→∞} μ(t,r) = −∞ for r ≤ 2M, and the timelike lines r = c, where c ∈ [0,2M], are incomplete … (c) … lim_{s→∞} γ*(s) = 2M, and every radially outgoing null geodesic γ with γ(0) > r* is future complete".

Corollary 2.3: inner part may be a static solution f_s; the solution then "exists for all r ≥ 0, t ≥ 0 and coincides with the static solution f_s for all r ≤ γ⁺(t)" — global existence in Schwarzschild time for non-small data.

Mechanism / causal argument. General-matter version Theorem 3.1 under (DEC), (NNP), (GLO), and "(GCC) There exists a constant c₁ > 0 such that ρ ≤ −c₁ j in D" (matter ingoing). Key sentences: "by (DEC) any geodesic … satisfies dR/ds ≤ e^{(μ−λ)}" (3.3), and "In view of (3.3) a possible break down of solutions at r = 0 will have no influence on the outer domain D." Also: "The reason why we need some matter in the region r ≤ r₀ is to ensure that initially ingoing matter continues to be ingoing for all times". LRR 2011 §3.7 paraphrase: "it is possible to choose the parameters for the data such that the particles of the outer matter part continue to move inward for all Schwarzschild time as long as the particles do not interact with the inner part." Note: no trapped surface is asserted (Schwarzschild coordinates "do not admit trapped surfaces", LRR §3.7); the paper's own comparison: Christodoulou's scalar-field conditions allow 2m/r ∈ (0,1) "whereas our conditions always require 2m/r to be quite close to one."

8. Andréasson & Rein 2010 — VERIFIED-WITH-CORRECTION #

Source: https://arxiv.org/abs/0910.1254 (PDF converted), J. Hyperbolic Differ. Equ. 7, 707–731.

Abstract: "We find explicit conditions on the initial data which guarantee the formation of a trapped surface in the evolution which in particular implies that weak cosmic censorship holds for these data. … we show that the event horizon is future complete. Furthermore we find that the apparent horizon and the event horizon do not coincide. … The analysis is carried out in Eddington-Finkelstein coordinates." Metric ds² = −a b² dv² + 2b dv dr + r²dΩ²; trapped ⇔ a<0.

Theorem 4.1 (verbatim): data f̊ = f_s + f̊_out with supp f̊_out ⊂ [R₀,R₁]×[p₋,p₊]×[0,L₊] (4.1), P := −max{p¹ …} (4.2); "Let data f_s + f̊_out be given such that R₀ ≥ 2M > r₀, L₊ := 12m₀², 2P ≥ 1 + L₊/r₀² (4.4), and such that there exists V > 0 with the property that (2P²/(1+L₊/r₀²)) exp(−(2M/r₀)V) > R₁ − 2M (4.5), ½(1 + L₊/r₀²)(1/p₋²) exp((2M/r₀²)V + (4M/r₀)V) < R₀ − r₀ (4.6). Then the solution launched by f_s + f̊ forms a trapped surface at some advanced time v < V." "Remark. Note that this result implies that weak cosmic censorship holds for these data in view of the results [12, 13]." Contrast with Rendall 1992: "the proof in [28] rests on a continuity argument and it is not possible to tell whether a given initial data set will evolve into a spacetime containing a trapped surface".

9. Andréasson, Living Rev. Relativ. 14 (2011) 4 — VERIFIED-WITH-CORRECTION #

Source: https://arxiv.org/abs/1106.1367 (ar5iv) = doi:10.12942/lrr-2011-4 (PMC5255633).

