{"id":"89e6c77d-e5ca-4ee9-ac01-5dc6b798e4a2","arxiv_id":"2501.11435","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"Dark matter admixture increases the central Ricci, Kretschmann, and Ricci-tensor curvature of quarkyonic neutron stars, while quarkyonic matter stiffens the EOS and reduces curvature.","lead":"This paper computes how dark matter and quark matter inside neutron stars change spacetime curvature, using a hybrid model that combines nuclear, quark, and dark matter equations of state. The main finding is that dark matter raises central curvature while quarkyonic matter lowers it, offering a new diagnostic for exotic neutron star interiors.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central curvature trends are inherited from an unpublished quarkyonic+DM EOS; without the EOS tables or a self-consistent derivation of Eqs. (8)-(9), the claimed DM-driven increase in central curvature cannot be independently verified.","rationale":"The GR curvature post-processing in Eqs. (12)-(15) and the TOV solutions are standard, and if the input EOS is correct, the qualitative trends likely follow from the relative stiffness of the baryonic, quarkyonic, and DM-admixed EOS. However, the manuscript does not contain the EOS construction itself: it relies on the companion paper [3] for the quarkyonic transition, the Gibbs conditions, and the DM coupling, and it provides no numerical tables or code. This is a load-bearing gap because the central claim is entirely a claim about the output of those TOV solutions. I also note the manuscript's self-referential dependence on [3], which the reader correctly identified as the weakest assumption. The concern is not about the standard curvature formulas but about the unverifiability and potential thermodynamic inconsistency of the summed EOS in Eqs. (8)-(9), especially the possibility of double-counting momentum states between Eqs. (1) and (3). The proposed test is a concrete way to settle whether the DM-induced curvature increase is real or an artifact of the unpublished EOS assembly. Since the reader's conditional verdict already captures this risk, the verdict remains unchanged.","tokens_in":12559,"tokens_out":11895,"duration_ms":134797,"concrete_test":"Request from the authors the numerical EOS tables for the three compositions in Fig. 1 (baryonic; baryonic+QM; baryonic+QM+DM) at n_t=0.3 fm^-3, Lambda_cs=800 MeV, and k_DM=0.03 GeV for the G3 parameter set, then independently integrate the TOV equations for M=1.4 M_sun and M_max and evaluate Eq. (14) at the stellar center. Confirm that K_center(no QM) < K_center(QM) < K_center(QM+DM) and that the resulting EOS satisfies dP/depsilon >= 0 and cs^2 <= 1. If the ordering or thermodynamic consistency fails, the central claim depends on the unshown EOS assembly rather than on the curvature formalism.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central result is that central K, R, and J increase when DM is added and decrease when quarkyonic matter is present. But these curves are only as reliable as the EOS input, and the manuscript does not actually present that EOS. Eqs. (8)-(9) merely sum E_BM, E_QM, and E_DM, while the quarkyonic transition density n_t, the QCD confinement scale Lambda_cs, beta equilibrium, charge neutrality, and the Gibbs construction are all delegated to the companion paper [3]. The text explicitly says 'A full detailed procedure is present in our previous work [3]'. This matters because the sign of the DM effect can depend on how the quark and nucleon Fermi seas are combined. In Eqs. (1) and (3) of the present manuscript, both the nucleon and quark integrals run from k=0 to the respective Fermi momenta; if the quark contribution is simply added on top of a nucleon EOS that still fills the low-momentum states, those states are double-counted. That double-counting would change the stiffness of the EOS and thus the TOV-derived curvature ordering. Similarly, Eq. (5) couples DM to nucleons and quarks through the Higgs field h, but the mean-field value of h is not solved for or displayed, so it is unclear whether the DM contribution in Eqs. (6)-(7) is thermodynamically consistent with the baryonic and quark sectors. Without the EOS tables or a reproducible construction, the claimed monotonic ordering K_c(baryonic) < K_c(quarkyonic) < K_c(quarkyonic+DM) cannot be checked.