{"id":"2192c51d-df6c-487b-9cde-38a3287efff3","arxiv_id":"1909.02685","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"At large D, RN-dS and GB-dS black holes evolve to stationary lumpy solutions on the instability threshold, and in the unstable region their mass localizes into spot-like or ring-like configurations.","lead":"Using a large dimension expansion, this paper shows that two families of de Sitter black holes become nonlinearly unstable exactly where linear theory predicts, and that at the threshold they settle into lumpy shapes. In the unstable region the mass density concentrates at poles or the equator, resembling black spots and rings and hinting at topology-changing transitions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Spot/ring end state is derived from a leading-order 1/D solution in a regime the paper itself says is invalid; printed equations also contain a sign inconsistency in the p-m relation.","rationale":"The reader's weakest assumption identified that the leading-order large-D effective equations may not capture the true nonlinear horizon dynamics, especially after the mass grows to order e^D. This is indeed the most load-bearing concern: the paper's own Sec. 4 concedes that the 1/D expansion breaks down before the end state is reached, so the localized spot/ring configurations are suggestive but not demonstrated. My stress-test sharpens this with the specific time scale t_* ~ (1/omega) log(D/a) at which the neglected 1/D terms become order one, showing that the regime in which the spot/ring classification is made is precisely the regime where the effective theory is uncontrolled. I also noticed an internal sign inconsistency in the printed equations: the static lumpy solution (8) and the analytical solution (17)-(18) with p = m' (or p = m' - tan z partial_t m) do not satisfy Eq. (6) as printed, yielding a factor of two instead of zero. This is likely a typo in the sign of p, but as written it prevents a reader from verifying the central analytical solution. The threshold result (P2 = 2) is a solid extension of the linear analysis and the paper is appropriately cautious, so I do not recommend overturning the CONDITIONAL verdict. The same concern is the reader's weakest assumption, so I agree; the verdict remains unchanged.","tokens_in":11501,"tokens_out":11191,"duration_ms":116794,"concrete_test":"Re-derive Eqs. (5)-(7) from the original derivation in Tanabe (arXiv:1511.06059), fix the sign of p in (8) and (18), and verify whether (15) with omega from (16) satisfies all three effective equations; then add next-to-leading-order 1/D corrections (as done for black strings in [15]) and evolve the same initial data. If the growth of m saturates or the profile does not localize before m ~ e^D, the three-way spot/ring classification is an artifact of the leading-order truncation.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim that unstable RN-dS/GB-dS black holes evolve toward localized black spots/rings rests on Eq. (15), which is presented as an exact solution of the leading-order large-D effective equations (5)-(7). But the paper itself states in Sec. 4 that when m grows to order e^D, the 1/D expansion is no longer valid. For the dominant growing mode, the exponent a e^{omega t} cos^2 z reaches order D at t_* ~ (1/omega) log(D/a); at that time the neglected O(1/D) terms become order one. The 'late time' at which the spot/ring classification applies therefore lies beyond the regime where the effective equations are controlled, so the three-way classification is an extrapolation of a truncated theory, not a demonstrated fate. Moreover, the printed equations are internally inconsistent: substituting the static lumpy ansatz (8) with p = m' into Eq. (6) yields 2 cot z m' != 0 unless m' = 0, and substituting the analytical ansatz (17)-(18) into Eq. (6) gives the same failure. The sign of p in (8) and (18) does not match Eqs. (5)-(6), making the claimed exact solution (15) unverifiable as printed and weakening confidence in the nonlinear threshold and end-state results.