{"id":"2e72a08b-1e34-43bd-80c9-d4f5ac8d07db","arxiv_id":"2608.07313","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A one-parameter geometric dark energy model fits the newest DESI, Planck, ACT, and supernova data as well as or slightly better than the standard cosmological model, with a small negative extra parameter.","lead":"Conformal Killing Gravity is a geometric model where dark energy comes from a symmetry of space-time rather than from a fitted formula. Tested against the newest DESI, Planck, ACT, and supernova data, it performs comparably to the standard model and may slightly favor an evolving dark energy.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The paper constrains CKG using CAMB but never derives the linear perturbation equations for the geometric dark fluid, so the reported constraints and the claimed strong Bayesian evidence rest on an unstated and potentially incorrect clustering assumption.","rationale":"The reader's weakest-assumption analysis identifies exactly the same load-bearing concern: the paper computes CMB and LSS observables with CAMB without deriving the CKG perturbation equations. My stress-test confirms this is the central issue. The paper's strongest claim is that CKG is statistically preferred over LambdaCDM, but that preference is built on constrained parameters whose likelihoods depend on the unstated perturbation treatment. If the geometric dark fluid clusters differently than assumed, the Omega_D constraints, the S8 comparison, and the Bayes factors all change. I agree with the reader's conditional verdict: the paper should be accepted only after the perturbation equations are derived and the analysis is rerun, or at least after the authors explicitly justify the smooth-fluid approximation from the theory. I do not see an internal inconsistency in the background derivation, and the datasets are appropriate, so a full rejection is not warranted. The CONDITIONAL verdict already captures this, so no change to the reader's verdict is needed.","tokens_in":18079,"tokens_out":4535,"duration_ms":46077,"concrete_test":"Derive the first-order perturbed field equations of CKG by perturbing the divergence-free conformal Killing tensor condition and the modified Einstein equations around the FLRW background. Express the result in terms of the effective-fluid perturbation variables (delta_D, theta_D, pi_D) or as modifications to the standard metric perturbation equations. Implement these equations in a modified Boltzmann code (e.g., a branch of CAMB or hi_class) and rerun the CMB+DESI DR2+Pantheon+ analysis. If the predicted CMB TT, P(k), or f_sigma8 at the best-fit Omega_D differ by more than the observational precision, or if the Omega_D posterior shifts by more than 0.02, then the assumption that CKG modifies only the background expansion is invalid and the reported constraints and Bayes factors require revision.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section II derives only the homogeneous background Friedmann equation (Eq. 19) for CKG, with the effective dark fluid described by Eqs. (9)-(10). However, Section III states that all theoretical predictions, including the CMB TT spectrum, P(k), and f_sigma8, are computed with CAMB. CAMB requires a full perturbation specification for every additional energy component. For a geometric modification such as the divergence-free conformal Killing tensor K_ab, the first-order perturbation delta K_ab is not automatically zero; it must be obtained from the perturbed conformal Killing equation and the perturbed FLRW metric. The paper neither derives delta K_ab nor specifies the dark-fluid sound speed or anisotropic stress used in the CAMB runs. The effective fluid in Eqs. (9)-(10) has no independent dynamics; its perturbations are fixed by the theory, not by an arbitrarily chosen w(z). If the fluid is not perfectly smooth, or if it sources the Poisson equation differently, the P(k) and f_sigma8 predictions in Figs. 1-2 and the low-ell CMB TT would change, directly shifting the Omega_D posterior in Table II and the Bayes factors in Table II. The 'strong evidence' claim is also fragile for a second, independent reason: the uniform prior on Omega_D has width ~1.02, while the posterior width is ~0.03, so the Bayes factor is heavily prior-volume dependent and should be recomputed with nested sampling and prior sensitivity checks. The perturbation issue is the more load-bearing concern because if the CMB and LSS likelihoods were computed with the wrong perturbation equations, every derived constraint in the paper is invalid.