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper computes, for the first time, the radial profiles of the Ricci scalar R, the contracted Ricci tensor J, the Kretschmann scalar K, and the Weyl scalar W for neutron stars described by a quarkyonic equation of state (EOS) with an admixture of non-annihilating WIMP dark matter. The EOS is built in a companion paper [3] by combining an E-RMF baryonic sector, a McLerran-Reddy/Zhao-Lattimer quarkyonic sector, and a Higgs-portal DM sector; the present work solves the TOV equations with that EOS and evaluates the standard curvature scalars of a static, spherically symmetric perfect-fluid star. The main reported findings are that adding DM softens the EOS, increases the central density, and thereby raises the central values of R, J, and K, while a larger quarkyonic contribution stiffens the EOS and lowers these central values; the compactness ratio and surface Kretschmann curvature also vary systematically with the DM Fermi momentum and transition density.","tokens_in":12937,"tokens_out":7315,"duration_ms":63023,"significance":"If the underlying quarkyonic+DM EOS is reliable, the paper offers a concrete and falsifiable diagnostic: the central and surface curvature scalars respond monotonically to the presence of DM and quark matter, which could in principle be used to constrain exotic-matter scenarios from future curvature-sensitive observations. The strength of the paper is that the curvature calculation itself is standard GR post-processing of TOV solutions (Eqs. 12-15), with no circular use of the target curvature observables. The significance is tempered, however, by the complete dependence of the results on the EOS of the companion paper [3], which is not reproduced or tabulated here, and by the ad hoc choice of DM parameters (k_f^DM = 0.03 and 0.04 GeV).","major_comments":[{"comment":"The text states that nucleons occupy a finite Fermi shell with a minimum Fermi momentum k_{f0} and an upper momentum k_{fn,p}, but the displayed nucleon integral in Eq. (1) still runs from k=0 to k_{fi}, while the quark integral in Eq. (3) also runs from k=0 to k_{fj}. If the total EOS is literally the sum (8)-(9), the low-momentum states are double-counted between the baryonic and quark sectors. The authors must either provide the correct shell-modified integrals actually used in the calculation, or explicitly state that Eqs. (1)-(3) are only schematic building blocks and that the full momentum-space partitioning is given in [3]. As written, the construction is not self-consistent and the curvature results cannot be reproduced from this manuscript alone.","section":"Section II.B, Eqs. (1)-(3) and (8)-(9)"},{"comment":"The mean-field value h_0 of the Higgs field enters the DM energy density (6) and pressure (7), but no field equation or extremum condition is given for h_0. Since the DM sector couples to nucleons and quarks through the same Higgs field (via the coupling f in Eq. 5), h_0 must be obtained self-consistently from the total effective potential; otherwise the thermodynamic consistency of E_DM and P_DM with the baryonic and quark sectors is not established. The authors should provide the equation for h_0 (or state the value used) and show how it is determined together with the meson fields of the E-RMF model.","section":"Section II.C, Eqs. (5)-(7)"},{"comment":"The central quantitative claim — that DM raises the central Kretschmann, Ricci-scalar, and Ricci-tensor values while quarkyonic matter lowers them — is inherited entirely from the EOS constructed in the companion paper [3], including the beta-equilibrium and Gibbs construction. This manuscript does not present the EOS tables, the quark-nucleon momentum partitioning, or the parameter values of the quarkyonic transition beyond the three free parameters n_t, Λ_cs, and k_f^DM. Because the curvature scalars are deterministic functions of E_tot, P_tot, and m(r) from the TOV solution, the reader cannot independently verify the trends without access to [3] or the EOS data. The authors should include the actual EOS tables (or a public repository link) and a summary of the Gibbs construction sufficient to reproduce the input to Eqs. (10)-(11).","section":"Section II.D and Results (Figs. 1-5)"},{"comment":"The discussion surrounding Fig. 4 states that the addition of quarkyonic matter stiffens the EOS and produces a 'drastic fall of curvature at the core', while adding DM 'further increases' the curvature, attributing this to softening of the EOS and increased central density. This sentence is internally confusing: the first half correctly relates stiffness to lower central density and lower curvature, but the second half says the fall 'further increases' with DM, which reads as a contradiction. The authors should clarify whether they mean that the curvature increase due to DM reverses the quarkyonic-induced decrease, and should state the central density ordering explicitly. This is a presentation issue, but it affects the interpretation of the main result.","section":"Section III, Fig. 4 and accompanying text"}],"minor_comments":[{"comment":"The conclusion states that the Kretschmann scalar (K(r)) and Weyl tensor (W) are 'significant as they only exist inside the neutron star', which directly contradicts Section II.F where K and W are correctly stated to be non-zero in the exterior vacuum (e.g., the Schwarzschild Kretschmann scalar is 48M^2/r^6). This should be corrected.","section":"Section IV, Conclusions"},{"comment":"The paper should state the metric signature and units (G=c=1) assumed in the curvature formulas. The expressions are standard for a signature (+,−,−,−) perfect-fluid spacetime, but this convention is not stated, which may confuse readers.","section":"Section II.F, Eqs. (12)-(15)"},{"comment":"The caption lists panels (a), (b), (c) but the figure is organized by parameter sets G3 and IOPB-I and by maximum/canonical mass; the caption is unclear about which panel corresponds to which case. Please clarify the panel layout.","section":"Section III, Fig. 1 caption"},{"comment":"The DM Fermi momenta k_f^DM = 0.03 and 0.04 GeV are introduced as ad hoc values without a physical justification or a relation to the DM abundance inside the star. A sentence motivating this range (e.g., from DM capture or self-interaction constraints) would improve the paper.