Modified gravity from Weyl connection and the $f(R,\cal{A})$ extension

Emmanuel N. Saridakis; Gerasimos Kouniatalis

REVIEW 3 major objections 3 minor 2 cited by

Modified gravity from Weyl connection and the $f(R,\cal{A})$ extension

T0 review · 3 major / 3 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper claims that promoting the Weyl vector to a dynamical field in Weyl geometry produces ghost-free modified gravity theories, with the simplest class reproducing ΛCDM and richer classes yielding a dynamical dark energy of…

desk verdict A Weyl-connection construction with a genuinely new f(R~,A) action, but the Class II field equations are algebraically inconsistent with Class III and the ghost-free claim for the general case rests on an unjustified analogy. read the letter →

arxiv 2411.14380 v2 pith:2CDKQ5RU submitted 2024-11-21 gr-qc astro-ph.COhep-th

Gerasimos Kouniatalis , Emmanuel N. Saridakis This is my paper · ORCID

classification gr-qcastro-ph.COhep-th PACS 04.50.Kd98.80.-k95.36.+x

keywords Weylgeometrymodifiedgravityconnectiondarkenergyghost-freeOstrogradskyinstabilityf(RA)cosmology

topics Dark Energy

open problems Dark Energy

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

This paper tries to establish that Weyl geometry—in which the connection carries an extra vector field, the Weyl vector—can serve as the basis for modified gravity. The simplest action built only from the Weyl-connection Ricci scalar reproduces general relativity, but promoting the Weyl vector to a dynamical field, with a kinetic term and general functions of its trace, gives gravitational theories whose field equations are second order and therefore free of Ostrogradsky ghosts, the instabilities that usually come with higher-derivative equations. The same is claimed for the most general extension $f(\tilde R,\mathcal A)$, where the higher-derivative terms are attributed to the usual scalaron. Applied to cosmology, these theories produce an effective dark-energy sector of geometrical origin: the simplest class recovers $\Lambda$CDM with an effective cosmological constant, while richer classes yield a dynamical dark energy whose equation-of-state parameter can be phantom-like and can cross the phantom divide. A sympathetic reader would care because this offers a purely geometrical origin for dark energy and a late-time mechanism that could raise $H_0$.

What carries the argument

The load-bearing object is the Weyl connection, $\tilde\Gamma^\lambda_{\mu\nu}=\Gamma^\lambda_{\mu\nu}-(A_\mu\delta^\lambda_\nu+A_\nu\delta^\lambda_\mu-A^\lambda g_{\mu\nu})$, whose non-metricity is $\tilde\nabla_\mu g_{\alpha\beta}=2A_\mu g_{\alpha\beta}$. The identity that carries the argument is the relation between the Weyl and Levi-Civita Ricci scalars, $\tilde R=R+2(d-1)\nabla_\nu A^\nu-(d-1)(d-2)A_\mu A^\mu$, which in $d=4$ reads $\tilde R=R+6\nabla_\nu A^\nu-6A_\mu A^\mu$. This identity guarantees that the new degrees of freedom enter with at most first derivatives, so the field equations stay second order. The auxiliary structure is the trace $\mathcal A\equiv A_\mu A^\mu$, used to build the functions $f(\mathcal A)$, $h(\mathcal A)$, and finally $f(\tilde R,\mathcal A)$, together with the kinetic term $-\frac14 F_{\mu\nu}F^{\mu\nu}$ that makes the Weyl vector dynamical.

What would settle it

Take the quadratic truncation $f(\tilde R,\mathcal A)=\alpha \tilde R^2+\beta\mathcal A$ together with the $-\frac14 F_{\mu\nu}F^{\mu\nu}$ kinetic term and compute the Hamiltonian in flat space. If the constraint algebra leaves a negative kinetic mode, or if the linearized metric field equation still contains fourth-order time derivatives after the conformal transformation used in the paper, the ghost-free claim for the general class fails.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that the Weyl-connection Ricci scalar differs from the Levi-Civita one by terms containing at most first derivatives of the Weyl vector, with $\tilde R = R + 6\nabla_\mu A^\mu - 6A_\mu A^\mu$ in four dimensions. Because of this identity, an action built from $\tilde R$ plus at most first-derivative couplings of $A_\mu$ leads to second-order equations for both the metric and the Weyl field. The paper therefore upgrades the Weyl vector to a genuine dynamical degree of freedom—adding $-\frac14 F_{\mu\nu}F^{\mu\nu}$, a potential $f(\mathcal A)$, and couplings $h(\mathcal A)$—and obtains theories with three extra propagating modes beyond general relativity; in the classes linear in $\tilde R$ it identifies these theories with the $L_3$ subclass of generalized Proca theories. For the most general $f(\tilde R,\mathcal A)$ action it claims the additional higher-derivative terms signal a scalaron and can be removed by a conformal transformation, so that class is also ghost free. In cosmology, Class II yields an effective cosmological constant $\Lambda_{\rm eff}=-C/2$ and hence $\Lambda$CDM, while Class III gives a conserved dynamical dark-energy sector; the explicit example with $h(\mathcal A)=\beta/\mathcal A$ and $f(\mathcal A)=\gamma$ reproduces the matter-then-dark-energy thermal history, puts the deceleration-to-acceleration transition at $z\approx0.6$, and gives a phantom dark-energy equation of state.

