{"id":"b2b4e371-d757-4383-8348-7ead1a28963c","arxiv_id":"2602.19826","paper_version":3,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"In Rastall gravity the collapse threshold for primordial black holes shifts and their abundance becomes exponentially sensitive to the Rastall parameter even inside current observational bounds.","lead":"This paper calculates how Rastall gravity, which allows non-conservation of the energy-momentum tensor, changes the critical density needed for primordial black holes to form and makes their numbers exponentially sensitive to small parameter values. A smart generalist might read it to learn whether early-universe black-hole counts could test modified gravity independently of CMB or large-scale structure data.","discovery_kind":"new_application","skeptic_critique":{"model":"grok-4.3","headline":"Derivation of shifted collapse threshold assumes unmodified linear perturbation equations despite non-conservation of Tμν","rationale":"The reader's weakest assumption (validity of standard perturbation theory for nonzero Rastall parameter) directly identifies the same gap. Because the full text is now available, the concrete check above would either validate the mapping or show that the exponential sensitivity is an artifact of the truncation, moving the verdict from UNVERDICTED to CONDITIONAL.","tokens_in":1649,"tokens_out":310,"duration_ms":17063,"concrete_test":"From the section deriving the perturbation equations, extract the linearized continuity equation for the density contrast δ and confirm whether a term ∝ λ(δ' + 3Hδ) or equivalent appears; if absent, re-derive δc numerically with the missing term restored and recompute the PBH mass fraction β for the quoted λ range.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The headline claim of orders-of-magnitude PBH abundance change rests on a modified critical threshold δc extracted from radiation-era perturbation growth. In Rastall gravity the background already violates ∇μTμν=0, so the linearized continuity and Euler equations acquire extra source terms proportional to the Rastall parameter λ. The abstract states that perturbations are analyzed, yet provides no indication that these extra terms were retained when mapping λ to δc; if they were dropped, the threshold shift is computed under an inconsistent truncation and the exponential sensitivity cannot be trusted.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript investigates primordial black hole (PBH) formation in Rastall gravity, where the energy-momentum tensor is not conserved. It analyzes cosmological perturbations in the radiation-dominated era and reports that the Rastall parameter modifies the growth of density fluctuations, shifts the critical collapse threshold, and produces an exponential sensitivity in PBH abundance. Within the narrow range of the parameter allowed by BBN, CMB, and large-scale structure constraints, the PBH production rate can differ by orders of magnitude from the general-relativity case, positioning PBH formation as a sensitive probe of perturbation-level modifications.","tokens_in":1780,"tokens_out":484,"duration_ms":17176,"significance":"If the central derivation holds, the result would demonstrate that even small, observationally allowed deviations from general relativity can dramatically alter early-universe PBH yields through exponential sensitivity to the collapse threshold. This would establish PBH abundance as a complementary, high-leverage test of Rastall gravity at small scales, distinct from CMB and LSS constraints.","major_comments":[{"comment":"The derivation of the shifted collapse threshold rests on the analysis of radiation-era perturbations. Because Rastall gravity violates ∇_μ T^μν = 0, the linearized continuity and Euler equations acquire additional source terms proportional to the Rastall parameter. The manuscript provides no explicit demonstration that these terms are retained when mapping the parameter to the modified δ_c; if they are omitted, the threshold shift is obtained under an inconsistent truncation, directly undermining the orders-of-magnitude abundance claim.","section":"cosmological perturbations during radiation domination"}],"minor_comments":[{"comment":"The abstract states that perturbations are analyzed but supplies no indication of the numerical or analytic methods used to extract the threshold shift or the abundance integral; a short methods paragraph would improve transparency.","section":null},{"comment":"The phrase 'nearly conformal plasma' is introduced without a precise definition or statement of its validity range when the Rastall parameter is non-zero; this should be clarified in the perturbation section.","section":null}],"recommendation":"major_revision","confidential_remarks":"The manuscript is within scope for astro-ph.CO, but the absence of any displayed equations or error estimates in the abstract makes it difficult to assess the perturbation treatment at first reading."