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REVIEW 4 major objections 5 minor 31 references

In curved spacetimes with a Killing horizon, the late-time decay of a thermal two-point function switches at a critical temperature from a temperature-set rate e^{-2πTt} to a constant quasinormal-mode rate e^{-2πTc t}.

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

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For thermal Bose fields in static de Sitter or a planar BTZ black hole, the late-time correlation decay switches from temperature-limited to quasinormal-mode-limited at a critical temperature Tc.

T0 review reviewed 2026-08-05 challenge →

load-bearing objection The planar BTZ computation is a clean new result, but the de Sitter part misses the light-field (imaginary ν) regime, where the critical temperature is mass-dependent. the 4 major comments →

arxiv 2509.02181 v1 pith:MFWK4YTR submitted 2025-09-02 hep-th

On the late time behavior of thermal two point function in curved space-time

Dmitrii V. Diakonov , Gleb S. Zverev This is my paper
classification hep-th
keywords thermal two-point functionKilling horizonquasinormal modesMatsubara frequencieslate-time decayrelaxation timede Sitter spaceBTZ black hole
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The reading

The paper tries to establish that a thermal Bose gas in a spacetime with a Killing horizon does not relax the way familiar thermal systems do. Its late-time two-point function decays as e^{-2πTt} at low temperature, saturating the thermal bound on relaxation time, but above a critical temperature Tc the decay rate stops following the temperature and pins to the imaginary part of the lowest quasinormal mode — the geometry's own damped oscillation frequency — giving e^{-2πTc t}. The relaxation time therefore saturates to 1/ω_I at high temperature instead of vanishing, opposite to the usual gas behavior, and the switch at Tc looks like a phase transition in the correlation function. The authors verify the pattern by residue computation in two concrete cases, the static de Sitter patch and the planar BTZ black hole, and they check that the de Sitter free energy has no corresponding change at Tc, so the transition is a property of the correlation function, not of thermodynamics.

Core claim

In a static spacetime with a Killing horizon, the thermal two-point function of a free Bose field has two families of singularities: thermal (Matsubara) poles at ω = 2πik/β, whose imaginary parts shrink as temperature rises, and quasinormal-mode poles — the damped characteristic oscillations of the field in that geometry — at fixed imaginary frequencies. The paper claims (Eqs. (1.15), (3.1)) that late-time decay, W_β(t → ∞) ~ e^{-2πTt} for T below a critical temperature Tc, switches to e^{-2πTc t} above Tc, so the relaxation time saturates at 1/ω_I instead of vanishing with increasing temperature. The explicit computations give Tc = (d−1)/(4π) in the static de Sitter patch (units R = 1) and

What carries the argument

The load-bearing object is the integral representation (2.3): W_β(t) ~ ∫dω e^{iωt} sinh(πω)/(e^{βω}−1) ∏_i 1/(ω−ω_i), a product of simple poles at the quasinormal frequencies ω_i multiplied by the Bose-Einstein weight. This form turns the late-time question into a residue problem: the poles closest to the real axis dominate the t → ∞ tail. Matsubara poles at ω = 2πik/β move toward the real axis as T grows; the quasinormal poles sit at fixed imaginary parts. Their crossing defines Tc. The sinh factor erases Matsubara poles at β = 2πk/n. In de Sitter the poles come from gamma-function factors in the mode functions; in planar BTZ the sum over angular modes is collapsed by a table integral (2.22

Load-bearing premise

The argument depends on the two-point function in a horizon spacetime being exactly a product of simple poles — quasinormal-mode poles times Matsubara poles — with no branch cuts, double poles, or continuous spectra closer to the real axis; the authors state this is not a general expansion for Killing-horizon spacetimes, only true in the cases they compute.

What would settle it

Compute the late-time tail of the thermal two-point function in a static horizon spacetime whose Green function is not a pure product of simple poles — for example a field whose quasinormal spectrum has a continuum, or a mass value producing double poles — and check whether the decay rate saturates at a temperature-independent value for T ≫ Tc. The pole picture predicts saturation; a T-dependent rate or a power-law tail would falsify it. Within the paper's own two examples, one can test the degenerate case where a Matsubara pole coincides with a quasinormal pole: the pure exponential should ac

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

If this is right

  • Above Tc the relaxation time is effectively geometric: τ → 1/ω_I as T → ∞, so no matter how hot the gas is, it forgets its initial state on the quasinormal timescale.
  • At temperatures β = 2πk/n the Matsubara poles partially cancel, so the late-time tail is again quasinormal-dominated; at the canonical temperature β = 2π the decay is purely quasinormal, which for BTZ reproduces the known CFT thermal two-point function.
  • The switch is invisible in thermodynamics: the de Sitter free energy's leading large-mass behavior changes at β = 2π, not at β = 4π/(d−1), so the sharp correlation-function change does not mark a genuine phase transition.
  • The high-temperature tail has a thermal form with effective temperature Tc — the "tOmperature" — meaning an infinitely hot horizon gas still looks thermally populated at finite Tc.
  • The authors expect the same two-regime pattern in any static horizon spacetime whose two-point function admits the pole-product form, but they state no general criterion for when that form holds.

