{"id":"c9a17c13-8f6e-4227-a32b-a32131df6ed4","arxiv_id":"2505.24070","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"Dark matter annihilation inside Earth would melt a substantial fraction of the inner core for cross sections previously allowed by surface heat-flow limits.","lead":"A new simulation of dark matter annihilation inside Earth shows that, for dark matter models that earlier heat-flow measurements did not exclude, the inner core would heat up enough to melt a large part of it. The result makes Earth's solid core a more sensitive dark matter probe and highlights that exoplanet searches and planet-formation constraints must account for how slowly heat escapes from a planet's center.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The paper does not quantify the thermalization timescale behind the isothermal DM profile of Eq. (1); if it were comparable to Earth's age, the heat would spread and the melt limit would weaken. A quick estimate suggests it is short, but the calculation is missing.","rationale":"The reader's weakest_assumption correctly identifies the unverified thermalization timescale as the most load-bearing step. The entire argument that a small total power of 3.4 TW melts the inner core depends on the DM density being concentrated within a scale radius rchi of about 150 km for TeV masses. If captured DM instead populated a volume comparable to the whole Earth, the steady-state central temperature rise at r=400 km would drop by roughly the ratio of effective volumes, and the limiting power would become tens of TW or more, erasing the claimed factor of 6-13 improvement. The paper does not provide the needed timescale calculation, so the reader's CONDITIONAL verdict is justified. At the same time, an order-of-magnitude estimate of the collision rate in the core, using n ~ 10^29 m^-3, v ~ 10 km/s, and sigma ~ 10^-37 cm^2, gives a scattering time of order a year and, even for the worst-case heavy-mass energy-loss fraction, a thermalization time below 10^5 yr, far shorter than Earth's age. Thus the concern is a missing verification rather than a demonstrated failure; the proposed check using the authors' public code should settle it. Secondary caveats, such as the capture-annihilation equilibrium assumption and the fact that the parameter space is already excluded by direct detection, are acknowledged in the paper and do not change the verdict. The heat equation model, the uncertainty scans over k and T0, and the released code provide good independent support for the framework; the conditional verdict remains appropriate pending the thermalization calculation.","tokens_in":11301,"tokens_out":34833,"duration_ms":362452,"concrete_test":"Use the authors' public DarkInferno and Asteria packages to compute the thermalization timescale for representative points on the Fig. 6 limit curve, e.g., mchi = 10 GeV, 1 TeV, and 10^5 GeV at the corresponding sigma_chiN limits. Sample captured orbits and integrate the orbit-averaged energy loss dE/dt = <n sigma v DeltaE> until the particle kinetic energy falls to (3/2) k_B T_c. If the 90% thermalization time is below 1 Gyr for all points, the isothermal profile is justified and the melt limit stands; if not, recompute the melt limit using the time-dependent non-thermal annihilation profile and reassess the factor-of-6-13 claim.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step lies in Sec. II.C: Eq. (1) assumes captured DM has thermalized to the isothermal Gaussian profile with scale radius rchi (Eq. 2), concentrating essentially all annihilation heat within the innermost few hundred km for mchi larger than about 100 GeV. The paper states that 'given enough time' DM thermalizes, but it neither computes nor cites a thermalization timescale. If thermalization took longer than the age of the Earth, captured DM would remain on large extended orbits, the annihilation profile would be much broader than Eq. (1), and the central temperature at r=400 km would rise far less for a given total power injection. In that case the 3.4 TW limiting power, and the claimed factor of 6-13 improvement over previous heat-flow limits, would be an underestimate of the required power, weakening the derived sigma_chiN limits. A rough collision-rate estimate for mchi=1 TeV and sigma_chiN=10^-36.9 cm^2 gives an orbit-averaged scattering timescale of order a year and an energy-loss timescale of order 10^2 to 10^4 yr, so the assumption is plausible; nevertheless, the paper's central quantitative claim rests on this unverified step.