Load-bearing premise

The assumption everything rests on is that the higher-derivative terms in the general $f(\tilde R,\mathcal A)$ field equations are harmless—just the usual extra scalar mode of $f(R)$ gravity that a conformal transformation can remove—even though this theory also couples that scalar to the Weyl vector through derivatives; no Hamiltonian check is given.

Editorial extensions

If this is right

In Class II the Weyl-field equation forces $f(\mathcal A)=6\mathcal A+C$, which turns the extra terms into an effective cosmological constant $\Lambda_{\rm eff}=-C/2$; the Friedmann equations then coincide with $\Lambda$CDM.
In Class III the effective dark-energy sector is conserved and dynamical, with an equation-of-state parameter that can be quintessence-like, phantom-like, or cross the phantom divide depending on $f(\mathcal A)$ and $h(\mathcal A)$.
The explicit model $h(\mathcal A)=\beta/\mathcal A$, $f(\mathcal A)=\gamma$ reproduces the sequence of matter and dark-energy eras, places the deceleration-acceleration transition at $z\approx0.6$, and yields a phantom $w_{\rm DE}$ today.
The general $f(\tilde R,\mathcal A)$ action adds three propagating vector modes plus the scalaron, and the paper claims these extra modes do not introduce Ostrogradsky ghosts.

Reading between the lines

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

The paper leaves the ghost-free status of $f(\tilde R,\mathcal A)$ at the level of an argument from $f(R)$ conformal rescalings; a Hamiltonian or full constraint-algebra check is the natural next step, and until it is done the claim for the most general class should be treated as open.
Because Classes II and III fit inside the $L_3$ subclass of generalized Proca theory, independent bounds on vector-tensor theories—gravitational-wave speed, solar-system tests, cosmological perturbations—can be imported to restrict $f(\mathcal A)$ and $h(\mathcal A)$; the paper does not carry out that projection.
The phantom behavior found in the worked example is exactly the kind of late-time effect that can raise $H_0$; fitting SN~Ia, BAO, CMB and $H(z)$ data would show whether the parameter region easing the Hubble tension remains observationally viable.

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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

Quick take: the f(R~,A) class is new but the paper as written has a load-bearing algebraic error in the Class II field equations and an unsupported ghost-freeness claim for the general case. Treat the equations with suspicion until corrected.

What's genuinely there: the Weyl-geometry identity (11) is standard and clearly presented. The idea of upgrading the Weyl vector to a dynamical field and building actions from R~, f(A), and eventually f(R~,A) is a reasonable way to generate vector-tensor theories from the connection. Class I correctly reduces to general relativity. The acknowledgement that Classes II and III fall within generalized Proca (L3) is honest, and the f(R~,A) class does appear to go beyond that, so there is a new formal kernel here. The cosmology section is a standard FLRW application, and the specific example is worked out consistently.

The problem: compare Class II and Class III. Set h=0 in (21); the action is identical to Class II. But (22) with h=0 contains the Maxwell stress tensor (1/2)F_{\nu\alpha}F^{\alpha\mu}, while (17) does not. Instead (17) has a term (1/4)F_{\alpha\beta}\nabla^\alpha A^\beta that is not produced by varying -1/4F^2. So the paper's own equations contradict each other for the same action. That is load-bearing, not a typo-level annoyance.

The ghost-free claim for f(R~,A) is also not established. Eq (26) is full of third and fourth derivative terms, including \nabla_\alpha\nabla_\nu\nabla_\mu A^\alpha, \nabla^2 R, etc. The defense is the f(R) scalaron analogy, but with f depending on both R~ and A, a conformal transformation generates derivative couplings between the scalaron and the vector; the analogy is not automatic. No Hamiltonian analysis or kinetic-matrix check is given. The A equation being second order does not control the metric sector. This is an assertion, not a proof.