Where Pith is reading between the lines

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

  • If the pole-product form holds in other horizon geometries, measuring decay exponents at the special temperatures β = 2πk/n would isolate individual quasinormal frequencies — a correlation-based spectroscopy of the horizon.
  • The paper does not discuss coincidences where a Matsubara pole lands exactly on a quasinormal pole; there the residue estimate degenerates and the pure exponential should acquire a polynomial-in-time prefactor. That is a clean place to test the mechanism.
  • Rindler space has a Killing horizon but, for a free field, no discrete quasinormal spectrum; the same logic would predict no saturation transition there, which would isolate exactly what the pole structure contributes.
  • Because saturation makes relaxation slower (not faster) than the thermal bound, horizon systems with discrete quasinormal spectra would forget no faster than the geometric timescale 1/ω_I — a constraint that could matter for thermalization bounds in holographic settings.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper studies the late-time decay of the thermal two-point function for a free scalar in two static, horizon-bearing spacetimes: the static patch of de Sitter and planar BTZ. Using the pole structure of the integrand after writing the thermal correlator in a product form (2.3), the authors derive closed-form asymptotics (2.13) and (2.24). They conclude that the decay changes at a critical inverse temperature β_c: for β>β_c the decay is e^{-2πt/β} (the thermal bound), while for β<β_c it is controlled by the lowest quasinormal mode, e^{-ω_I t}. In de Sitter they find β_c=4π/(d-1), and in planar BTZ β_c=2π/Δ. The paper interprets this as a temperature-independent high-temperature relaxation time, or 'tOmperature'. The analysis is presented as applying to Bose gases in such spacetimes, with the caveat that (2.3) is not claimed to be general.

Significance. If correct, the paper gives a clean, concrete realization of the idea that thermal correlators in horizon spacetimes can exhibit a crossover between a thermal-bound regime and a quasinormal-mode regime, with the high-temperature relaxation time saturating at 1/ω_I. The derivation is explicit: pole counting, residue arguments, and product formulae are used without any fitted parameters. The BTZ example connects to CFT thermal correlators, and the de Sitter example provides a simple testbed. However, the significance is tempered by the fact that the analysis is restricted to two examples and, as shown below, one of the two examples is not handled correctly for light scalar fields.

major comments (4)
  1. [Sec. 2.1, Eqs. (2.8), (2.11), (2.13)] The critical inverse temperature β_c=4π/(d-1) and the QNM decay e^{-(d-1)t/2} in (2.13) are derived assuming ν is real. For m^2 < ((d-1)/2)^2, ν = i μ with μ>0, and the pole list (2.11) itself gives the nearest QNM at i((d-1)/2 - μ), not at i(d-1)/2. The Matsubara pole crosses that pole at β_c = 4π/(d-1-2μ), not at 4π/(d-1). For β in (4π/(d-1), 4π/(d-1-2μ)), the claimed low-temperature formula e^{-2πt/β} is not the leading late-time decay; the QNM pole dominates. In the massless limit μ→(d-1)/2 the decay exponent tends to zero, so the claimed universal Tc and the high-temperature saturation fail for a valid mass range. The abstract, Eq. (2.13), and the de Sitter part of the conclusions therefore need either a mass restriction ν∈R (m^2≥((d-1)/2)^2) or a mass-dependent β_c.
  2. [Eqs. (2.13), (2.14), (2.24)] The piecewise formulas are not uniform in β. For β=2π k/n, Eq. (2.14) cancels the Matsubara poles, so the e^{-2πt/β} branch in (2.13) and (2.24) fails even when β is on the low-temperature side of β_c. The text acknowledges this in prose, but the displayed asymptotic formulas and the 'sharp transition' statement do not carry the exception. As written, (2.13) is literally false e.g. for β=2π in d=4, where the late-time decay is set by the QNM pole, not by e^{-t}.
  3. [Sec. 2.1, T=Tc] At β=β_c the Matsubara pole and the QNM pole coincide for ν=0 (and for ν real they share the same imaginary part). The residue analysis then generically produces a t e^{-...} factor from a double pole (or, at least, equal-weight superposition of two poles with the same imaginary part). The paper does not discuss the exact critical temperature; it only gives the two sides. This is a technical gap in the claimed 'sharp transition', and should be addressed or explicitly qualified.
  4. [Sec. 2, Eq. (2.3)] The central working assumption is the product form (2.3). The paper correctly states that this is not a general expansion, and the two examples are clear. I do not regard the lack of a general proof as a flaw, but the abstract's wording 'for a Bose gas in a curved space-time with a Killing horizon' overstates the scope. The title and abstract should make clear that the statement is demonstrated only for the two treated geometries.
minor comments (5)
  1. [Sec. 2.1, Eq. (2.13)] The coefficients C+, C-, Cβ are not given. It would be helpful to state at least whether they are nonzero; the distinction matters when β=2πk/n because of the cancellations in (2.14).
  2. [Sec. 2.1, Eq. (2.11)] The notation ± i(...) ± ν is ambiguous for complex ν. A short clarification of the branch (e.g. ν defined by the principal square root and the implied combination of signs) would prevent the reader from misreading the pole locations for imaginary ν.
  3. [Sec. 2, Eq. (2.2)] The statement that the single-particle spectrum is independent of mass is terse. Near the horizon g00 diverges, but the argument as written does not cover all modes; a more precise statement of the mode spectrum would be useful.
  4. [Sec. 2.2, Eq. (2.22)] The Barnes integral (2.22) should have its convergence conditions stated. As written, the replacement of the sum over n by an integral over k is not fully justified.
  5. [Sec. 3] The phrase 'tOmperature' is used without definition. A parenthetical definition, e.g. the effective temperature at infinite temperature, would help the reader.