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies the thermal effect of dark matter annihilation inside the Earth's inner core. The authors model captured dark matter with an isothermal Gaussian density profile and solve the spherically symmetric heat equation in the inner core with a fixed temperature at the inner-core boundary. After checking that the core comes to thermal equilibrium on a timescale of about a gigayear, they use the steady-state temperature profile to ask whether dark matter heating would melt the innermost 400 km of the inner core. For TeV-scale dark matter they find that annihilation power of about 3.4 TW is sufficient, which is a factor of 6-13 smaller than previous Earth heat-flow limits; this is converted into limits on the spin-independent dark matter-proton cross section. They also discuss implications for exoplanet heating and provide a public code, DarkInferno.","tokens_in":11565,"tokens_out":17524,"duration_ms":181947,"significance":"If correct, the paper identifies a new and substantially more sensitive observable for dark matter annihilation in terrestrial planets, improving on existing Earth heat-flow limits by roughly an order of magnitude. The numerical setup is clearly specified, the uncertainty scan over thermal conductivity and initial temperature gives only a factor of about 1.6 variation in the limit, and the public code makes the results reproducible. The main caveat is that the isothermal dark matter profile that concentrates all heating in the innermost core is assumed without a quantitative thermalization timescale; this is the step on which the quantitative limit most directly depends.","major_comments":[{"comment":"The isothermal Gaussian profile in Eq. (1) concentrates essentially all annihilation heat within the innermost few hundred km for mχ ≳ 100 GeV, and the derived 3.4 TW limit depends on this concentration. The paper states that 'given enough time' captured dark matter thermalizes, but it does not compute or cite a thermalization timescale. For the benchmark mχ = 1 TeV, σχN = 10^-36.9 cm2, an order-of-magnitude estimate from the local scattering rate gives a collision timescale of about a year and an energy-loss timescale of order 10^2-10^4 years, so the assumption is plausible; however, this estimate should appear in the text, together with a statement of how it scales over the mass and cross-section range of the limit. Without it, the central claim rests on an unverified step.","section":"Sec. II.C, Eq. (1)"},{"comment":"The steady-state limit is justified by showing that equilibrium is reached after about 1 Gyr for the single benchmark of mχ = 1 TeV (Fig. 4). Because the equilibration time depends on the spatial distribution of the heat source and on the thermal conductivity, the paper should demonstrate that the time-independent solution is valid across the mass range and k range used in Figs. 5-6, or explain why the 1 TeV case is the slowest one. If equilibrium has not been reached for some part of the parameter space, the derived limits would need to be weakened.","section":"Sec. II.E and Fig. 4"}],"minor_comments":[{"comment":"The dependent variable u in Eq. (3) is never defined; it should be identified as the temperature (or the thermal energy density) entering the subsequent plots.","section":"Sec. II.C, Eq. (3)"},{"comment":"The melt criterion uses a pure-conduction model and does not include the latent heat of fusion or the possibility that a molten inner core would convect. The chosen 10^4 K threshold is conservative relative to reported iron melting points, but the authors should state explicitly why these simplifications do not break the limit (e.g., a rough latent-heat energy budget) and whether a phase-boundary treatment would strengthen or weaken the constraint.","section":"Sec. II.D"},{"comment":"The text refers to seismic 'porosity' measurements, whereas Ref. [51] constrains small-scale heterogeneity from scattered waves; the wording should be aligned with the cited measurement.","section":"Sec. II.D"},{"comment":"Since the conclusion notes that the constrained parameter space is already ruled out by direct detection, showing a representative direct-detection exclusion curve in Fig. 6 would make the comparison transparent for the reader.","section":"Sec. III.A / Fig. 6"},{"comment":"The limit assumes Γcap = Q̇χ, which requires a sufficiently large annihilation cross section; the paper should state the assumed annihilation cross section (or note that the limit applies only to models in which equilibrium is reached) so that the σχN–mχ exclusion is not misinterpreted as applying to models with arbitrarily small ⟨σv⟩.","section":"Sec. II.C"}],"recommendation":"major_revision","confidential_remarks":"The paper is well written and the code release is a clear strength. The missing thermalization timescale is the main technical gap; once the authors add a short estimate and confirm the steady-state assumption over the parameter range, I would be satisfied. The phase-change and convection simplifications are likely conservative and should be discussed rather than fully modeled. I do not see grounds for rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The real advance here is the shift from instantaneous surface heat-flow limits to a time-dependent treatment of the inner core. The authors show that prior limits based on 20-44 TW surface flow are too weak by about an order of magnitude: if dark matter deposits its heat deep inside, the inner core melts long before the surface notices. That is a clean, physical insight, and the paper supports it with a transparent finite-difference solution, a released code, and a robustness scan over k and T0 that shifts the limit by only about 1.6. The seismic comparison is also conservative: they allow melting out to 400 km, where data lose sensitivity. Credit where due: this is a solid application of standard heat transport, not a contrived fit.