The Lambda-CDM recovery is also weaker than advertised. f(A)=6A+C gives an effective cosmological constant with C a free integration constant; that is just putting Lambda in by hand in different notation. The specific example imposes Omega_DE0 and Omega_m0 and fixes beta and gamma, so the resulting w_DE behavior is an illustration, not a prediction.

Who is this for? People working on modified gravity and Weyl geometry might want to see the f(R~,A) construction and the cosmological setup. But the paper needs correction before it can be relied on. I would send it to a referee -- the new kernel deserves scrutiny, and an expert can quickly verify the algebra and request the missing Hamiltonian analysis -- but I would not cite it in its present form.

Recommendation: send to peer review, but expect major revision.

Referee Report

3 major / 3 minor

Summary. The manuscript constructs modified gravity theories based on a Weyl connection, in which the Weyl vector A_mu is promoted to a dynamical field. It presents four classes of actions: Class I uses only the Weyl-connection Ricci scalar and is shown to reduce to general relativity; Class II adds f(A) and a Maxwell kinetic term; Class III adds a derivative self-coupling h(A)A_mu A_nu grad_tilde_mu A_nu; Class IV is the general f(tilde R, A) extension. The paper claims that the resulting metric and Weyl field equations are second order and free from Ostrogradsky ghosts, and it applies the theories to FLRW cosmology, obtaining an effective cosmological constant in Class II and a dynamical dark-energy sector in Class III, with a specific numerical example of the dark-energy equation-of-state parameter.

Significance. The construction is geometrically motivated and, if correct, would provide a new class of modified gravity theories with a vector degree of freedom emerging from the connection rather than being added by hand. The Weyl-connection identities in Section II.A and the recovery of GR in Class I are standard and correctly presented, and the proposed relation to generalized Proca theories is plausible. However, the central claims are not currently supported: the Class II metric field equations are algebraically inconsistent, the f(tilde R, A) metric field equations contain higher-derivative terms that the paper does not eliminate, and the claimed ghost-freeness rests on an unproved conformal-transformation analogy. Because the cosmological applications inherit these problems, the paper in its present form does not establish its headline results.

major comments (3)

[II.B.2, Eqs. (16)-(17)] The Class II field equations do not follow from action (15). Varying the Maxwell term -1/4 F_mu_nu F^mu_nu with respect to the metric produces the stress-energy tensor T^A_mu_nu = F_mu_alpha F_nu^alpha - (1/4) g_mu_nu F^2, which should appear in K^mu_nu. Equation (17) contains no such term and instead includes a term (1/4) F_alpha_beta grad^alpha A^beta delta^mu_nu that is not a metric variation of any term in (15). The inconsistency is confirmed by Eq. (22) of Class III, which does contain the standard Maxwell stress term (1/2) F_nu_alpha F^alpha_mu. The field equations for Class II therefore need to be re-derived before the cosmological results based on them can be trusted.
[II.B.4, Eq. (26) and following] The claim that the f(tilde R, A) theory is second order and ghost free is contradicted by Eq. (26), which contains explicit fourth-order metric terms such as grad_nu grad_mu R, g_mu_nu grad^2 R, grad_alpha grad_nu grad_mu A^alpha, and (grad R)^2. The paper asserts that these can be eliminated by a conformal transformation, but no such transformation is exhibited and no Hamiltonian or perturbative stability analysis is provided. In pure f(R) gravity the standard scalaron argument works because the conformal factor depends only on the Ricci scalar; here f_tilde R depends also on A, so the transformation would generate new derivative couplings between the scalar and vector sectors with no demonstrated canonical structure. The abstract and Section IV repeat the ghost-free claim, but as written it is an unsupported assertion.
[III.A.1, Eqs. (29)-(31)] The statement that Class II 'recovers Lambda-CDM' conflates a model choice with a prediction. Equation (29), f(A) = 6A + C, is obtained by imposing the cosmological Weyl-field equation, but f(A) is an arbitrary function in the action; restricting it to this linear form is a selection of the theory, not a consequence of the geometry. The integration constant C then enters as Lambda_eff = -C/2. Similarly, the specific example in Section III.B fixes beta and gamma by hand and imposes Omega_DE0 and Omega_m0 from data, so the resulting w_DE(z) behavior in Figs. 1 and 2 is an illustrative fit rather than a falsifiable prediction.

