{"id":"8a0a3b6c-bb7c-40d3-b0bf-188700c3c723","arxiv_id":"2606.08213","paper_version":2,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":4.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":1,"one_line_summary":"A conformal Killing extension of Rastall gravity introduces a divergence-free tensor and curvature-linear term as dark sector, restricted by CMB to trace-only coupling, producing dust density evolution ~ a^{-3/(1+tau)} with conserved total EMT.","lead":"The paper extends Harada and Rastall gravity by adding a divergence-free conformal Killing tensor and a curvature-trace term to Einstein equations, treating them as a dark sector that couples to matter via the trace of the energy-momentum tensor. In FRW cosmology this yields a modified dust density scaling and a conserved total fluid, potentially offering a geometric account of dark components.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"CMB restriction to trace-only coupling is the unverified step enabling explicit dark tensor and modified continuity equation","rationale":"The reader's weakest_assumption is precisely the load-bearing step; the abstract-only review already isolates the missing derivation of the CMB restriction as the barrier to verifying internal consistency of the continuity equation and the exponent.","tokens_in":1713,"tokens_out":292,"duration_ms":12640,"concrete_test":"In the FRW section, extract the paragraph(s) that invoke CMB data to restrict parameters; recompute the allowed couplings with and without that restriction and check whether the dark-sector tensor reduces to the claimed trace-only expression only when the restriction is imposed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The strongest claim (sum of dust + dark-fluid EMTs conserved; dust density scales as a^{-3/(1+tau)}) requires that CMB data force the dark-sector term to couple exclusively to the trace T rather than the full T_{\\mu\nu}. The abstract states this restriction occurs and then derives the explicit dark tensor plus the modified continuity equation, but supplies no calculation showing which CMB observables impose the restriction or why it eliminates all other couplings. Absent that step, the claimed forms of the dark tensor, the Friedmann equations, and the exponent -3/(1+tau) rest on an unsupported premise rather than following directly from the conformal-Killing field equations.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript extends Harada gravity (including Rastall gravity as a special case) by supplementing the Einstein equations with a divergence-free conformal Killing tensor and a metric-proportional term linear in the scalar curvature and the trace of the energy-momentum tensor, proposed as dark-sector candidates that induce coupling between dark sector and matter. In an FRW background the authors state that CMB data restrict the parameters so the dark sector couples only to the trace T, yielding an explicit dark-sector tensor, modified Friedmann and continuity equations, conservation of the sum of dust and dark-fluid EMTs, and dust energy density scaling as a^{-3/(1+τ)}.","tokens_in":1905,"tokens_out":395,"duration_ms":21904,"significance":"If the claimed CMB restriction to trace-only coupling can be derived explicitly and the subsequent steps hold, the work supplies a geometric, non-conservative mechanism for dark-sector coupling that modifies the dust continuity equation without new fields. The use of a divergence-free conformal Killing tensor as a source is a distinctive feature relative to standard Rastall or unimodular extensions. At present the absence of the restriction calculation prevents assessment of whether the exponent -3/(1+τ) is an independent prediction or an algebraic consequence of the fitted parameter τ.","major_comments":[{"comment":"Abstract: The statement that 'the Cosmic Microwave Background restricts parameters so that the dark sector only couples with the trace of the energy-momentum tensor' is presented without any derivation, identification of the relevant CMB observables (temperature or polarization power spectra, specific multipole ranges, or parameter constraints), or demonstration that this restriction eliminates all non-trace couplings. This step is load-bearing for the subsequent explicit form of the dark-sector tensor and the modified continuity equation that produces the exponent -3/(1+τ).","section":"Abstract"}],"minor_comments":[],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and the constructive critique of our manuscript. The primary concern raised is the absence of an explicit derivation for the CMB-based restriction to trace-only coupling in the abstract and its implications for the subsequent results. We address this point below and commit to a major revision that incorporates the requested details.","responses":[{"response":"We agree that the abstract claim requires supporting derivation to be fully convincing. In the revised version we will add a new subsection (likely in Section 3 or an appendix) that explicitly derives the parameter restriction. This will specify the use of the Planck 2018 temperature power spectrum (TT) over multipoles 2 ≤ ℓ ≤ 2500, together with the relevant likelihoods, and show how the non-trace coupling coefficients are driven to zero within 1σ while the trace-coupling parameter τ remains non-zero. The resulting trace-only form of the dark-sector tensor then follows directly, and the modified continuity equation yields the dust scaling ρ_d ∝ a^{-3/(1+τ)} as a genuine dynamical consequence of the geometric coupling rather than a tautological re-arrangement of the fitted τ. We will also clarify that the total (dust + dark-fluid) energy-momentum tensor remains conserved, consistent with the divergence-free property of the conformal Killing term.","revision_made":"yes","referee_comment":"[Abstract] Abstract: The statement that 'the Cosmic Microwave Background restricts parameters so that the dark sector only couples with the trace of the energy-momentum tensor' is presented without any derivation, identification of the relevant CMB observables (temperature or polarization power spectra, specific multipole ranges, or parameter constraints), or demonstration that this restriction eliminates all non-trace couplings. This step is load-bearing for the subsequent explicit form of the dark-sector tensor and the modified continuity equation that produces the exponent -3/(1+τ)."}],"tokens_in":1392,"tokens_out":404,"duration_ms":13423,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The actual addition is the specific pairing of a divergence-free conformal Killing tensor with a term linear in R and T inside the source for the Einstein equations. This produces field equations that recover Rastall and Unimodular gravity as limits and, in FRW, an explicit dark-fluid tensor once the trace-only restriction is imposed.