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constrains Conformal Killing Gravity (CKG), a modified-gravity framework in which a divergence-free conformal Killing tensor acts as an effective dark-energy fluid. Starting from the RW metric, the authors derive a modified Friedmann equation containing an extra density parameter Ω_D with redshift dependence (1+z)^{-2}, and then run MCMC analyses with Cobaya/CAMB using Planck PR4 CMB, ACT DR6 lensing, DESI DR2 BAO, and three Type Ia supernova compilations. They report negative Ω_D values, a quintessence-like effective equation of state, a future critical redshift z_c in the range roughly -0.8 to -0.7 where H(z_c)=0, and Bayesian evidence that they describe as strongly favoring CKG over ΛCDM.","tokens_in":18401,"tokens_out":5776,"duration_ms":58062,"significance":"If the main claims hold, this is an interesting result: a geometric dark-energy model with a single extra parameter that is observationally competitive with ΛCDM, with no ad hoc equation-of-state parametrization and with a concrete prediction for the future evolution. The paper is also transparent in using public likelihoods and standard MCMC tools, and the background derivation in Section II is clear and internally consistent. However, the two load-bearing pillars of the paper—the computation of CMB and matter-power-spectrum predictions from a purely geometric modification, and the claimed strong Bayesian evidence—are not presently established. The perturbation equations for the geometric dark fluid are never given, and the reported Bayes factors appear to be dominated by the arbitrary width of the Ω_D prior rather than by improved fit. The paper therefore needs substantial revision before its central conclusions can be accepted.","major_comments":[{"comment":"The paper derives only the homogeneous background Friedmann equation for CKG, yet Section III states that all theoretical predictions (CMB TT, P(k), fσ8) are computed with CAMB. CAMB requires a complete perturbation specification for every additional energy component. The perturbed conformal Killing tensor δK_ab, the effective sound speed, and the possible anisotropic stress of the geometric dark fluid are never stated. If the fluid is not exactly smooth, or if it sources the Poisson equation differently, the P(k), fσ8, CMB lensing, and low-ℓ TT predictions in Figs. 1-2 and the posteriors in Table II will change. Please derive δK_ab from the perturbed conformal Killing equation, or otherwise specify and justify the perturbation prescription used in the CAMB runs, and test the sensitivity of the Ω_D constraints to that prescription.","section":"Section II and III, Eqs. (16)-(19), Fig. 1 and Fig. 2"},{"comment":"The reported Bayesian evidence is not supported by the maximum-likelihood differences and is strongly prior-volume dependent. The prior on Ω_D is uniform over [-1, 0.02], a width of about 1.02, while the posterior widths in Table II are roughly 0.03. The logarithm of the Bayes factor therefore receives an Occam penalty of order ln(1.02/0.03) ≈ 3.5 from the prior volume alone, and the values ln B ≈ -5.5 to -6.8 cannot be interpreted as strong evidence independent of that arbitrary prior choice. Moreover, the CMB+DESI combination has Δχ²_MAP = +1.13, i.e., ΛCDM fits better, while ln B = -5.53 still favors CKG; this contradicts the paper's claim of agreement between the minimized chi-square and the Bayesian evidence. Please recompute the evidence with a prior-sensitivity analysis (for example, varying the Ω_D prior width or using nested sampling) and moderate the strength-of-evidence claims accordingly.","section":"Table I and Table II"},{"comment":"The prior Ω_D ∈ [-1, 0.02] is asymmetric and truncates almost all positive values. Since Eq. (26) maps positive Ω_D to w_D < -1 (phantom-like behavior), the conclusion that CKG shows 'no evidence for phantom crossing' is partly enforced by the prior rather than by the data. The text should either widen the prior to include positive Ω_D values or explicitly state that the no-phantom-crossing conclusion is conditional on the adopted prior.","section":"Section IV, Eqs. (26)-(28), and Table I"},{"comment":"The 'reconstructed' equation of state w_D(z), the deceleration parameter q(z), and the future critical redshift z_c are deterministic functions of the fitted parameters Ω_D and Ω_Λ. They are therefore parameter transformations of the same data used in the likelihood, not independent predictions of the model. In particular, the statement that CKG 'predicts' z_c in the range -0.8 to -0.7 should be rephrased as a derived constraint from the posterior, and the paper should clarify that a genuine prediction would require a dataset or observable not already used in the fit.","section":"Section IV, Eqs. (26)-(28), Fig. 4"}],"minor_comments":[{"comment":"Fig. 2 overlays observational fσ8 data, but the dataset list in Section III does not mention an RSD likelihood; please clarify that the growth-rate comparison is illustrative and not part of the MCMC