\n\nThe soft spots are real but not showstoppers. The load-bearing assumption is that captured dark matter thermalizes into the compact isothermal profile of Eq. (1). The paper neither computes nor cites a thermalization timescale. The stress-test estimate suggests it is short (years to 10^4 yr), but that estimate is not in the paper, and the melting limit depends directly on the heat being concentrated in the innermost 400 km. If thermalization were slower, the profile would broaden and the 3.4 TW limit would weaken. This needs to be addressed, ideally with an explicit timescale calculation or a parameterized broader profile. The other gaps, such as ignoring latent heat, holding the inner-core boundary at T0, and neglecting phase-change dynamics, are likely order-unity effects and the authors are upfront about them; the melt threshold itself is chosen conservatively.\n\nThe exoplanet angle is the most interesting part. The finding that the core takes around 1 Gyr to reach thermal equilibrium is directly relevant to young exoplanet constraints and to using planet formation to set dark matter limits. The authors acknowledge that the Earth parameter space is already closed by direct detection, which dampens the significance for actual dark matter, but the observable is novel and the code makes it easy to apply elsewhere.\n\nThis is a well-built paper with one genuine hole and several conservative approximations. It deserves a serious referee. My recommendation: send it to peer review, with a request that the authors add an estimate or citation for the thermalization timescale and show how the limit shifts if the profile is broadened. Conditional acceptance after such a revision is reasonable.","headline":"Takes the heat equation seriously instead of assuming instantaneous surface equilibrium, and the inner-core melting probe is genuinely more sensitive; main gap is an unquantified thermalization timescale.","tokens_in":699,"tokens_out":2548,"would_cite":true,"duration_ms":35445,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper argues that if dark matter annihilates inside Earth, it would melt a large part of the inner core, giving a new way to constrain dark matter scattering.","keywords":["dark matter capture","Earth's inner core","annihilation heating","heat flow limits","seismic constraints","exoplanet dark matter","spin-independent scattering","thermal conduction"],"falsifier":"Compute the dark-matter thermalization time for $\\sigma_{\\chi N}\\sim10^{-37}\\,\\mathrm{cm}^2$ and $m_\\chi\\sim1$ TeV; if it is longer than Earth's age, or if seismic data ever show solid material within 400 km of the center for parameters above the quoted limit, the central claim would be contradicted.","tokens_in":11084,"feed_emoji":"🔥","tokens_out":5986,"duration_ms":58173,"temperature":0.7,"pith_summary":"Earlier work compared dark-matter heat with Earth's total heat flow at the surface; this paper instead asks what that heat does to the interior, solving the spherically symmetric heat equation with dark matter energy deposition concentrated near the center. It finds that for TeV-mass dark matter with spin-independent cross sections near $10^{-37}\\,\\mathrm{cm}^2$, annihilation would push the central core above $10^4$ K and melt the inner 400 km, in conflict with seismic observations that the inner core is solid. The new limiting heat injection is 3.4 TW, six to thirteen times smaller than the 20--44 TW used in earlier work, so the constraint on dark matter-proton scattering is correspondingly stronger. If correct, this turns the state of Earth's core into a dark matter observatory and shows that planetary interior structure must be included in dark matter heating arguments.","feed_headline":"Dark matter annihilation could melt Earth's inner core","feed_subtitle":"A melt-based test tightens dark matter scattering limits by 6–13 times over earlier heat-flow bounds.","key_machinery":"The argument runs on the isothermal dark matter density profile $n_\\chi(r) \\propto e^{-r^2/r_\\chi^2}$ with scale radius $r_\\chi = \\sqrt{3k_BT_c/(2\\pi G\\rho_c m_\\chi)}$, which concentrates heavy dark matter deep in the core, and on the spherical heat equation $\\frac{1}{\\alpha}\\frac{\\partial u}{\\partial t} = \\frac{\\partial^2 u}{\\partial r^2} + \\frac{2}{r}\\frac{\\partial u}{\\partial r} + \\frac{1}{k}\\dot q$ with volumetric heating $\\dot q \\propto n_\\chi^2$. Heat deposition is normalized by the capture rate, and the temperature profile is evolved until it reaches steady state; varying the uncertain conductivity (20--224 W/m/K) and initial core temperature (4000--7000 K) shifts the resulting cross-section limit by less than about 30%.","core_discovery":"The paper's central claim is that the existence of Earth's solid inner core is a dark matter observatory. For masses above roughly 100 GeV, essentially all annihilation heat is dumped within the innermost 400 km, and the steady-state temperature there exceeds the conservative $10^4$ K melting threshold before the total heat flow reaches the levels allowed by previous bounds. Therefore any model with capture--annihilation equilibrium and a spin-independent cross section at the level of a few times $10^{-37}\\,\\mathrm{cm}^2$ for TeV-scale masses would melt a substantial fraction of the inner core, which seismic data rule out. The paper also shows that the time for this heat to conduct outward is about $10^9$ years, comparable to the age of the core, so thermal equilibrium assumptions must be checked when