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper applies the large-D (1/D) expansion to study the nonlinear fate of the 'Λ instability' of Reissner-Nordstrom-de Sitter (RN-dS) and Gauss-Bonnet-de Sitter (GB-dS) black holes. Working with the effective, first-order-in-time equations (5)-(7) and (21)-(22), the authors identify the instability threshold with the linear condition P2=2, construct static 'lumpy' solutions, and present an analytic time-dependent solution whose linearized spectrum reproduces the quasinormal-mode spectra of Refs. [21,22]. By numerically evolving random small perturbations, they report that on the threshold the solutions settle to stationary lumpy black holes, while in the unstable region the mass density grows exponentially and, at late times, resembles single-spot, double-spot, or black-ring configurations. The paper also reports stability of the corresponding flat and AdS cases.","tokens_in":11763,"tokens_out":9398,"duration_ms":97330,"significance":"The paper contains useful and, if corrected, publishable results: the analytic solution connecting the nonlinear evolution to the known quasinormal-mode spectra is a valuable consistency check, and the identification of the nonlinear threshold with P2=2 is a physically relevant statement. The authors also deserve credit for explicitly acknowledging that the 1/D expansion breaks down when m is of order e^D and for discussing the limitations of their end-state classification. However, the printed equations contain a sign inconsistency that invalidates the static and analytic solutions as stated, and the central late-time spot/ring claim lies outside the controlled regime of the effective theory. The manuscript requires correction and re-verification before the results can be accepted.","major_comments":[{"comment":"The static lumpy solution (8) is not a solution of the printed effective equations. Substituting q = Q m/(1+Q^2) and p = m' into Eq. (6) at partial_t m = 0 gives 2 cot z m' = 0, which is nonzero for generic P1 and P2; Eq. (5) likewise requires p = -m'. The same problem occurs for the analytic solution: inserting (17) and (18) into (6) yields 2 cot z m' instead of zero. The correct relation appears to be p = -m' - tan z partial_t m (with p = -m' in the static case). Please correct the sign convention in the effective equations or in the momentum ansatz, and re-check the derivation of the static-solution branches and of the exact solution, since as printed the central analytic results are unverifiable.","section":"§3.1, Eqs. (5)-(8), (17)-(19)"},{"comment":"The late-time single-spot/double-spot/black-ring classification is an extrapolation beyond the domain of validity of the 1/D expansion. As the paper itself states in Sec. 4, once m grows to order e^D the expansion is no longer valid; for the mode in Eq. (15) this occurs at t_* ~ omega^{-1} log(D/a). The snapshots in Fig. 4 are taken in this late-time regime, so the effective equations do not control the claimed localized configurations. The abstract's statement that the unstable solutions 'resemble fully localized black spots and black ring' is therefore not established by the present calculation; it should be presented as a speculative analogy unless the localized regime is treated with a separate large-D rescaling such as that of Ref. [36].","section":"§3.1, Eq. (15); Sec. 4"},{"comment":"The numerical evidence for the threshold and for the three late-time classes lacks convergence tests and error estimates. The manuscript reports only that NDSolve was used with random initial amplitudes up to 10^{-3}; no grid resolution, time-stepping, or consistency checks are given. In view of the sign inconsistency in the printed equations, it is not even clear whether the code implements Eq. (6) or a corrected version. Please state the numerical scheme, the exact sign convention used, and provide convergence data (e.g., Richardson extrapolation or residual checks) for at least one representative parameter point.","section":"§3.1, Figs. 2-4"}],"minor_comments":[{"comment":"The display of omega in Eq. (16) is ambiguous because the square-root bracket is not closed. Please write, for example, omega = (sqrt(2(1+Q^2)^2 Lambda_hat + 2Q^2 - 1) - 1)/(1+Q^2) and confirm that P2=2 gives omega=0.","section":"§3.1, Eq. (16)"},{"comment":"Equation (22) is typeset with unbalanced parentheses and missing bracket closures in the coefficients of partial_z p and partial_z m; it should be reformatted so that the PDE can be read unambiguously.","section":"§3.2, Eq. (22)"},{"comment":"The phrase 'P2 is an positive integer' contains a typo ('an' should be 'a'), and