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a companion application of the authors' own EOS paper [3]. The curvature formalism is standard and the qualitative trends are plausible, but the paper currently cannot stand alone: the EOS construction is not reproducible from the information given, and the displayed equations for the nucleonic and quark energy densities are inconsistent with the described shell picture. The issues are fixable within the manuscript's scope by adding the correct momentum-space integrals, the equation for the Higgs mean field, and EOS tables or a data link. I would not reject, but the revision must address the consistency of Eqs. (1)-(9)."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick read: this is a competent post-processing paper. The genuinely new piece is that nobody has plotted the Ricci scalar, Ricci tensor, Kretschmann scalar, and Weyl tensor for a quarkyonic-plus-dark-matter EOS before. The curvature formulas are standard GR identities, and the qualitative trends—DM softens the EOS, quark matter stiffens it, so central curvature tracks central density—are physically sensible. Running the same analysis with G3 and IOPB-I and getting the same ordering is a plus.\n\nThe soft spot is that the result is inherited from the companion paper. The EOS is not self-contained here. As printed, Eqs. (1) and (3) integrate both nucleons and quarks from k = 0 up to their respective Fermi momenta. If the quarkyonic EOS really just adds those two terms, the low-momentum states are double-counted, which would change the stiffness and therefore the curvature ordering. The text says nucleons occupy a finite Fermi shell, so the companion paper may handle this correctly, but this manuscript does not show it. That is the load-bearing gap.\n\nThe DM sector is under-reported too: the mean-field value of h in Eq. (5) is never solved for, and M_chi* is used without definition. The DM Fermi momenta (0.03 and 0.04 GeV) are plausible but ad hoc, and there is no uncertainty or sensitivity analysis. No EOS tables or code are provided, so an independent referee cannot check the central claim from this paper alone.\n\nMinor issues: the text says the Weyl tensor \"reached negative values\" even though W is defined as a square root and is therefore non-negative—likely a sign convention in the plot, but it is sloppy. The conclusion that quarkyonic stars are \"the most compact objects in nature\" overreaches; that is model-dependent and does not follow from these calculations.\n\nThe bottom line: this is a useful model-mapping exercise, not a new result about nature. For someone working on curvature diagnostics or exotic-matter signatures in compact stars, the figures are informative. For peer review, it deserves engagement—the calculation is straightforward and the trends are testable—but it should come back with the EOS construction made explicit, the double-counting question resolved, data or code shared, and the language tightened. Send it to review with expectations of major revision.","headline":"A competent curvature post-processing of a quarkyonic+DM EOS, but the central input is delegated to a companion paper and the printed equations raise a double-counting question that needs resolving before the trends can be trusted.","tokens_in":13504,"tokens_out":3554,"would_cite":false,"duration_ms":35956,"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":"Adding dark matter raises the central spacetime curvature of quarkyonic neutron stars.","keywords":["dark matter","neutron stars","quarkyonic matter","spacetime curvature","Kretschmann scalar","relativistic mean-field theory","Tolman-Oppenheimer-Volkoff equations","compactness"],"falsifier":"A precise, model-independent mass–radius measurement of a heavy neutron star (from a future X-ray timing or gravitational-wave event) that falls outside the mass–radius band predicted by this paper's DM-admixed quarkyonic equation of state would falsify the input equation of state and with it the curvature shift; a direct sign check is equally decisive—if an independent calculation of the same hybrid model found that adding dark matter lowers rather than raises the central density, the central claim would reverse.","tokens_in":12362,"feed_emoji":"🌌","tokens_out":15753,"duration_ms":134997,"temperature":0.7,"pith_summary":"The paper argues that even a small admixture of