{"id":"6c4a9e85-ee04-4929-a024-a65cefe60e0c","arxiv_id":"1908.02368","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Closed-form analogue Hawking temperatures are derived for power-law velocity profiles in dispersive sonic and optical media, with numerical verification.","lead":"This paper derives formulas for the temperature of fake 'Hawking' radiation emitted when waves travel through a medium whose speed changes like an analogue black hole. The formulas cover several realistic velocity profiles and are checked against numerical integration.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The optical master formula Eq. (23) uses the wrong energy variable: Table I and the paper's own Eq. (45) require ω′, not ω, in the numerator, so Tables III/V are not established as written.","rationale":"The paper's strongest claim is that Eqs. (22)-(23) yield the closed-form spectra in Tables II-V. The sonic half (Eq. 22) has external support from Ref. [19], and the Appendix residue derivations for n=1,2,3 are self-consistent, so the sonic tables are plausible. The numerical integration, however, only verifies the contour integrals against themselves; it does not validate Eq. (23) as the physical Hawking temperature. The load-bearing weakness is precisely the optical master formula. The reader located this as an unproven topological equivalence; I agree that is a gap, but there is a sharper internal inconsistency: Table I and Eq. (45) require the optical energy to be ω′, while Eq. (23) writes ℏω. Since ω′/ω = 1 − n(ω)/n_{g0} ≠ 1 generally, the optical spectra in Tables III/V would change unless the paper implicitly redefines temperature, which it never states. This is not a matter of external consensus or parameter fitting; it is an internal consistency check that can be settled by recomputing one case. Because the issue is correctable but blocks the optical claim as written, the conditional verdict stands unchanged.","tokens_in":26137,"tokens_out":28940,"duration_ms":300992,"concrete_test":"Re-evaluate the optical linear profile δn(τ)=δn0 τ/a using Eq. (23) as written and with the numerator changed to ℏω′, computing ∮τ(ω)dω for the same contour. Check which version reproduces the standard linear-profile Hawking temperature Eq. (47), T = ℏ/(2π k_B) (1/δn) dδn/dτ|_h. If the as-written Eq. (23) yields T/T0 = ω/ω′, while the ω′ version yields unity, then Eq. (23) must be corrected and Tables III/V recomputed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section III presents Eq. (23), k_B T = iℏω ∮_Γ τ(ω)dω, as the optical analogue of the sonic formula (22), obtained by 'following the transformations in Section II D'. But Table I maps the sonic energy ω to the optical comoving frequency ω′, and the wavenumber k to −ω. Consistency therefore requires ℏω′ (not ℏω) in the numerator of the optical formula. The paper itself confirms this in Section IV C: the optical second-order expression, Eq. (45), is T = ℏω′/(4π k_B)[...]^{-1}. If Eq. (23) is taken literally, every optical closed-form spectrum in Tables III and V and the optical panels of Fig. 5 is multiplied by the frequency-dependent factor ω/ω′ = [1 − n(ω)/n_{g0}]^{-1} relative to the comoving-frame temperature; for the linear τ profile the result would no longer be T = T0. No statement in the paper defines a separate 'lab-frame effective temperature' or carries this conversion, so the optical half of the central claim is not supported as written. The additional assertion that the complex-plane topology of τ(ω) matches that of z(k) is only illustrated in Fig. 4, not proven; this is a second, independent gap in the same formula.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript extends the Leonhardt-Robertson complex-contour formula for the Hawking temperature in dispersive sonic analogs to optical analogs, and uses Cauchy's residue theorem to evaluate the contour integrals for power-law velocity profiles z^{±1/n} and τ^{±1/n} (n = 1, 2, 3, and numerically for 1/4 and 1/5). It also derives an infinite-series expression for a second-order profile u = α1 z + α2 z^2, studies its convergence, and applies it to tanh and sech^2 profiles. The authors report agreement between the analytic spectra and numerical contour integration to about 10^-15.","tokens_in":26426,"tokens_out":5232,"duration_ms":55004,"significance":"If the optical results are correct, the paper supplies closed-form effective Hawking temperatures for several experimentally relevant sonic and optical velocity profiles and a systematic second-order correction beyond the linear-profile approximation; the sonic closed forms and the high-precision numerical cross-checks are useful. Strengths include explicit residue calculations in the appendix, parameter-free expressions with no fitting to data, and a clearly stated convergence criterion for the series. However, the central optical master equation contains an energy-variable error that affects the optical tables and figures as written.","major_comments":[{"comment":"The optical master formula uses the wrong energy variable. Table I maps the sonic pair (k, ω) to (−ω, ω′), with ω′ defined in Eq. (15); applying that mapping to Eq. (22) gives k_B T = iℏω′ / ∮_Γ τ(ω)dω, not iℏω / ∮_Γ τ(ω)dω. The paper itself uses ℏω′ in the second-order optical result, Eq. (45). Taken literally, Eq. (23) multiplies every optical closed-form spectrum in Tables III and V and the optical panels of Fig. 5 by the frequency-dependent factor ω/ω′ = [1 − n(ω)/n_g0]^{-1}; for the linear τ profile the table entry T/T0 = 1 would no longer hold. No separate lab-frame effective temperature is defined. Because the numerical integrations in Sec. V evaluate the same Eq. (23), they verify the algebra of the contour integral but not the physical optical spectrum. This is the load-bearing formula for the optical half of the paper.","section":"Sec. III, Eq. (23) and Table I"},{"comment":"The claim that the complex-plane topology of τ(ω) is the same as that of z(k) is only illustrated, not proven. The optical dispersion β(ω) = (ω/c)√(1 + ω^2/ω0^2) and the Kerr-induced δn(τ) define τ(ω) through a different analytic function than the sonic z(k); branch cuts or additional poles of τ(ω) inside the chosen contour would change the residue sum in Eq. (23). The authors should either provide an explicit analytic-continuation argument (or proof that the relevant Riemann surfaces are homeomorphic away from branch cuts) or state the topological equivalence as an assumption and verify the optical table entries against an independent wave-equation simulation.","section":"Sec. III, Eq. (23) and Fig. 4"}],"minor_comments":[{"comment":"The summation index is inconsistent: the series is over m, but the derivative is written as ∂^{j−1}_ω; the index j is undefined and should be m.","section":"Sec. IV C, Eq. (45)"},{"comment":"The first term in the square brackets, k∞^2 v_g^2(k∞)/(2 v_g(k∞)), is not dimensionless; it should presumably be k∞ v'_g(k∞)/(2 v_g(k∞)) to match the structure of the corresponding Table IV entry.","section":"Appendix A.6, Eq. (A.28)"},{"comment":"The statement that 'the real part of the integral in Eqs. 22 and 23 is zero' should be justified: for a general contour it is not automatic, and the authors should state the contour properties that guarantee it.","section":"Sec. V, paragraph after Eq. (23)"}],"recommendation":"major_revision","confidential_remarks":"The sonic part appears sound and the numerical checks are precise, but the optical master formula Eq. (23) appears to use the wrong energy variable (ω instead of ω′), which propagates into Tables III and V and Fig. 5. I would ask the authors to correct Eq. (23) and regenerate all optical results, and to add an explicit argument for the claimed topological equivalence, before acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Have you looked at this? It extends Leonhardt and Robertson's contour-integral method to power-law velocity profiles and adds a second-order series for quadratic profiles. The sonic side is careful: the analytic evaluations are internally consistent and the numerical contour integrals match to ~1e-15. The second-order series for u = alpha1 z + alpha2 z^2, and its application to tanh and sech^2 profiles, is new and practically useful.\n\nThe optical side has a real problem. Eq. (23) writes k_B T = i hbar omega / contour tau(omega) domega, but the mapping in Table I and the paper's own Eq. (45) both say the energy variable in this formula should be the comoving frequency omega', not the lab frequency omega. If Eq. (23) is taken literally, every optical closed-form result in Tables III and V picks up a frequency-dependent factor omega/omega', and the linear-tau case would no longer be thermal. The authors almost certainly used the right variable in their actual derivations - the table entries look like the correct results - but as written the manuscript is internally inconsistent. That needs a correction, not a rethink.\n\nTwo smaller issues. First, the claim that the complex-plane topology of tau(omega) matches that of z(k) is asserted from Fig. 4 rather than argued; given that the optical formula is being imported from the sonic derivation, a few sentences on why the branch cuts behave would close the gap. Second, the numerical checks integrate the same master formula, so they verify the algebra but not the physical formula; an independent wave-equation simulation would be stronger, but that is a minor complaint in this literature. No code or data is deposited, so the 1e-15 claim is not independently checkable.\n\nThe citation pattern looks honest, and I see no parameter fitting or circular logic. The paper is incremental - it doesn't change the framework - but it gives experimenters closed forms that aren't in the cited papers, which is real value.