Conformal Killing Gravity: New Constraints from DESI DR2 BAO datasets

Carlo Alberto Mantica; Dhruba Jyoti Gogoi; Ghulam Mustafa; Himanshu Chaudhary; Luca Guido Molinari; Salvatore Capozziello

REVIEW 4 major objections 4 minor 59 references

Conformal Killing Gravity: New Constraints from DESI DR2 BAO datasets

T0 review · 4 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2608.07313 v1 pith:BRQFNHDO submitted 2026-08-07 astro-ph.CO gr-qchep-th

Himanshu Chaudhary , Salvatore Capozziello , Carlo Alberto Mantica , Luca Guido Molinari , Dhruba Jyoti Gogoi , Ghulam Mustafa This is my paper · ORCID

classification astro-ph.COgr-qchep-th

keywords conformalKillinggravitygeometricdarkenergyDESIDR2baryonacousticoscillationsBayesianmodelcomparisonquintessencecosmologicaltensionsmodified

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

The 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.

Second, 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.

Third, 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.

These 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.

Referee Report

4 major / 4 minor

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.

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 (4)

[Section II and III, Eqs. (16)-(19), Fig. 1 and Fig. 2] 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.
[Table I and Table II] 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 IV, Eqs. (26)-(28), and Table I] 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 IV, Eqs. (26)-(28), Fig. 4] 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.

minor comments (4)

[Section III, dataset list] 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 III, Table I] 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.
[Eq. (19)] 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.
[Throughout] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the CKG constraints and reported derived quantities are standard model consequences, not fits disguised as predictions.

full rationale

The derivation chain is not circular. The conformal Killing tensor Kab is derived from the RW conformal Killing vector and its divergence-free condition (Eqs. 7-10), and the modified Friedmann equation (Eq. 19) follows algebraically from the definition Tab = Rab - (1/2)R gab - Kab. The free parameters Omega_D and Omega_Lambda are then fitted to external cosmological data; the reported wD(z), q(z), and zc (Eqs. 26-28) are model consequences evaluated at the best-fit point, not independent data fits subsequently relabelled as predictions. Refs. [14,23,26] are self-citations for the CKG reformulation, but the present paper restates the core derivation, so the self-citations are not load-bearing. The main weakness is a completeness/correctness gap rather than circularity: Section II derives only the background equations, while Section III invokes CAMB for the CMB TT spectrum, P(k), and f_sigma8 without presenting the perturbed conformal Killing equations or the perturbation properties of the geometric dark fluid. That gap affects how the reported constraints should be interpreted, but it does not make the derivation equivalent to its inputs.

Assumptions & free parameters 2 free parameters · 5 assumptions · 1 invented entities

The central claim rests on the CKG field-equation postulate, the assumed form of the conformal Killing tensor, the decoupling of matter from the geometric fluid, and an unstated choice to keep LambdaCDM perturbation theory in a modified background. The only genuinely new fitted parameter is Omega_D; the other fitted parameters are standard cosmological parameters.

free parameters (2)

Omega_D (CKG geometric dark-energy density parameter) = Omega_d > -0.0413 (95% CMB+DESI DR2); -0.043 +/- 0.032, -0.053 +/- 0.031, -0.069 +/- 0.043 for the Pantheon+…
New CKG parameter; prior U[-1, 0.02] (Table I); fitted to CMB, BAO, and SNe data and central to the model comparison.
LambdaCDM baseline parameters (Omega_cdm h^2, Omega_b h^2, 100 theta_MC, ln(10^10 A_s), n_s, tau) = Posterior means in Table II
Standard six cosmological parameters fitted simultaneously; not introduced by this paper but required for the CKG fit and the H0 and S8 comparisons.

assumptions (5)

domain assumption CKG is equivalent to Einstein equations supplemented by a divergence-free conformal Killing tensor acting as a source.
Invoked at Eq. (12) and attributed to Ref. [14]; the paper sketches but does not independently prove this equivalence.
domain assumption The RW space-time admits the divergence-free conformal Killing tensor Kab of Eq. (7) with integration constants C and Lambda.
Section II, Eq. (7); follows from the conformal Killing vector xi = a u and the divergence-free condition, a model postulate from prior CKG literature.
domain assumption Baryonic and cold dark matter and radiation are separately conserved, with no interaction with the geometric dark fluid.
Section II, 'Conservation of the matter and radiation fluids implies...'; this decoupling is assumed, not derived.
ad hoc to paper Standard LambdaCDM linear perturbation theory remains valid with only H(z) modified in CKG.
Section III uses CAMB for CMB TT, P(k), and f_sigma8 (Fig. 2) without stating CKG perturbation equations; this is a modeling choice specific to this analysis.
domain assumption Spatial flatness, Omega_k = 0.
Section II states 'our analysis is restricted to a spatially flat FLRW universe'; standard but it removes the curvature-like degeneracy with the Omega_D term.

invented entities (1)

Geometric dark-energy fluid (effective perfect fluid from the conformal Killing tensor Kab)
purpose: To source late-time accelerated expansion without an empirical dark-energy parametrization or a cosmological constant.
The fluid is defined by the CKG model, and its density and pressure are fixed by the fitted Omega_D; there is no observational handle on the fluid independent of the model fits.

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Pith. "Pith review of Conformal Killing Gravity: New Constraints from DESI DR2 BAO datasets." pith.science (2026). https://pith.science/paper/BRQFNHDO

@misc{pith2026260807313,
  author       = {Pith},
  title        = {Pith review of: Conformal Killing Gravity: New Constraints from DESI DR2 BAO datasets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BRQFNHDO}},
  note         = {Machine review of arXiv:2608.07313}
}

abstract

We investigate a geometric approach referred to as the Conformal Killing Gravity (CKG), in which the dark energy sector emerges naturally from the conformal Killing symmetry of the Robertson--Walker space-time. Within this framework, the divergence-free conformal Killing tensor behaves as an effective perfect fluid, giving rise to a dynamical dark-energy component whose density, pressure, and equation of state are uniquely determined by the underlying geometry, without introducing any empirical dark-energy parametrization. The resulting CKG model extends the standard $\Lambda$CDM cosmology through a single additional parameter while recovering the $\Lambda$CDM limit in the absence of the geometric contribution. We constrain the model using Planck PR4 (NPIPE) CMB temperature, polarization, and lensing observations, ACT DR6 CMB lensing, DESI DR2 baryon acoustic oscillation measurements, and the Pantheon$+$, DES-Dovekie, and Union3 Type Ia supernova compilations. The analysis shows that the CKG favors a quintessence-like dark-energy evolution with no evidence for phantom crossing, while the reconstructed equation of state rapidly approaches the cosmological constant at earlier cosmic times. Furthermore, the model predicts a future critical redshift in the range $-0.8 \lesssim z_c \lesssim -0.7$, indicating that the present cosmic expansion eventually reaches a turning point before the formal singular limit at $z=-1$. Since the geometric contribution modifies only the post-recombination expansion history, the sound horizon remains essentially unchanged and the model does not provide a complete solution to the $H_0$ tension. Our results demonstrate that the CKG provides a simple, physically motivated, and observationally favored framework for describing the late-time accelerated expansion of the Universe.

Figures

Figures reproduced from arXiv: 2608.07313 by the authors.

**Figure 2.** FIG. 2: This figure shows the evolution of the [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** FIG. 3: The figure shows the corner plot of the CKG model at 68% ( [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: FIG. 4: The figure shows the evolution of [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

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