Circularity Check

0 steps flagged

No circularity: the late-time asymptotics follow from residue analysis of exact prior two-point functions; no fitted parameter is renamed as a prediction.

full rationale

The claimed result (1.15)/(3.1) is a theorem about the pole structure of the thermal two-point functions in the static de Sitter patch and planar BTZ. The inputs are the exact mode-sum expressions (2.9) and (2.21), taken from [12,14,24,27]. These exact expressions do not contain the late-time transition; the transition temperature Tc=(d−1)/(4π) for dS and Tc=Δ/(2π) for BTZ is obtained by comparing the imaginary parts of the Matsubara poles 2πi/β with those of the quasinormal-mode poles (2.11) and (2.24) and applying the residue theorem. No parameter is fitted, and the decay exponents are not imposed; they are read off from known pole locations. The product representation (2.3) is an explicitly stated assumption, not an imported ansatz: the paper says 'This is not a general expansion... but in the cases that we will consider, it is.' The self-citations to [12,14] supply exact quantum-field-theoretic mode sums and free-energy results, not the target late-time behavior, so they are independent evidence rather than circular support. The tOmperature comment merely cites [30,31] for nomenclature and is not load-bearing. The light-mass concern raised by ν=iμ (Eq. (2.8)) is a physical correctness issue about the ordering of poles for m^2<((d−1)/2)^2; it is not an instance of circular reasoning because Eq. (2.13) is derived from the same pole list and would fail (if it fails) on mathematical grounds, not because the result was assumed. The paper also honestly states in Section 3 that it lacks a general criterion and has only verified the specific examples, which further supports that the derivation is self-contained given the stated inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

No free parameters or new entities. The computation relies on mode-sum results from cited works and on a meromorphic product structure that is asserted for the two examples.

axioms (4)
  • standard math Residue theorem and gamma-function product formula (2.12) are used without proof.
    Invoked in Section 2.1 and 2.2 to convert the frequency integrals (2.10) and (2.23) into sums of residues at poles (2.11) and the BTZ analog; standard background.
  • domain assumption The exact single-particle modes form a complete set and the thermal state is the Planckian density matrix over these modes at arbitrary inverse temperature β.
    The two-point functions (2.9) and (2.21) are taken from refs [14,24,27] and are the entire input of the computation; the authors note that for β ≠ 2π/κ the state has strong back-reaction and may be unstable below the Hawking temperature (ref [17]).
  • ad hoc to paper The integrand of the thermal two-point function can be brought to the product form (2.3) with only quasinormal and Matsubara poles.
    The authors state this is not a general expansion and holds only in the cases considered; the residue-based late-time asymptotics and the claim of a critical-temperature transition depend directly on this meromorphic structure.
  • domain assumption In the planar BTZ limit the sum over the angular quantum number n can be replaced by an integral.
    Section 2.2, before Eq. (2.22); the planar limit decompactifies the angular direction and is standard for the planar BTZ black hole.

reviewed 2026-08-05 · how reviews work

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Cite this review

Pith. "Pith review of On the late time behavior of thermal two point function in curved space-time." pith.science (2026). https://pith.science/paper/MFWK4YTR

@misc{pith2026250902181,
  author       = {Pith},
  title        = {Pith review of: On the late time behavior of thermal two point function in curved space-time},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MFWK4YTR}},
  note         = {Machine review of arXiv:2509.02181}
}
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read the original abstract

We investigate the late-time behavior of the thermal two-point function for a Bose gas in a curved space-time with a Killing horizon. We demonstrate that the late-time behavior undergoes a sharp transition at a critical temperature, reminiscent of a phase transition. Contrary to expectations for typical gases, the relaxation time does not vanish at high temperatures but instead saturates to a constant value that depends on the imaginary part of the lowest quasinormal modes.

discussion (0)

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Reference graph

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This paper was first reviewed by deepseek-v4-flash on August 5, 2026.