the quantization condition should also state explicitly that P2 must be a positive integer for regularity at z=pi/2.","section":"§2, after Eq. (8)"},{"comment":"In the sentence 'the non-linear evolution of the Myers-Perry black holes agrees qualitatively well with the full numerical stimulation,' 'stimulation' should be 'simulation.'","section":"§4"},{"comment":"The caption states that the vertical axes are not uniformized; please specify the normalization or make clear that the plots are schematic, since otherwise the shapes of m(z) are not quantitatively interpretable.","section":"Fig. 1 caption"}],"recommendation":"major_revision","confidential_remarks":"The sign problem in Eqs. (5)-(8) is likely a typographical sign error (p should be -m' in the static case), but because the static and analytic solutions are central to the paper, the correction must be made and the text rechecked. The numerical results also need reproducibility details. If the authors fix these and soften the end-state claim, the paper could be suitable for publication. The paper's topic fits the journal, and the stress-test concern about the sign inconsistency is valid and load-bearing."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a worthwhile extension of the large-D stability program, but as printed it has a sign error in the p–m relation that breaks the central 'exact' solution. I would send it back for major revision, not desk-reject it.\n\nWhat is new: first nonlinear time evolution of the RN-dS and GB-dS effective equations, a threshold consistent with linear analysis, a classification of late-time states into single spot, double spot, and ring, and an analytic solution whose exponents reproduce the QNM spectra of [21,22]. The paper is honest about the 1/D breakdown and about not determining the true endpoint. Citation practice looks appropriate.\n\nSoft spots. First, the sign issue. Eq. (6) reads ∂t m + cot z ∂z m + cot z p = 0. For the static lumpy ansatz (8) with p = m', this gives 2 cot z m' = 0, so no non-uniform static solution exists. The same problem appears for the analytic solution (17)-(19). Either the sign in (6) should be negative or the p relation should be p = -m'. As printed, the claimed exact solutions do not solve the effective equations, so the nonlinear threshold and late-time classification are unverified as they stand. This is likely a typo, but it is load-bearing.\n\nSecond, the paper itself concedes that once m grows to order e^D, the 1/D expansion is no longer valid, and the spot/ring classification is made in that regime. That is a serious caveat, though they phrase it as suggestive. Still, the headline claim 'fate of instability' overstates what the truncated theory can actually show.\n\nThird, the numerics: NDSolve, no convergence tests, no error estimates, and a fitted amplitude a. For a paper whose main evidence is numerical evolution, that is thin.\n\nWho it is for: people working in large-D effective theories and black hole instabilities. The ideas are worth pursuing, but the printed version needs careful revision before it can be trusted. I would send it to peer review, with the requirement that the authors fix the sign, rerun the numerics with convergence checks, and soften the endpoint claims. If the sign is just a typo, this could be a solid contribution; as is, I would not cite it.","headline":"A useful large-D nonlinear study, but a load-bearing sign error in the printed equations makes the central exact solution unverifiable as written.","tokens_in":12287,"tokens_out":3376,"would_cite":false,"duration_ms":34389,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The nonlinear instability of de Sitter black holes drives the horizon toward localized spots or a black ring, the large-$D$ effective equations show.","keywords":["de Sitter black holes","large-D expansion","black hole instability","Reissner-Nordstrom-de Sitter","Gauss-Bonnet-de Sitter","lumpy black holes","black spots","black ring"],"falsifier":"A full nonlinear numerical evolution of an unstable RN-dS or GB-dS black hole at moderate dimension, say $D=8$, would settle the claim: the prediction fails if the instability threshold is not at the $P_2=2$ parameter line or if the horizon does not approach single-spot, double-spot, or ring-like localized mass configurations. A cheaper internal test is to evolve