constraints.","section":"Section III, dataset list"},{"comment":"The text defines the baseline parameter vector as {Ω_cdm h^2, Ω_b h^2, 100θ_MC, ln(10^10 A_s), n_s, τ}, but Table I also lists a prior on H0; please state explicitly whether H0 is sampled directly or derived from the other parameters.","section":"Section III, Table I"},{"comment":"Eq. (19) omits the radiation term Ω_R(1+z)^4 without comment after Eq. (16) introduced it; please state explicitly where radiation is neglected in the background and where it is included in the CAMB computation.","section":"Eq. (19)"},{"comment":"There are several minor grammatical slips (for example, 'Both models shows' and the title 'datasets' with no space), and some references contain rendering artifacts; a careful copyedit would improve readability.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper is a competent observational analysis of an interesting geometric dark-energy model, and the background derivation is solid. The missing perturbation treatment is the main scientific gap: without it, the CAMB-based constraints are conditional on an unstated clustering assumption. The Bayes-factor claim is also fragile because the prior on Ω_D dominates the evidence. I would not reject the paper, because the central model may well be defensible after the authors supply the perturbation equations or restrict the claims to background observables, and after the evidence is recomputed with prior-sensitivity tests. The present version, however, overstates both the observational support and the predictive power of the model."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a real data analysis, not a crank submission, but the two headline claims are shaky. What is actually new is the updated MCMC constraints on Omega_D from DESI DR2 BAO plus Planck PR4/ACT lensing and three SNe compilations. The background equations (Eqs. 16-19) are internally consistent, and the authors are appropriately cautious about H0, saying explicitly that the model leaves the sound horizon unchanged and does not resolve the tension. The S8 discussion is also measured. Credit where earned: this is a competent extension of prior work.\n\nThe soft spots, in order of severity. First, the perturbation sector is never specified. Section II derives only the homogeneous Friedmann equation, yet Section III computes the CMB TT spectrum, P(k), and f_sigma8 with CAMB, and Figs. 1-2 present these as CKG predictions. For a divergence-free conformal Killing tensor, the first-order perturbation is not automatically zero; you need to perturb the conformal Killing equation and specify the dark-fluid sound speed and anisotropic stress. The paper does neither. Without that, the Omega_D constraints in Table II rest on an unstated clustering assumption.\n\nSecond, the Bayes factors are not credible as reported. For CMB+DESI DR2, the paper's own Delta-chi2_MAP is +1.13 (CKG best fit worse than LambdaCDM), yet ln B = -5.53, which they call strong evidence for CKG. A nested model with a worse maximum likelihood cannot produce strong evidence in its favor unless the evidence calculation is dominated by prior-volume effects. The broad U[-1,0.02] prior on Omega_D with a posterior width around 0.03 is a red flag; the Bayes factors should be recomputed with nested sampling and prior sensitivity checks. Relatedly, the prior excludes almost all positive Omega_D, so the \"no phantom crossing\" conclusion is partly baked into the prior, not a data-driven result.\n\nThird, the reconstructed w(z) and the future critical redshift are deterministic functions of the fitted Omega_D and Omega_Lambda. They are transforms of the fit, not independent predictions.\n\nThese are fixable issues, not necessarily fatal ones. The background is sound and the data handling looks careful. The paper overstates what it has shown: the \"strong evidence\" language should be dropped until the Bayes factors are redone, and the perturbation equations (or an explicit statement of the CAMB implementation) must be provided. A serious referee should engage with it, because the updated constraints on a geometric dark-energy model in light of DESI DR2 are of real interest. But as it stands, the central quantitative claims need revision.","headline":"A legitimate but overstated constraints paper: the background derivation is clean and the H0 discussion is honest, but the perturbation treatment is unstated and the claimed strong Bayes factor is contradicted by the paper's own Delta-chi-squared numbers.","tokens_in":19026,"tokens_out":3785,"would_cite":false,"duration_ms":37204,"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":"Conformal Killing Gravity derives dark energy from a spacetime symmetry and reports strong Bayesian evidence for it over ΛCDM using DESI DR2 and CMB