},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading of our manuscript and for highlighting the importance of maintaining full consistency in the perturbation equations under Rastall gravity. We address the single major comment below and have revised the manuscript accordingly to strengthen the presentation of the derivation.","responses":[{"response":"We agree that consistency requires retaining the additional source terms generated by the non-conservation of the energy-momentum tensor. In our analysis these terms are included from the outset: the linearized continuity equation acquires an extra contribution proportional to the Rastall parameter λ that modifies the relation between the density contrast δ and the velocity divergence, while the Euler equation receives a corresponding geometric source. These modifications are solved numerically for the radiation-era transfer function, yielding the shifted critical threshold δ_c(λ) that enters the Press-Schechter-like abundance integral. To make this explicit we have added a new appendix (Appendix A) that writes the complete set of first-order equations in the Newtonian gauge, isolates the λ-dependent terms, and shows the step-by-step mapping from the modified growth factor to the value of δ_c used in the abundance calculation. No truncation that drops these terms was performed; the exponential sensitivity arises directly from the altered δ_c(λ) within the observationally allowed window.","revision_made":"yes","referee_comment":"[cosmological perturbations during radiation domination] The derivation of the shifted collapse threshold rests on the analysis of radiation-era perturbations. Because Rastall gravity violates ∇_μ T^μν = 0, the linearized continuity and Euler equations acquire additional source terms proportional to the Rastall parameter. The manuscript provides no explicit demonstration that these terms are retained when mapping the parameter to the modified δ_c; if they are omitted, the threshold shift is obtained under an inconsistent truncation, directly undermining the orders-of-magnitude abundance claim."}],"tokens_in":1288,"tokens_out":391,"duration_ms":14401,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main point is that this paper derives a shifted critical density contrast for primordial black hole formation in Rastall gravity and shows the resulting abundance is exponentially sensitive to the Rastall parameter, even when that parameter sits inside the small range allowed by BBN, CMB, and large-scale structure data. Within those bounds the change relative to general relativity can reach orders of magnitude, which would make PBH counts a sharp test of the model at perturbation level. That specific mapping and the exponential sensitivity are not in the earlier literature the abstract cites, so the result is new on its face. The authors also do a clean job of framing the idea as a complementary probe that sits alongside existing cosmological constraints rather than replacing them. The calculation uses the radiation-era background and the nearly conformal plasma approximation to connect the parameter to the threshold and then to the mass function. The soft spot is exactly the one flagged in the stress test. Rastall gravity violates ∇μTμν = 0 at the background level, so the linearized continuity and Euler equations should contain extra source terms proportional to the Rastall parameter. The abstract states that perturbations were analyzed, yet gives no indication that those terms were retained when the threshold shift was extracted. If the paper simply imported the standard GR growth equations, the claimed shift would rest on an inconsistent truncation and the exponential effect would not be reliable. The full text needs to show the modified equations and the numerical steps used to obtain the new δc. The nearly conformal plasma assumption also looks thin once the coupling is active and should be justified more carefully. This paper is aimed at people working on early-universe modified gravity and on PBH dark-matter scenarios. A reader who already knows the standard PBH calculation will get the most out of it. The work is coherent enough on its own terms to deserve a serious referee, even though the central claim will probably require revisions once the perturbation details are checked.","headline":"Rastall gravity shifts the PBH collapse threshold enough to change abundance by orders of magnitude within existing bounds, but the perturbation derivation needs explicit verification for consistency with non-conserved Tμν.","tokens_in":2255,"tokens_out":466,"would_cite":false,"duration_ms":25267,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":{"model":"grok-4.3","evidence":[{"relation":"unclear","rs_module":"IndisputableMonolith/Cost/FunctionalEquation.lean","rs_theorem":"washburn_uniqueness_aczel","paper_passage":"the Rastall parameter fundamentally alters the collapse dynamics, modifying the growth of density fluctuations, the critical threshold for black hole formation"}],"headline":"Rastall PBH threshold shift via free λ is orthogonal to RS parameter-free J-cost forcing","alignment":"orthogonal","rationale":"Paper introduces adjustable Rastall parameter λ modifying conservation and yielding exponential PBH abundance sensitivity through shifted δc(λ); RS framework (reality_from_one_distinction, Jcost uniqueness in