Sections: 3.7 "Formation of black holes and trapped surfaces", 3.8 "Numerical studies on critical collapse". §3.7 quotes: "The first result in this direction was obtained by Rendall [149] … a trapped surface forms in the evolution."; "Dafermos [62] has proven that, if a spherically-symmetric spacetime contains a trapped surface and the matter model satisfies certain hypotheses, then weak cosmic censorship holds true. In [64] it was then shown that Vlasov matter does satisfy the required hypotheses."; "[24] … explicit conditions on the initial data … guarantee the formation of trapped surfaces … Eddington–Finkelstein coordinates … control the life span of the solution to ensure that there is sufficient time to form a trapped surface before the solution may break down."; "[20] … Schwarzschild coordinates. Note that these coordinates do not admit trapped surfaces. The initial data in [20] consist of two separate parts of matter…"; "[23] … all the matter do cross the event horizon asymptotically in Schwarzschild time." §3.8 (complete): "In [147] a numerical study on critical collapse … was initiated … Rein and Rodewis [148] … convergence properties … The conclusion of [147] is that Vlasov matter is of type I. There are two other independent numerical simulations … [128, 21]. In these simulations, maximal-areal coordinates are used rather than Schwarzschild coordinates as in [147]. The conclusion of these studies agrees with the one in [147]." §3.1: "Rendall uses maximal-isotropic coordinates in [156] … double null coordinates in [64, 63]. Maximal-areal coordinates and Eddington–Finkelstein coordinates are used in [21, 17], and in [24] respectively." Corrections: "Oppenheimer" does not appear in the review; the only dust mentions concern cosmological late-time asymptotics and shell-crossings. Andréasson–Rein–Rendall 2003 = "On the Einstein–Vlasov system with hyperbolic symmetry", Math. Proc. Camb. Phil. Soc. 134, 529–549 (gr-qc/0110089) — not an OS analogue. The Oppenheimer–Snyder-type Vlasov result is Andréasson & Rein, "Oppenheimer-Snyder type collapse for a collisionless gas", CMP 406:284 (2025), arXiv 2410.06701: "when the corresponding initial data are suitably approximated by data for a collisionless gas … then a trapped surface forms" (Painlevé–Gullstrand coordinates).

10. Hayward 1994 — VERIFIED #

Source: https://arxiv.org/abs/gr-qc/9303006 (ar5iv), PRD 49, 6467. Definition: "A trapping horizon is the closure H̄ of a 3-surface H foliated by marginal surfaces on which θ₋|_H ≠ 0 and ℒ₋θ₊|_H ≠ 0 … outer if ℒ₋θ₊|_H < 0, inner if ℒ₋θ₊|_H > 0, future if θ₋|_H < 0 and past if θ₋|_H > 0." Theorem 2: "If the null energy condition holds, a trapping horizon is null if and only if the internal shear and normal energy density vanish. Otherwise, an outer trapping horizon is spatial and an inner trapping horizon is Lorentzian." Theorem 3: "If the null energy condition holds, the area form of a future outer or past inner trapping horizon is non-decreasing … in all cases being constant if and only if the horizon is null." NEC as used: φ₊ ≥ 0, ρ ≥ 0 (eqs 4a,b). Not spherical-symmetry-specific; "trapped region cannot shrink" must be argued from Theorems 2–3 (achronal outer horizon + non-decreasing area), it is not a verbatim claim.

11. Hellaby & Lake 1985 — VERIFIED via secondary #

Citation confirmed (Hellaby–Krasiński gr-qc/0510093): "C. Hellaby, K. Lake, 'Shell Crossings and the Tolman Model', Astrophys. J. 290, 381-387 (1985) [+ erratum: Astrophys. J. 300, 461 (1986)]." Original text not retrievable. Sussman 2010, arXiv 1005.0717, Appendix C (verbatim; convention Ṙ² = 2M/R + E, c t_bb = bang time): "Parabolic and hyperbolic models or regions: R′ > 0 ⇔ {M′ ≥ 0, E′ ≥ 0, t′_bb ≤ 0} (C.1); Elliptic models or regions: ±R′ > 0 ⇔ ±M′ ≥ 0, ±t′_bb ≤ 0, ±[M′/M − (3/2)E′/E + c t′_bb |E|^{3/2}/(2πM)] ≥ 0 (C.2) … The equal sign holds only at symmetry centers and at values of r where R′ = 0." Sussman's elliptic form (C.8): c t′_bb/(3R′_i/R_i) ≤ 0, c t′coll/(3R′i/R_i) ≥ 0, δ_i^{(m)} ≥ −1, i.e. "two of the three conditions in (C.8) are sign conditions on the gradients of the coordinate locus of the central singularity (t′bb, t′coll)". Hellaby–Krasiński gr-qc/0510093 (Sec. IV): "the set of necessary and sufficient conditions for no shell crossings in an elliptic model is {t{B,r}/M{,r} ≤ 0, t{C,r}/M{,r} ≥ 0}". (My check: with t_C − t_B = 2πM/(−E)^{3/2}, t_C′ ≥ 0 is algebraically the third condition of (C.2).)