past $m\\sim e^D$ and see whether the growth formula (15) breaks before any localized profile forms.","tokens_in":11293,"feed_emoji":"🕳️","tokens_out":11656,"duration_ms":106149,"temperature":0.7,"pith_summary":"This paper asks what actually happens when a higher-dimensional de Sitter black hole is unstable, using the large-dimension expansion to follow the full nonlinear evolution rather than only linearized perturbations. For both charged Reissner-Nordström–de Sitter and Gauss–Bonnet–de Sitter black holes, it finds that the nonlinear instability threshold coincides exactly with the linear criterion $P_2=2$. Right on the threshold, generic perturbations settle down to stationary “lumpy” black holes with mass density $(1+Q^2)e^{a\\cos^2 z}$. Just beyond the threshold, the evolution does not settle: the mass density grows exponentially in time and localizes near a single pole, both poles, or the equator, configurations the paper names single spot, double spot, and black ring. The upshot is that the “$\\Lambda$ instability” may drive a topology-changing transition to localized black objects in higher dimensions.","feed_headline":"Unstable de Sitter black holes localize into spots or a ring","feed_subtitle":"Nonlinear large-D evolution matches the linear threshold and leaves mass concentrated at poles or the equator.","key_machinery":"The load-bearing object is the large-$D$ effective description obtained by integrating out the radial direction: for RN-dS the horizon functions $m(t,z)$, $q(t,z)$, $p(t,z)$ satisfy the first-order system (5)–(7), and the GB-dS system is the analogous pair (21)–(22). Because these equations are first order in the polar angle, they allow non-smooth profiles at the equator $z=\\pi/2$, which is how the two hemispheres can evolve with different amplitudes. The analytic solution (19), $m(t,z)=c\\exp\\left[\\sum_{\\ell\\ge 2}(c_{\\ell+}e^{-i\\omega_{\\ell+}t}+c_{\\ell-}e^{-i\\omega_{\\ell-}t})\\cos^\\ell z\\right]$, carries the argument: its exponents $\\omega_{\\ell\\pm}$ are exactly the scalar gravitational quasinormal-mode frequencies, the $\\ell=2$ mode has the largest growth rate, and the growing mode yields the spot and ring shapes.","core_discovery":"The central claim is that the “$\\Lambda$ instability” of RN-dS and GB-dS black holes has a sharp nonlinear threshold at $P_2=2$, matching the linear onset, and that the unstable evolution is captured by an explicit time-dependent solution of the large-$D$ effective equations. On the threshold the system relaxes to stationary lumpy black holes of the form $m(z)=(1+Q^2)e^{P_1(\\cos z)^{P_2}}$, which at the threshold is $m(z)=(1+Q^2)e^{a\\cos^2 z}$. In the unstable region $P_2>2$, the mass distribution obeys $m(t,z)=(1+Q^2)e^{a e^{\\omega t}\\cos^2 z}$ with $\\omega=\\left(\\sqrt{2(1+Q^2)^2\\hat\\Lambda+2Q^2-1}-1\\right)/(1+Q^2)$, so the mass concentrates at the north pole, the south pole, both poles, or the equator depending on the initial perturbation. These late-time profiles are argued to resemble fully localized black spots ($S^{D-2}$) and black rings ($S^1\\times S^{D-3}$). The paper also gives an exact nonlinear solution whose exponents reproduce the scalar quasinormal-mode spectrum, showing why the $\\ell=2$ mode dominates and why the threshold is set by $P_2=2$.","pith_inferences":["Because the equator non-smoothness comes from the first-order-in-$z$ form of the large-$D$ equations, a natural next step is to include $1/D$ corrections and check whether the single/double spot and ring patterns survive once second derivatives restore smoothness.","If the spot and ring end states exist at finite $D$, the $\\Lambda$ instability would be a topology-changing mechanism distinct from the black-string instability, and testing it would bear on the conjecture that the black-string instability is the only topology-changing mechanism in vacuum gravity.","The exponential-growth template $e^{a e^{\\omega t}\\cos^2 z}$ may describe other large-$D$ instabilities with a dominant $\\ell=2$ mode, such as ultraspinning and bar-mode instabilities, whose endpoint shape could be read off from the same formula.","A finite-$D$ numerical simulation with the same charge and cosmological-constant parameters would provide a sharp test: if its threshold does not lie on $P_2=2$, the large-$D$ effective equations are not quantitatively