\n\nI'd send it to peer review. The reviewer should ask for the optical formula to be fixed and the omega/omega' issue resolved; then it's publishable. For my own work, I wouldn't cite the optical results until that's cleaned up.","headline":"Sonic results are solid and the second-order series is genuinely new; the optical master formula has a frequency mismatch that looks like a typo but needs fixing before the optical tables can be trusted.","tokens_in":26989,"tokens_out":5811,"would_cite":false,"duration_ms":60143,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper derives closed-form Hawking temperatures for dispersive analog black holes, both sonic and optical.","keywords":["analog gravity","Hawking radiation","Hawking temperature","dispersive media","optical fibers","Bose-Einstein condensates","Kerr effect","contour integration"],"falsifier":"Numerically integrate Eq. (23) along a contour that encloses the origin but crosses the assumed branch cuts of $\\tau(\\omega)$ for the optical dispersion $\\beta(\\omega)=\\frac{\\omega}{c}\\sqrt{1+\\omega^2/\\omega_0^2}$; any discontinuous change in the result would falsify the topological-equivalence assumption. As a second check, compare the closed-form Tables III and V with direct numerical integration of Eq. (23) for a growing optical profile such as $\\tau^{1/5}$, the case where the paper reports unexpectedly near-thermal behavior.","tokens_in":25916,"feed_emoji":"🕳️","tokens_out":10770,"duration_ms":100576,"temperature":0.7,"pith_summary":"The paper aims to make the Hawking temperature of analog black holes analytically computable when the medium is dispersive, meaning that wave speed depends on frequency, for both sonic and optical systems. It claims that two contour integrals, one in wavenumber for sound and one in frequency for light, give the effective temperature, and that for power-law velocity profiles these integrals are exactly solvable by Cauchy's residue theorem. The resulting closed-form spectra, given for $z^{\\pm 1}$, $z^{\\pm 1/2}$, $z^{\\pm 1/3}$ and their optical counterparts, are thermal at low frequencies but generally drift from thermality as dispersion grows. Because real experiments in Bose-Einstein condensates and optical fibers cannot ignore dispersion, closed formulas are what allow quantitative predictions of analog Hawking radiation. The paper also gives a second-order approximation, as an infinite series with a stated convergence condition, for profiles such as $\\tanh$ and $\\operatorname{sech}^2$.","feed_headline":"Closed-form Hawking spectra found for analog black holes","feed_subtitle":"Power-law velocity profiles in sonic and optical systems now have analytic temperatures, matching numerics to 10^-15.","key_machinery":"The load-bearing object is the complex-plane map from the conjugate variable to spacetime position: $z(k)$ for sound and $\\tau(\\omega)$ for light. Around the horizon the Hawking effect connects positive- and negative-norm modes through the poles and branch structure of this map, so the effective temperature is the residue of $1/z(k)$ (or $1/\\tau(\\omega)$) enclosed by the contour $\\Gamma$. The paper evaluates these residues for power-law profiles and normalizes the result by the surface gravity $\\alpha$ at the phase horizon, writing $T/T_0$ in terms of the dispersion relations $c(k)=c_0\\sqrt{1-k^2/k_0^2}$ and $\\beta(\\omega)=(\\omega/c)\\sqrt{1+\\omega^2/\\omega_0^2}$. The optical formula rests on the claim, made in Section III and Fig. 4, that the complex-plane topology of $\\tau(\\omega)$ is the same as that of $z(k)$.","core_discovery":"On the paper's own terms, the central discovery is that the effective Hawking temperature in a dispersive analog spacetime is fixed by a Cauchy contour integral of the inverse velocity profile: $k_B T = i\\hbar\\omega / \\oint_\\Gamma z(k)\\,dk$ in the sonic case and $k_B T = i\\hbar\\omega / \\oint_\\Gamma \\tau(\\omega)\\,d\\omega$ in the optical case. For profiles $u(z)\\propto z^{\\pm 1/n}$ and $\\delta n(\\tau)\\propto \\tau^{\\pm 1/n}$, these integrals evaluate in closed form, expressing the spectrum $T/T_0$ through the phase and group velocities of the medium (Tables II–V). The closed forms match direct numerical integration to about $10^{-15}$. A second-order profile $\\alpha_1 z + \\alpha_2 z^2$ leads to an infinite-series temperature whose first 60 terms obey a stated convergence condition and match numerics inside the convergence region. A direct consequence is that the Schwarzschild-like profile $z^{-1/2}$ is not thermal once dispersion is included.","pith_inferences":["Beyond the paper's examples, the closed-form spectra could serve as fitting templates: a measured frequency-dependent Hawking temperature in a fiber or condensate would allow the dispersion scale $k_0$ or $\\omega_0$ to be extracted directly.","If the claimed topological equivalence between the sonic and optical integrals is more than formal, the same residue method should transfer to other analog platforms, such as water waves, polaritons, or superconducting circuits, once their dispersion relations are cast in the same comoving form.","A testable extension is to replace power-law profiles with experimentally realistic pulse shapes such as Gaussians or super-Gaussians; the contour-integral method and the numerical routine in Section V can be applied unchanged to predict whether their spectra stay close to thermal.","The paper's remark that a Planck-scale