12. LTB central singularity nakedness — VERIFIED-WITH-CORRECTION #

• Singh & Joshi 1996, CQG 13, 559 (gr-qc/9409062), abstract (paraphrase from arXiv page): if the first density derivative at the centre is nonzero → naked; if ρ₁=ρ₂=0, ρ₃≠0 → depends on parameters; if the first three derivatives vanish → black hole.
• Singh 1999, CQG 16, 3307 (gr-qc/9808003), verbatim: naked "if ρ₁<0 or if ρ₁=0, ρ₂<0"; "If ρ₁=ρ₂=0 and ρ₃<0 the singularity is naked if the dimensionless quantity ζ = √3ρ₃/4ρ₀^{5/2} is less than or equal to −25.9904, and covered when ζ exceeds this value"; covered if "ρ₁=ρ₂=ρ₃=0" (marginally bound).
• Jhingan & Joshi 1997 (gr-qc/9701016; general f(r)): expansions F(r) = ΣF_n r^{n+3}, f(r) = Σ_{n≥2} f_n r^n (13); naked iff V(X)=0 (20) has a real positive root; "whenever α is greater than 3 we always have black hole"; "when all the first three terms ρ₁, ρ₂ and ρ₃ in the expansion of density profile are zero … We know that in the marginally bound case this situation always corresponds to a black hole [8]"; for f ≠ 0 with f₃=f₄=0, f₅≠0 they get α=3 and "we can have again both the possibilities, namely the black holes and strong curvature naked singularities depending on the choice of initial free functions." Fig. 2 caption: for ξ = F₃/F₀^{5/2} < −2 "the center is the first point to get trapped".
• Bound case, smooth even data: Ortiz & Sarbach 2014 (arXiv 1311.0268), abstract: "A spherical dust cloud which is initially at rest and which has a monotonously decaying density profile collapses … Provided the density profile is not too flat, meaning that its second radial derivative is negative at the center, this singularity is visible to local, and sometimes even to global observers." Conditions (i)–(viii) incl. (vi) bounded collapse, (viii) non-degeneracy; cites Christodoulou CMP 93, 171 (1984): "for regular, time-symmetric generic initial data there exist infinitely many radial light rays emanating from the central singularity". Correction: none of the accessible sources phrase the classification as t_C(r) = t₀ + t_n rⁿ; they use density (or F, f) expansions. The "n ≥ 4 covered" statement is established for marginally bound data; for E<0 with an odd velocity-profile term it can fail (JJ97 eq. 77–78).

C. Newtonian cold collapse #

13. Fillmore–Goldreich 1984 / Bertschinger 1985 — VERIFIED via secondary #

Originals (ApJ 281, 1; ApJS 58, 39) not retrievable (ADS 405, IOP 404, S2 abstracts elided).

• Sikivie–Tkachev–Wang 1997 (astro-ph/9609022, PRD 56, 1863), verbatim: "An analytical treatment of the radial infall model using adiabatic invariants predicted [11] the behaviour of the density near the origin to be: ρ ∝ r^{−9ǫ/(3ǫ+1)} in the range 2/3 ≤ ǫ ≤ 1 and ρ ∝ r^{−2} in the range 0 < ǫ ≤ 2/3. These predictions agree very well with our results." ([11] = FG84.) With angular momentum: "For ǫ < 2/3 we find ρ ∝ r^{−2} outside the first inner caustic (the same as with j = 0) but ρ ∝ r^{−γ} inside with γ = 9ǫ/(3ǫ+1)." Abstract: "the presence of angular momentum produces an effective core radius, i.e. it makes the contribution of the halo to the rotation curve go to zero at zero radius" — a shallower inner cusp, not a constant-density core.
• Nusser 2001 (astro-ph/0008217, MNRAS 325, 1397): radial: M ∝ r for 0<ε≤2/3, M ∝ r^{3/(1+3ε)} for ε>2/3; with L = ℒ√(GM_r_) at turnaround: "M ∝ r^{3/(1+3ε)} for all ε>0, in the region r/r_t ≪ ℒ" (⇒ ρ ∝ r^{−9ε/(1+3ε)}); no core formation.
• Xun Shi lecture notes (MPA): FG84 result Υ = 3s/(s+3) for s ≤ 3/2, Υ = 1 for s ≥ 3/2 with s = 1/ε, i.e. inner slope −9ε/(1+3ε) vs −2; ε=1 ⇒ ρ ∝ r^{−9/4} (Bertschinger).
• Subramanian 1999 (astro-ph/9909280): α = 9ε/(3ε+1); "α < 2 can only obtain if the system has non-radial velocity dispersions"; static cores with α<1 need tangential > radial dispersion.
• Zukin & Bertschinger 2010 (arXiv 1008.0639, PRD 82, 104044): FG84/B85 "2<ν<2.25"; Nusser: "0<ν<2.25"; with tidal-torque parameter ϖ they obtain ρ ∝ r^{−1} for galactic haloes.