faithful."],"forward_implications":["Nonlinear and linear thresholds coincide: whenever $P_2=2$ the perturbation is marginal, and for $P_2>2$ there is no static lumpy endpoint, only time-dependent evolution.","On the threshold the attractor is a one-parameter family of stationary lumpy holes, with the amplitude selected by the initial data, so thermodynamics at leading order in $1/D$ does not fix the final shape.","In the unstable region the effective mass density grows as $e^{\\omega t}$ until the $1/D$ expansion breaks down, after which a localized spot or ring is the natural configuration at finite $D$.","In asymptotically flat or AdS backgrounds, both charged and Gauss–Bonnet black holes are stable, confirming that the instability needs a positive cosmological constant plus charge or Gauss–Bonnet coupling.","The $\\ell=2$ dominance in the nonlinear solution gives a concrete prediction: the final lumpy or localized shape is controlled by the quadrupole harmonic of the initial perturbation."],"supporting_citations":[{"why":"supplies the large-$D$ expansion method and the effective-equation scheme used throughout.","marker":"[10]"},{"why":"gives the linear quasinormal-mode spectrum and static lumpy solutions for RN-dS that the nonlinear evolution is compared with.","marker":"[21]"},{"why":"gives the GB-dS effective equations, static solutions, and quasinormal-mode spectrum used in Sec. 3.2.","marker":"[22]"},{"why":"provides the finite-$D$ nonlinear black-string evolution that motivates asking what the final state of the instability is.","marker":"[5]"},{"why":"shows how the large-$D$ method follows black-string evolution to its endpoint, the template adapted here.","marker":"[15]"},{"why":"identifies the Gauss–Bonnet–de Sitter instability whose nonlinear fate is studied here.","marker":"[8]"},{"why":"supplies the lumpy-black-hole terminology and the idea of a lumpy-to-localized transition.","marker":"[40]"},{"why":"is the source for naming the localized configurations single spot, double spot, and black ring.","marker":"[41]"}],"fun_headline_variants":["Large-D de Sitter black holes split into spots or rings","de Sitter black holes in large D: instability yields spots or rings","Nonlinear instability of de Sitter black holes spawns spots or a ring","At large D, de Sitter black holes go lumpy or localized","Threshold of de Sitter black hole instability: spots or ring"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument rests on the leading-order large-$D$ effective equations faithfully encoding the true nonlinear horizon dynamics, and the paper itself concedes that once $m$ grows to order $e^D$ the $1/D$ expansion is no longer valid, so the localized spot and ring shapes may be artifacts of the regime where the effective theory has broken down.","fun_headline_variants_meta":{"raw":{"variants":["Large-D de Sitter black holes split into spots or rings","de Sitter black holes in large D: instability yields spots or rings","Nonlinear instability of de Sitter black holes spawns spots or a ring","At large D, de Sitter black holes go lumpy or localized","Threshold of de Sitter black hole instability: spots or ring"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000659,"raw_usage":{"total_tokens":3021,"prompt_tokens":961,"completion_tokens":2060,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":577,"completion_tokens_details":{"reasoning_tokens":1967}},"tokens_in":577,"tokens_out":2060,"duration_ms":15894,"temperature":1.0,"reasoning_tokens":1967,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T04:42:41.298870+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A full nonlinear numerical evolution of an unstable RN-dS or GB-dS black hole at moderate dimension, say $D=8$, would settle the claim: the prediction fails if the instability threshold is not at the $P_2=2$ parameter line or if the horizon does not approach single-spot, double-spot, or ring-like localized mass configurations. A cheaper internal test is to evolve past $m\\sim e^D$ and see whether the growth formula (15) breaks before any localized profile forms.","supporting_citations":[{"cited_title":"Static Gauss-Bonnet Black Holes at Large $D$","cited_arxiv_id":"1703.06381","evidence_quote":"gives the GB-dS effective equations, static solutions, and quasinormal-mode spectrum used in Sec. 3.2."}],"review_version":1}