data.","keywords":["conformal Killing gravity","geometric dark energy","DESI DR2","baryon acoustic oscillations","Bayesian model comparison","quintessence","cosmological tensions","modified gravity"],"falsifier":"Derive and solve the full linear perturbation equations for the conformal Killing fluid and compare the predicted $f\\sigma_8(z)$ and matter power spectrum with the redshift-space distortion measurements quoted in the paper; a deviation larger than current errors would show that the reported $\\Omega_D$ constraints depend on the unmodified-perturbation assumption. Alternatively, a high-redshift measurement of $w_D(z)$ at $z \\gtrsim 1$ that excludes $w = -1$ would falsify the model's predicted return to ΛCDM-like behavior.","tokens_in":17845,"feed_emoji":"🌌","tokens_out":8218,"duration_ms":72664,"temperature":0.7,"pith_summary":"Conformal Killing Gravity (CKG) derives dark energy from a geometric symmetry of the Robertson–Walker spacetime: a divergence-free conformal Killing tensor acts as an effective fluid whose density, pressure, and equation of state are fixed by geometry, not chosen by hand. The paper claims that this one-parameter extension of ΛCDM is statistically preferred in a joint analysis of Planck PR4 CMB, ACT DR6 lensing, DESI DR2 BAO, and three Type Ia supernova compilations. The fitted geometric parameter is negative, giving a quintessence-like dark energy with no phantom crossing, while the sound horizon is essentially unchanged, so the H0 tension is not resolved. Bayesian model comparison is reported as strong evidence in favor of CKG over ΛCDM for every dataset combination, and the model predicts a future epoch where the expansion halts.","feed_headline":"Dark energy from spacetime symmetry favored in DESI fit","feed_subtitle":"Conformal Killing Gravity's single extra parameter is strongly preferred by CMB, BAO, and supernova data.","key_machinery":"The load-bearing object is the divergence-free conformal Killing tensor $K_{ab} = g_{ab}\\left(\\frac{5}{6} C a^2 - \\Lambda\\right) + u_a u_b \\frac{C a^2}{3}$ on the Robertson–Walker background, which the field equations reinterpret as an effective perfect fluid $(p_D + \\mu_D) u_a u_b + p_D g_{ab}$. The fluid's density $\\mu_D = -\\frac{1}{2} C a^2 + \\Lambda$ and pressure $p_D = \\frac{5}{6} C a^2 - \\Lambda$ are fixed by the geometry, so no equation-of-state parametrization is introduced. Substituting into the modified Friedmann equations yields the single extra term $\\Omega_D (1+z)^{-2}$ in the expansion rate, and every late-time prediction of the paper—the quintessence-like $w_D(z)$, the unchanged sound horizon, and the future turning point where $H(z_c)=0$—follows from that term.","core_discovery":"The central claim is that the dark energy driving late-time acceleration can be an emergent geometric component rather than a cosmological constant or a phenomenological parametrization. In the CKG framework the divergence-free conformal Killing tensor of the Robertson–Walker spacetime behaves as a perfect fluid with density $\\mu_D = -\\frac{1}{2} C a^2 + \\Lambda$ and pressure $p_D = \\frac{5}{6} C a^2 - \\Lambda$, which contributes to the Friedmann equation only as $\\Omega_D/(1+z)^2$. With $\\Omega_D < 0$, the effective equation of state $w_D(z) = -1 - \\frac{2}{3}\\frac{\\Omega_D}{\\Omega_D + \\Omega_\\Lambda(1+z)^2}$ is above $-1$ at late times and approaches $-1$ at high redshift, so the expansion history differs from ΛCDM only after recombination and never crosses the phantom divide. The fits yield $\\Omega_D$ negative (bounded below by $-0.0413$ for CMB+DESI DR2 alone, with means around $-0.04$ to $-0.07$ when supernovae are added), an unchanged sound horizon of about $147.8$ Mpc, a future critical redshift $z_c \\approx -0.7$ to $-0.8$ at which $H(z_c)=0$, and log Bayes factors of $-5.5$ to $-6.8$ favoring CKG.","pith_inferences":["The paper's evidence estimates could be checked against the prior volume: a uniform prior $\\Omega_D \\in [-1, 0.02]$ may penalize ΛCDM (whose $\\Omega_D = 0$ lies near the boundary), so reporting Bayes factors with a scale-invariant or shrinkage prior would test whether the strong preference is robust.","Deriving the linear perturbation equations for the conformal Killing fluid would turn the background-level preference into a full theory: if the fluid has non-zero sound speed or anisotropic stress, the $P(k)$ and $f\\sigma_8$ predictions of the paper's Boltzmann-solver calculation would change and could be confronted with redshift-space distortion and weak-lensing data.","The model's unique late-time signature is the eventual halt of expansion; precise low-redshift $H(z)$ measurements from standard sirens or cosmic chronometers can constrain $z_c$ even though the turning point lies in the future.","Interpreting the geometric