D. Oscillatons and SP → Vlasov #

14. Oscillaton maximum mass — VERIFIED-WITH-CORRECTION #

• Alcubierre et al. 2003 (gr-qc/0301105, CQG 20, 2883), verbatim: "we express the gravitational constant in terms of he Planck mass: G = 1/m_Pl²" (non-reduced); "The maximum mass M_c = 0.607 m_Pl²/m_Φ is reached for a central value φ₁c(0) = 0.48 … fundamental frequency Ω = 0.864"; later "the critical mass M_c ≃ 0.606 (m_Pl²/m)". Seidel–Suen [1] cited only for discovery.
• Grandclément–Fodor–Forgács 2011 (arXiv 1107.2791): "A maximum mass of M_max = 0.60535 is attained for ω_min = 0.8608"; "M_max = 1.6085×10²⁰ kg eV/(mc²)" (Eq. 45; consistent with m_Pl = G^{−1/2}); "consistent with … where the authors found ω_min = 0.864 and M_max = 0.607".
• Liebling & Palenzuela LRR (arXiv 1202.5809 §3.4): "M_max = 0.607 M_Planck²/m".
• Ureña-López 2002 (gr-qc/0104093): "maximum mass M_max ≃ 0.5522 m_Pl²/m with σ_c(0) ≃ 0.235" — inconsistent with the later 0.605–0.607 (likely truncation-level difference; not verified).
• Seidel & Suen 1991 PRL 66, 1659: only the abstract was accessible; it gives no number. The attribution of 0.605–0.607 to Seidel–Suen is not verified.

15. Rigorous semiclassical limits — VERIFIED (abstract level) #

• Lions & Paul 1993, Rev. Mat. Iberoam. 9, 553–618 (EUDML abstract): Wigner measures "satisfy, in linear situations (Schrödinger equations) or nonlinear ones (time-dependent Hartree equations), transport equations of Liouville or Vlasov type." Saffirio review (2307.07762): "convergence of the Hartree dynamics towards the Vlasov equation was proven in weak topology, including singular potentials such as the Coulomb interaction, using compactness methods" [Lions–Paul; Markowich–Mauser]. Mixed states, d=3, weak, non-quantitative.
• Zhang, Zheng & Mauser 2002, CPAM 55, 582–632: title "The limit from the Schrödinger–Poisson to the Vlasov–Poisson equations with general data in one dimension" — 1D, general (pure-state/measure) data; abstract not retrievable (403). Only 1D result covering non-smooth/caustic-type data.
• Golse & Paul 2017, ARMA 223, 57 (arXiv 1510.06681): quantitative Hartree→Vlasov and N-body→Vlasov via a quantum Monge–Kantorovich pseudo-distance; potential "bounded … of class C^{1,1}" (Lipschitz gradient) — Coulomb excluded; all t ≥ 0 (exponential growth).
• Lafleche 2019, J. Stat. Phys. 177, 20 (arXiv 1809.04544): "quantitative version of the semiclassical limit from the Hartree to the Vlasov equation with singular interaction, including the Coulomb potential" via propagation of moments (mixed states; rate √ℏ in W₂,ℏ).
• Saffirio 2019 CMP 373, 571 (arXiv 1903.06001): strong (trace/HS) convergence for |x|^{−a}, a ∈ (0,½), mixed states. Saffirio 2020 SIMA 52, 5533 (arXiv 1903.06013): Coulomb/gravitational, trace-norm convergence "for a special class of mixed quasi-free states".
• Lafleche & Saffirio 2023, APDE 16, 891 (arXiv 2003.02926): "general singular interaction potential including the Coulomb and gravitational interactions … explicit bounds in the strong topologies of Schatten norms … for general initial data in some Sobolev space and any fixed time interval" (needs W^{s+1,∞} ∩ H^{s+1} regularity of the Vlasov solution ⇒ caustics not covered; d = 2,3).
• Chong, Lafleche & Saffirio (arXiv 2103.10946, JEMS): many-body → Hartree–Fock → Vlasov with singular potentials incl. Coulomb, local in time for N^{−1/2} ≪ ℏ ≤ N^{−1/3}.
• Newer: Leopold & Saffirio 2023 SIMA 55, 1676 (arXiv 2203.03031): relativistic Hartree–Fock → relativistic Vlasov with singular potentials; Lafleche–Saffirio 2023 review of uniqueness criteria (arXiv 2303.10634); Lafleche 2024 (arXiv 2401.05773); Smith 2026 (arXiv 2607.22490): uniform-in-time strong convergence near Penrose-stable states, regular kernels only. All 3D results assume regular Vlasov solutions (or weak/measure limits without identified equation for Coulomb); none covers gravitational caustics or strong-field collapse.