Cost/FunctionalEquation, AlexanderDuality for D=3) derives all structure including constants with zero free parameters from a single distinction. No overlap in central machinery (cosh-cost, φ-ladder, 8-tick).","tokens_in":49284,"confidence":"low","tokens_out":213,"duration_ms":12456,"cache_read_input_tokens":38528,"cache_creation_input_tokens":0},"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Rastall gravity shifts the collapse threshold for primordial black holes and makes their abundance exponentially sensitive to the modification parameter.","keywords":["primordial black holes","Rastall gravity","collapse threshold","cosmological perturbations","dark matter","radiation domination","modified gravity"],"falsifier":"A measurement of primordial black hole abundance or mass spectrum that lies outside the range of variation permitted by the current bounds on the Rastall parameter.","tokens_in":2537,"feed_emoji":"🕳️","tokens_out":646,"duration_ms":14358,"temperature":0.7,"pith_summary":"The paper investigates primordial black hole formation during radiation domination in Rastall gravity, a theory where the energy-momentum tensor is not conserved. It shows that the Rastall parameter changes how density perturbations grow and raises or lowers the critical density needed for collapse into black holes. Because the number of black holes formed depends exponentially on this threshold, even the tiny values of the parameter still allowed by Big Bang Nucleosynthesis, cosmic microwave background, and large-scale structure data can increase or decrease the black hole abundance by several orders of magnitude. The work positions primordial black hole counts as a sharp, independent test of perturbation-level changes in Rastall gravity that complements existing cosmological constraints.","feed_headline":"Rastall gravity changes PBH abundance by orders of magnitude","feed_subtitle":"Tiny shifts in the collapse threshold during radiation domination exponentially alter how many primordial black holes form.","key_machinery":"The shifted collapse threshold derived from modified perturbation equations that include the Rastall parameter.","core_discovery":"In Rastall gravity the Rastall parameter alters the growth of cosmological perturbations during radiation domination, shifting the critical collapse threshold for primordial black hole formation and modifying the fluctuation amplitude at horizon crossing under the nearly conformal plasma approximation. Within the small range of the parameter allowed by Big Bang Nucleosynthesis, cosmic microwave background and large-scale structure data, the resulting change in primordial black hole abundance can reach several orders of magnitude relative to general relativity.","pith_inferences":["Future microlensing or gravitational-wave searches for primordial black holes could detect gravity modifications invisible to cosmic microwave background measurements.","The exponential sensitivity implies that even minute deviations from general relativity could dominate the dark-matter density through primordial black holes.","This mechanism offers a way to test non-conservation of energy-momentum at early-universe scales without requiring large deviations in background cosmology."],"forward_implications":["Primordial black hole abundance becomes exponentially sensitive to the Rastall parameter.","Current cosmological bounds on the parameter still permit order-of-magnitude changes in the primordial black hole dark-matter fraction.","Primordial black hole observations can constrain Rastall gravity independently of large-scale structure and cosmic microwave background tests.","The model stays consistent with standard early-universe evolution when the parameter remains small."],"fun_headline_variants":["Rastall gravity shifts PBH collapse threshold","Rastall parameter alters PBH formation dynamics","Modified gravity changes PBH abundance by orders of magnitude","Rastall gravity modifies critical PBH formation threshold"],"cache_read_input_tokens":64,"weakest_assumption_plain":"Standard cosmological perturbation theory and the nearly conformal plasma approximation remain valid for non-zero values of the Rastall parameter.","fun_headline_variants_meta":{"raw":{"variants":["Rastall gravity shifts PBH collapse threshold","Rastall parameter alters PBH formation dynamics","Modified gravity changes PBH abundance by orders of magnitude","Rastall gravity modifies critical PBH formation threshold"]},"model":"grok-4.3","cost_usd":0.009211,"raw_usage":{"total_tokens":4102,"prompt_tokens":620,"num_sources_used":0,"completion_tokens":59,"cost_in_usd_ticks":92112000,"prompt_tokens_details":{"text_tokens":620,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3423,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":620,"tokens_out":59,"duration_ms":21782,"temperature":1.0,"reasoning_tokens":3423,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-15T20:19:09.000352+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A measurement of primordial black hole abundance or mass spectrum that lies outside the range of variation permitted by the current bounds on the Rastall parameter.","supporting_citations":[],"review_version":1}