dark matter changes how spacetime is curved inside a neutron star built from quarkyonic matter. It combines a baryonic relativistic mean-field description, a quarkyonic phase in which a quark Fermi sea coexists with nucleons near the Fermi surface, and a dark-matter component, solves the Tolman–Oppenheimer–Volkoff equations, and then evaluates four curvature invariants—the Ricci scalar, the Ricci-tensor contraction, the Kretschmann scalar, and the Weyl tensor—as functions of radius. The central finding is that dark matter softens the equation of state and raises the central density and pressure, which pushes the central values of the Ricci scalar, the Ricci-tensor contraction, and the Kretschmann scalar upward; quarkyonic matter does the opposite by stiffening the equation of state. A reader should care because this gives a concrete, calculable signature through which curvature measurements could in principle reveal whether dark matter and deconfined quarks are present in the densest observable stars.","feed_headline":"Dark matter raises central spacetime curvature in quarkyonic stars","feed_subtitle":"In a hybrid quarkyonic model, adding dark matter softens the equation of state and intensifies core spacetime curvature.","key_machinery":"The argument runs on two pieces: the hybrid equation of state and the curvature identities that turn it into geometry. The equation of state is assembled as $E=E_{\\rm BM}+E_{\\rm QM}+E_{\\rm DM}$ and $P=P_{\\rm BM}+P_{\\rm QM}+P_{\\rm DM}$, where the quarkyonic part is a phenomenological phase in which quarks occupy low-momentum states above a transition density $n_t$ with a QCD confinement scale $\\Lambda_{\\rm cs}$, and the dark-matter part is a non-annihilating fermion with Fermi momentum $k_f^{\\rm DM}$ coupled through a scalar mediator. Solving the Tolman–Oppenheimer–Volkoff equations gives the density, pressure, and enclosed-mass profiles, which then feed the four invariants: $R(r)=8\\pi(E_{\\rm tot}(r)-3P_{\\rm tot}(r))$, $J(r)=\\sqrt{(8\\pi)^2(E_{\\rm tot}^2(r)+3P_{\\rm tot}^2(r))}$, $K(r)=\\sqrt{(8\\pi)^2(3E_{\\rm tot}^2(r)+3P_{\\rm tot}^2(r)+2P_{\\rm tot}(r)E_{\\rm tot}(r))-128E_{\\rm tot}(r)m(r)/r^{3}+48m^{2}(r)/r^{6}}$, and $W(r)=\\sqrt{\\frac{4}{3}\\left(6m(r)/r^{3}-8\\pi E_{\\rm tot}(r)\\right)^2}$. These identities carry the conclusion: any change in the central density or pressure—dark matter raising them, quarkyonic matter lowering them—shows up directly in the central values of $R$, $J$, and $K$.","core_discovery":"On the paper's own terms, the central claim is that the interior spacetime geometry of a neutron star responds oppositely to dark matter and quarkyonic matter. For a star built from the quarkyonic effective field theory with the effective relativistic mean-field model, the central Kretschmann scalar $K$, Ricci scalar $R$, and Ricci-tensor contraction $J$ all increase when dark matter is added at Fermi momenta $k_f^{\\rm DM}=0.03$–$0.04$ GeV, because the dark-matter component softens the total equation of state and lifts the central density and pressure. Changing the quarkyonic parameters—the transition density $n_t$ and the QCD confinement scale $\\Lambda_{\\rm cs}$—toward a stiffer equation of state lowers these central curvature values. The same calculation shows that dark matter reduces compactness and central pressure, while higher transition densities raise compactness and central pressure, and the trends hold for both the G3 and IOPB-I nuclear parameter sets across canonical $1.4\\,M_\\odot$ and maximum-mass stars.","pith_inferences":["The same curvature machinery could be linked to tidal deformability and moment of inertia, turning gravitational-wave and pulsar-timing data into indirect constraints on the dark-matter fraction.","Applying the calculation to anisotropic pressure or to modified theories of gravity would add explicit terms beyond $E$ and $P$ in the curvature identities, providing a way to separate exotic-matter signals from gravity-model signals.","Since $R$ and $J$ vanish outside the star while $K$ and $W$ extend into the vacuum, matching external curvature proxies such as lensing and redshift to interior predictions could constrain the core equation of state without direct access to the core."],"forward_implications":["If the central claim is right, neutron stars with more dark matter should show higher central values of the Ricci scalar, Ricci-tensor contraction, and Kretschmann scalar than purely baryonic stars of the same mass.","Quarkyonic matter, by contrast, lowers central curvature and makes the radial profile smoother, so the two exotic components pull curvature in distinguishable directions.","The effects are strongest at the maximum mass and nearly flat for canonical $1.4\\,M_\\odot$ stars, so the most massive neutron stars are the place to look for DM and quark-matter curvature signatures.","Higher transition densities make stars more compact with higher central pressure, while dark matter reduces compactness, so measurements of compactness alone can separate