16. Burnett's conjecture and high-frequency limits — VERIFIED #

• Huneau–Luk 2018, Duke Math. J. 167, 3315 (arXiv 1706.09501): polarized U(1); local-in-time small-data Einstein–multiple-null-dust solutions arise as weak limits of vacuum.
• Huneau–Luk, Ann. Sci. ENS (2024) (arXiv 1907.10743): abstract quoted; Burnett's conjecture proven for U(1)-symmetric C⁴ metrics in elliptic gauge; limit = Einstein–massless Vlasov via microlocal defect measure.
• Guerra & Teixeira da Costa (arXiv 2107.00942, ARMA 2024): "streamlined proof of a stronger result … remove the need for control on higher derivatives"; wave maps.
• Huneau–Luk 2024 (arXiv 2403.03470): "Burnett's conjecture … when the metrics satisfy a generalized wave coordinate condition" ⇒ Einstein–massless Vlasov.
• Touati, CMP 2023 (arXiv 2206.12318): high-frequency vacuum family without symmetry converging to Einstein–null dust; Touati 2024 (arXiv 2402.17530): reverse Burnett for any number of null dusts.
• Huneau–Luk 2025 (arXiv 2506.21779): "Given suitable small, localized, U(1)-symmetric solutions to the Einstein-massless Vlasov system in an elliptic gauge, we prove that they can be approximated by high-frequency vacuum spacetimes."
• Huneau–Luk review (arXiv 2404.07659): Conjecture 1.1/1.2 stated; Section 8 open problems include "8.5 Large solutions", "8.9.1 Formation of trapped surfaces", "8.11 Semi-classical limits for the Einstein–Klein Gordon system"; constructions are "small-data and local-in-time".
• Massive case: no Einstein–Klein–Gordon → Einstein–(massive) Vlasov or → dust theorem found. Closest: Salvi 2025, Ann. Henri Poincaré (arXiv 2502.06622), flat-space massive Klein–Gordon–Maxwell → relativistic pressureless Euler–Maxwell "and show how this system … is related to the relativistic massive Vlasov-Maxwell equations". Strong-field collapse is not covered anywhere.

Files (all absolute): extracted texts in /tmp/claude-0/-root-PBH/a34e6be4-0ee0-4598-bfd0-88626255d779/scratchpad/ — webfetch-1790190461761-coh6zz.txt (RRS98), webfetch-1790190466103-swufid.txt (OC02), webfetch-1790190470054-1mwve1.txt (AKR11), ak2014.txt, ar2010.txt, gunther2021.txt, stw1997.txt, alcubierre2003.txt, sussman2010.txt, jj1997.txt, os2014.txt, xunshi.txt, hellaby2009.txt.