dark fluid as a physical component requires specifying its fluctuations and energy conditions; without that, CKG is currently a background-only model, and any claim about structure growth rests on the unmodified ΛCDM perturbation equations."],"forward_implications":["If the preference is real, dark energy needs no new scalar field or cosmological constant; a purely geometric source with one parameter reproduces the late-time data.","The model predicts $w_D(z) \\geq -1$ with convergence to $w = -1$ by $z \\approx 1$, so high-redshift BAO and supernova measurements can distinguish it from evolving dark energy that crosses the phantom divide.","Since the sound horizon is barely changed, the H0 tension must be addressed by pre-recombination physics; CKG alone cannot raise H0 to the local distance-ladder value.","The future critical redshift in $-0.8 \\lesssim z_c \\lesssim -0.7$ means the current accelerating phase is temporary: the expansion rate passes through zero before the singular $z = -1$ limit.","The model's S8 values sit within about $0.1\\sigma$ of KiDS-Legacy and below $1.8\\sigma$ of DES Y6, so it does not aggravate the S8 tension."],"supporting_citations":[{"why":"Introduce Conformal Killing Gravity as a way to explain cosmic acceleration without exotic dark energy.","marker":"[12, 13]"},{"why":"Reformulate CKG as Einstein equations plus a divergence-free conformal Killing tensor, giving the $\\Omega_D$ term.","marker":"[14]"},{"why":"Earlier CKG constraints that this paper extends with DESI DR2 and updated likelihoods.","marker":"[23]"},{"why":"The DESI DR2 BAO measurements that drive the late-time expansion constraints.","marker":"[11]"},{"why":"The Planck PR4 CamSpec CMB likelihood used for temperature, polarization, and lensing.","marker":"[15, 16]"},{"why":"ACT DR6 CMB lensing data added to the CMB constraint.","marker":"[17, 18]"},{"why":"The Pantheon+ supernova compilation and its cosmology analysis.","marker":"[19, 20]"},{"why":"The DES-Dovekie supernova sample used as an independent supernova dataset.","marker":"[21]"},{"why":"The Union3 supernova compilation used in the combined fits.","marker":"[22]"}],"fun_headline_variants":["Dark energy from spacetime symmetry passes DESI test","Geometric dark energy model favored by DESI and CMB","Conformal Killing Gravity: dark energy without constant","Spacetime symmetry yields dark energy, new data agrees","One extra parameter: dark energy from geometry, no phantom"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the conformal Killing dark fluid changes only the expansion history, while the standard ΛCDM equations for density perturbations remain valid; if the geometric fluid clusters or modifies the Poisson equation, the computed CMB, matter-power, and $f\\sigma_8$ predictions—and hence the $\\Omega_D$ constraints and Bayesian evidence—would shift.","fun_headline_variants_meta":{"raw":{"variants":["Dark energy from spacetime symmetry passes DESI test","Geometric dark energy model favored by DESI and CMB","Conformal Killing Gravity: dark energy without constant","Spacetime symmetry yields dark energy, new data agrees","One extra parameter: dark energy from geometry, no phantom"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000256,"raw_usage":{"total_tokens":1689,"prompt_tokens":1170,"completion_tokens":519,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":786,"completion_tokens_details":{"reasoning_tokens":441}},"tokens_in":786,"tokens_out":519,"duration_ms":4856,"temperature":1.0,"reasoning_tokens":441,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T10:30:18.533906+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Derive and solve the full linear perturbation equations for the conformal Killing fluid and compare the predicted $f\\sigma_8(z)$ and matter power spectrum with the redshift-space distortion measurements quoted in the paper; a deviation larger than current errors would show that the reported $\\Omega_D$ constraints depend on the unmodified-perturbation assumption. Alternatively, a high-redshift measurement of $w_D(z)$ at $z \\gtrsim 1$ that excludes $w = -1$ would falsify the model's predicted return to ΛCDM-like behavior.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Reformulate CKG as Einstein equations plus a divergence-free conformal Killing tensor, giving the $\\Omega_D$ term."},{"cited_title":"Capozziello, C","cited_arxiv_id":null,"evidence_quote":"Earlier CKG constraints that this paper extends with DESI DR2 and updated likelihoods."},{"cited_title":"Abdul Karim, J","cited_arxiv_id":null,"evidence_quote":"The DESI DR2 BAO measurements that drive the late-time expansion constraints."},{"cited_title":"Rubin, G","cited_arxiv_id":null,"evidence_quote":"The Union3 supernova compilation used in the combined fits."}],"review_version":1}