dispersive spacetime would make Schwarzschild Hawking radiation nonthermal is an implication worth testing directly by evaluating the same contour integral with a Planckian dispersion relation."],"forward_implications":["For the linear profile $u\\propto z$, the spectrum is exactly thermal at the standard Hawking temperature even with dispersion; every other studied profile deviates from thermality as the wavenumber grows.","The Schwarzschild-like decaying profile $z^{-1/2}$ produces a spectrum that starts at $T_0$ and then decreases, so exact thermality is not expected from a Schwarzschild-type flow in a dispersive analog system.","Optical-fiber analogs with Kerr perturbations can use the optical formulas directly, since the temperature is expressed through the relative phase and group velocities $v_{pr}(\\omega)$ and $v_{gr}(\\omega)$ without solving the wave equation numerically.","For tanh and sech$^2$ profiles, a second-order expansion yields a convergent infinite series for the temperature; the first 60 terms match numerics wherever the stated convergence condition holds.","The approach to thermality with growing profiles is not monotonic in the exponent: $z^{1/5}$ is closer to thermal than lower powers, a trend the paper verifies both analytically and numerically."],"supporting_citations":[{"why":"Provides the analytical theory and the sonic contour integral (22) that this work extends.","marker":"[19]"},{"why":"Supports the topological connection between supersonic and subluminal dispersions used for the optical case.","marker":"[20]"},{"why":"Establishes the optical-fiber analog experiment whose comoving Kerr-perturbation setup the optical model follows.","marker":"[11]"},{"why":"Supplies the minimal optical dispersion model used for the horizon physics.","marker":"[12]"},{"why":"Gives the Hamilton-equation dynamics and Bogoliubov dispersion used to set up the velocity-profile problem.","marker":"[24]"},{"why":"Showed that the Schwarzschild-like profile is nonthermal under dispersion, which Tables IV–V reproduce.","marker":"[34]"},{"why":"Provides the dispersive wave-equation treatment underlying the modified dispersion relation.","marker":"[18]"}],"fun_headline_variants":["Exact Hawking temperatures for power-law analog black holes","Contour integrals yield exact analog Hawking spectra","Dispersive analog Hawking temperature solved to 10^-15","Schwarzschild-like analog black holes are not thermal","Analytic Hawking spectra for sonic and optical analogs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The optical calculation presumes that the singularities of $\\tau(\\omega)$—its poles and branch cuts—mirror those of $z(k)$ inside the integration contour; if that mirroring fails for the optical dispersion or for a particular Kerr profile, the optical Hawking temperature formula would need correction.","fun_headline_variants_meta":{"raw":{"variants":["Exact Hawking temperatures for power-law analog black holes","Contour integrals yield exact analog Hawking spectra","Dispersive analog Hawking temperature solved to 10^-15","Schwarzschild-like analog black holes are not thermal","Analytic Hawking spectra for sonic and optical analogs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000355,"raw_usage":{"total_tokens":1881,"prompt_tokens":847,"completion_tokens":1034,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":463,"completion_tokens_details":{"reasoning_tokens":955}},"tokens_in":463,"tokens_out":1034,"duration_ms":10326,"temperature":1.0,"reasoning_tokens":955,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:47:50.291005+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Numerically integrate Eq. (23) along a contour that encloses the origin but crosses the assumed branch cuts of $\\tau(\\omega)$ for the optical dispersion $\\beta(\\omega)=\\frac{\\omega}{c}\\sqrt{1+\\omega^2/\\omega_0^2}$; any discontinuous change in the result would falsify the topological-equivalence assumption. As a second check, compare the closed-form Tables III and V with direct numerical integration of Eq. (23) for a growing optical profile such as $\\tau^{1/5}$, the case where the paper reports unexpectedly near-thermal behavior.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the analytical theory and the sonic contour integral (22) that this work extends."},{"cited_title":"Drori, Y","cited_arxiv_id":null,"evidence_quote":"Supplies the minimal optical dispersion model used for the horizon physics."},{"cited_title":"Corley, Physical Review D 57, 6280 (1998)","cited_arxiv_id":null,"evidence_quote":"Gives the Hamilton-equation dynamics and Bogoliubov dispersion used to set up the velocity-profile problem."},{"cited_title":"Bermudez, in Journal of Physics Conference Series (2016) p","cited_arxiv_id":null,"evidence_quote":"Showed that the Schwarzschild-like profile is nonthermal under dispersion, which Tables IV–V reproduce."},{"cited_title":"Bermudez and U","cited_arxiv_id":null,"evidence_quote":"Provides the dispersive wave-equation treatment underlying the modified dispersion relation."}],"review_version":1}