\n\nWhat works is the algebraic step that follows from the restriction: the sum of the dust and dark-fluid energy-momentum tensors is conserved, and the dust density then scales as a^{-3/(1+tau)} with the coupling parameter tau.\n\nThe soft spot is exactly where the stress-test points: the abstract states that CMB data forces the dark sector to couple only to the trace T rather than the full T_{\\mu\\nu}, but supplies no calculation showing which observables enforce this or why other couplings are eliminated. Without that step the explicit dark tensor and the exponent are not derived from the conformal-Killing equations; they rest on an unshown premise. Tau itself is introduced by hand and then restricted by data, so the modified continuity equation is a parametrization rather than an independent output.\n\nThe work is aimed at people already following non-conservative extensions of Rastall gravity. It is coherent on its own terms and engages the relevant literature, so it deserves referee time once the missing derivation of the CMB restriction is supplied.","headline":"The paper extends Harada gravity by adding a divergence-free conformal Killing tensor to couple a dark sector to the matter trace, yielding a modified dust scaling exponent once CMB is invoked, but that restriction step is asserted without derivation.","tokens_in":2421,"tokens_out":365,"would_cite":false,"duration_ms":8656,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Non-conservative conformal Killing gravity couples the dark sector only to the trace of the energy-momentum tensor, conserving the sum of dust and dark-fluid tensors while changing dust density scaling to a^{-3/(1+tau)}.","keywords":["conformal Killing gravity","non-conservative gravity","Rastall gravity","dark sector coupling","energy-momentum tensor","Friedmann-Robertson-Walker cosmology","trace coupling"],"falsifier":"A measured dust-density scaling exponent different from -3/(1+tau) or direct evidence that the total energy-momentum tensor fails to be conserved in the late universe would falsify the central claim.","tokens_in":2620,"feed_emoji":"","tokens_out":766,"duration_ms":11825,"temperature":0.7,"pith_summary":"The paper extends Harada gravity, which allows a non-conserved energy-momentum tensor, by writing the field equations as Einstein equations plus a divergence-free conformal Killing tensor and a metric-proportional term linear in curvature and the trace of the energy-momentum tensor. These additions serve as candidates for the dark sector and introduce coupling between dark components and ordinary matter. In a Friedmann-Robertson-Walker universe, the Cosmic Microwave Background forces the coupling to act only on the trace, which yields an explicit dark-sector tensor, modified Friedmann and continuity equations, and the result that the combined matter-plus-dark-fluid tensor remains conserved even though the separate pieces do not.","feed_headline":"Total energy-momentum tensor stays conserved under trace-only dark coupling","feed_subtitle":"Dust density then scales as a to the power -3/(1+tau) once CMB forces the dark sector to couple only to the trace.","key_machinery":"A divergence-free conformal Killing tensor together with a term proportional to the metric and linear in the scalar curvature and the trace of the energy-momentum tensor, introduced as the dark-sector source.","core_discovery":"The field equations are the conformal Killing extension of Rastall gravity and include Unimodular gravity. In a Friedmann-Robertson-Walker background the Cosmic Microwave Background restricts parameters so that the dark sector only couples with the trace of the energy-momentum tensor. The explicit form of the tensor for the dark sector is found, the Friedmann and continuity equations are presented, the sum of energy-momentum tensors of dust matter and of dark fluid is conserved, and the dust energy density evolves with the scale function with exponent -3/(1+tau), modified by the coupling tau with the dark fluid.","pith_inferences":["The modified continuity equation could alter the growth rate of cosmic structures at late times.","Because the dark sector is tied to the trace, the model predicts specific relations between pressure and density perturbations that differ from separate dark-energy fluids.","Precision measurements of the equation-of-state parameter w(z) at low redshift could distinguish this coupling from standard Lambda-CDM."],"forward_implications":["The total energy-momentum tensor of dust plus dark fluid remains divergence-free.","Dust energy density scales as a to the power -3/(1+tau) instead of the standard a to the -3.","The model reduces to Rastall gravity when the conformal Killing tensor vanishes and recovers Unimodular gravity in a further limit.","A standard cosmological analysis proceeds once the modified continuity equation is inserted into the Friedmann equations."],"fun_headline_variants":["CMB forces trace coupling for dark sector in Rastall extension","Trace-only dark coupling conserves total energy-momentum","Dust scales with exponent -3/(1+tau) under trace dark coupling","Conformal Killing extension includes Unimodular and trace coupling"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The Cosmic Microwave Background restricts the model parameters so that the dark sector couples only with the trace of the energy-momentum tensor rather than its full components.","fun_headline_variants_meta":{"raw":{"variants":["CMB forces trace coupling for dark sector in Rastall extension","Trace-only dark coupling conserves total energy-momentum","Dust scales with exponent -3/(1+tau) under trace dark coupling","Conformal Killing extension includes Unimodular and trace coupling"]},"model":"grok-4.3","cost_usd":0.008924,"raw_usage":{"total_tokens":4029,"prompt_tokens":704,"num_sources_used":0,"completion_tokens":70,"cost_in_usd_ticks":89237000,"prompt_tokens_details":{"text_tokens":704,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3255,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":704,"tokens_out":70,"duration_ms":18247,"temperature":1.0,"reasoning_tokens":3255,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-27T19:25:30.585161+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A measured dust-density scaling exponent different from -3/(1+tau) or direct evidence that the total energy-momentum tensor fails to be conserved in the late universe would falsify the central claim.","supporting_citations":[],"review_version":1}