the two influences.","The surface ratio $K(R)/K_\\odot$ rises with dark-matter content, linking a comparatively accessible surface quantity to the presence of dark matter inside the star."],"supporting_citations":[{"why":"Supplies the hybrid baryonic–quarkyonic–dark-matter equation of state and the parameter choices ($n_t$, $\\Lambda_{\\rm cs}$, $k_f^{\\rm DM}$) used throughout.","marker":"[3]"},{"why":"Provides the beta-equilibrated, charge-neutral quarkyonic matter model that fixes the quark–nucleon Fermi-momentum relations.","marker":"[14]"},{"why":"Introduces the quarkyonic picture of matter with a quark Fermi sea beneath a nucleonic Fermi surface.","marker":"[61]"},{"why":"Gives the dark-matter Lagrangian and the mean-field energy density and pressure used to build the DM component.","marker":"[22]"},{"why":"Earlier study of dark-matter effects on neutron-star curvature that supplies the $K(R)/K_\\odot$ trend and methodology.","marker":"[29]"},{"why":"Previous general-relativistic curvature computation whose approach this paper follows for the four invariants.","marker":"[38]"},{"why":"Establishes the compactness–curvature relationship and symmetry-energy dependence motivating the analysis.","marker":"[35]"},{"why":"The Tolman–Oppenheimer–Volkoff equations that convert the equation of state into mass, radius, and radial profiles.","marker":"[67]"},{"why":"Provides the G3 nuclear parameter set that defines one of the two baryonic equations of state.","marker":"[30]"},{"why":"Provides the IOPB-I nuclear parameter set that defines the other baryonic equation of state.","marker":"[31]"}],"fun_headline_variants":["Dark matter boosts curvature in quarkyonic neutron stars","DM softens EOS, intensifies core curvature in stars","Quarkyonic model: DM raises central spacetime curvature","Dark matter shapes neutron star geometry, new model shows"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The calculation assumes that the mixed baryonic–quarkyonic–dark-matter equation of state, with its chosen transition density, confinement scale, and dark-matter coupling, faithfully represents matter inside a real neutron star; if that matter model is unrepresentative, every curvature profile inherits the error.","fun_headline_variants_meta":{"raw":{"variants":["Dark matter boosts curvature in quarkyonic neutron stars","DM softens EOS, intensifies core curvature in stars","Quarkyonic model: DM raises central spacetime curvature","Dark matter shapes neutron star geometry, new model shows"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000169,"raw_usage":{"total_tokens":1326,"prompt_tokens":1066,"completion_tokens":260,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":682,"completion_tokens_details":{"reasoning_tokens":194}},"tokens_in":682,"tokens_out":260,"duration_ms":2963,"temperature":1.0,"reasoning_tokens":194,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T18:16:22.261066+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A precise, model-independent mass–radius measurement of a heavy neutron star (from a future X-ray timing or gravitational-wave event) that falls outside the mass–radius band predicted by this paper's DM-admixed quarkyonic equation of state would falsify the input equation of state and with it the curvature shift; a direct sign check is equally decisive—if an independent calculation of the same hybrid model found that adding dark matter lowers rather than raises the central density, the central claim would reverse.","supporting_citations":[{"cited_title":"Dark Matter Influence on Quarkyonic Stars: A Relativistic Mean Field Analysis","cited_arxiv_id":"2401.02190","evidence_quote":"Supplies the hybrid baryonic–quarkyonic–dark-matter equation of state and the parameter choices ($n_t$, $\\Lambda_{\\rm cs}$, $k_f^{\\rm DM}$) used throughout."},{"cited_title":"Zhao and J","cited_arxiv_id":null,"evidence_quote":"Provides the beta-equilibrated, charge-neutral quarkyonic matter model that fixes the quark–nucleon Fermi-momentum relations."},{"cited_title":"Ciarcelluti and F","cited_arxiv_id":null,"evidence_quote":"Gives the dark-matter Lagrangian and the mean-field energy density and pressure used to build the DM component."},{"cited_title":"A similar kind of trend is observed here","cited_arxiv_id":null,"evidence_quote":"Earlier study of dark-matter effects on neutron-star curvature that supplies the $K(R)/K_\\odot$ trend and methodology."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Previous general-relativistic curvature computation whose approach this paper follows for the four invariants."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the compactness–curvature relationship and symmetry-energy dependence motivating the analysis."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the G3 nuclear parameter set that defines one of the two baryonic equations of state."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the IOPB-I nuclear parameter set that defines the other baryonic equation of state."}],"review_version":1}