applying the argument to younger planets.","pith_inferences":["If the inner core's melting point is closer to the reported $7600\\pm500$ K baseline than to the paper's conservative $10^4$ K threshold, the true exclusion region would extend to somewhat lower cross sections than plotted.","A decisive extension would be a dedicated calculation of the dark-matter thermalization time in Earth for $\\sigma\\sim10^{-37}\\,\\mathrm{cm}^2$; if that time exceeds Earth's age, the melt region would be smaller and the limit weaker.","Applying the melt criterion to planets with independent age and composition measurements, especially dense rocky exoplanets near the Galactic Center, could convert a single-planet bound into a population-level search."],"forward_implications":["The spin-independent dark-matter-proton cross-section limit for TeV-scale dark matter improves by a factor of 6--13 relative to previous Earth heat-flow limits.","Any planet-based dark matter search must account for internal heat transport: heat deposited in a core takes of order $10^9$ years to reach the outer core, so young planets may not yet show the steady-state signal.","The constraint applies to dark matter that annihilates locally into photons or charged particles; models with long-lived mediators that escape the planet evade this particular melt test.","The same calculation, applied to exoplanets near the Galactic Center, predicts melted cores and could serve as a new observable for dark matter capture in planetary populations."],"supporting_citations":[{"why":"Supplies the seismic observations of inner-core heterogeneity down to about 400 km from the center that define the solid region the limit protects.","marker":"[51]"},{"why":"Sets the 20 TW Earth heat-flow limit that this paper's melt-based bound is compared against and improves upon.","marker":"[41]"},{"why":"Provides a more recent 44 TW Earth heat-flow limit and the equilibrium capture--annihilation framework.","marker":"[43]"},{"why":"Combined with [43] it gives the 44 TW comparison line and a capture-rate calculation that differs from the one used here.","marker":"[61]"},{"why":"Supplies the capture-rate computation and Earth parameters that normalize the dark matter heat source.","marker":"[59]"},{"why":"Gives the measured thermal conductivity of iron at core conditions, the intermediate value the paper adopts.","marker":"[65]"},{"why":"Provides the melting point of iron at inner-core boundary conditions used to justify the $10^4$ K melt criterion.","marker":"[68]"},{"why":"Derives the isothermal regime and density profile shape the paper assumes for captured dark matter.","marker":"[72]"},{"why":"Supplies the Gaussian dark matter density profile formula used to compute annihilation heating.","marker":"[73]"}],"fun_headline_variants":["Earth's inner core as a dark matter thermometer","Melted core sets new dark matter limits","Dark matter heat could melt Earth's inner core","Solid core reveals dark matter's heat","Inner core melting: a dark matter test"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The limit assumes captured dark matter quickly settles into a compact Gaussian cloud centered on Earth, so all annihilation heat is deposited in the innermost core; if dark matter instead stayed on wide, slowly shrinking orbits over Earth's age, the melt region would be smaller and the bound weaker.","fun_headline_variants_meta":{"raw":{"variants":["Earth's inner core as a dark matter thermometer","Melted core sets new dark matter limits","Dark matter heat could melt Earth's inner core","Solid core reveals dark matter's heat","Inner core melting: a dark matter test"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000147,"raw_usage":{"total_tokens":1146,"prompt_tokens":868,"completion_tokens":278,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":484,"completion_tokens_details":{"reasoning_tokens":210}},"tokens_in":484,"tokens_out":278,"duration_ms":3717,"temperature":1.0,"reasoning_tokens":210,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T12:37:27.301792+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the dark-matter thermalization time for $\\sigma_{\\chi N}\\sim10^{-37}\\,\\mathrm{cm}^2$ and $m_\\chi\\sim1$ TeV; if it is longer than Earth's age, or if seismic data ever show solid material within 400 km of the center for parameters above the quoted limit, the central claim would be contradicted.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the seismic observations of inner-core heterogeneity down to about 400 km from the center that define the solid region the limit protects."},{"cited_title":"Terrestrial and Martian Heat Flow Limits on Dark Matter","cited_arxiv_id":"1909.11683","evidence_quote":"Provides a more recent 44 TW Earth heat-flow limit and the equilibrium capture--annihilation framework."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the measured thermal conductivity of iron at core conditions, the intermediate value the paper adopts."},{"cited_title":"Williams, R","cited_arxiv_id":null,"evidence_quote":"Provides the melting point of iron at inner-core boundary conditions used to justify the $10^4$ K melt criterion."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Derives the isothermal regime and density profile shape the paper assumes for captured dark matter."}],"review_version":1}