minor comments (3)

[II.B.4, Eq. (26)] The term '1/4 g_mu_nu F_alpha_beta grad^alpha_beta' is not well defined; the contracted index structure should be written out explicitly.
[Throughout] There are several typographical issues, including 'form now on' for 'from now on', the misspelling 'ans¨atze', and unrendered LaTeX tokens in the axis labels of Figs. 1 and 2 in the provided version, which make the figures difficult to read.
[III.B] The text says the model reproduces the thermal history of the Universe, but the support is a single numerical example with imposed present-day density parameters; the wording should make clear that this is an existence demonstration rather than a cosmological fit.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central derivations are direct field-equation consequences, and the ΛCDM recovery uses a free integration constant rather than a fitted or self-referential input.

full rationale

I walked the derivation chain and found no step in which a claimed prediction or first-principles result is equivalent by construction to an input. Class I genuinely reduces to general relativity because variation with respect to Aμ gives Aμ=0. Class II's field equations (16)-(19) are obtained by direct variation of action (15). The ΛCDM branch follows from the Weyl-field equation in the form f′(A)=6, integrating to f(A)=6A+C, and then Λeff is defined as −C/2. That C is an arbitrary integration constant of the chosen branch, not a quantity fitted to data or predicted from the theory; the statement that the model 'recovers ΛCDM' is therefore an existence/equivalence statement for a free parameter, not a circular reduction. Class III and the specific example are explicit toy models with β and γ chosen by hand, and imposing Ω_DE0≈0.7 as an initial condition is standard cosmological practice; the resulting wDE(z) is an output of the chosen model. Class IV's equations (26)-(27) are varied directly from action (25); the claim that the higher-derivative metric terms can be eliminated by a conformal transformation is an unsupported analogy and a stability/ghost-analysis risk, especially because the paper itself, in Sec. IV, postpones perturbative studies to 'separate projects.' However, an unsupported assertion is not circular: the paper does not use the ghost-free conclusion as an input to derive the field equations. The self-citations present are contextual reviews or applications in other frameworks (f(T), f(Q,C), etc.) and are not load-bearing for the Weyl-connection construction. Under the stated rules, this warrants score 0.

Assumptions & free parameters 4 free parameters · 4 assumptions · 1 invented entities

The central claims rest on arbitrary functions f(A), h(A), and f(R~,A), on hand-chosen constants (C, beta, gamma), and on imposed present-day density parameters. The 'effective cosmological constant' in Class II is an integration constant, not a derived value. The dynamical Weyl vector is a known geometric object, but its promotion to an independent field with arbitrary potentials is an additional modeling step with no independent observational handle.

free parameters (4)

Integration constant C = arbitrary; Lambda_eff = -C/2
In Class II, f(A)=6A+C and C acts as an effective cosmological constant; the value is not predicted.
beta = 0.1 in Fig. 1; 0, 0.1, 0.15 in Fig. 2
Coupling constant in the specific example h(A)=beta/A; chosen by hand.
gamma = -1 in Fig. 1; -0.5, -1, -2 in Fig. 2
Constant in f(A)=gamma; chosen by hand.
Omega_DE0, Omega_m0 = approximately 0.7, 0.3
Imposed from observational data to integrate the cosmological model.

assumptions (4)

domain assumption The matter action depends only on the metric and matter fields, not on the Weyl vector A_mu.
Stated in Section II.B.1 before Eq. (12); if matter coupled to A_mu, the field equations and conservation laws would change.
domain assumption Weyl integrability A_mu = partial_mu phi reduces the vector degrees of freedom from four to three.
Used in Section II.B.2 for the degree-of-freedom count, but the action does not impose this condition, so the count is not derived from the action.
ad hoc to paper Arbitrary functions f(A), h(A), and f(R~,A) are admissible even when they break Weyl gauge invariance, justified by EFT.
Section II.B.4 final paragraph; this allows the specific potentials and couplings chosen in the paper.
ad hoc to paper A conformal transformation eliminates the extra scalar degree of freedom without introducing Ostrogradsky ghosts in the vector sector.
Invoked in Section II.B.4 after Eq. (27); not proven for the vector-tensor case.

invented entities (1)

Dynamical Weyl vector A_mu
purpose: Provides the extra degrees of freedom that generate the effective dark energy sector.
The Weyl vector is a standard geometric object, but promoting it to a propagating field with arbitrary potential and couplings is a modeling choice; no mass, coupling strength, or observational signature is predicted.

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Cite this review

Pith. "Pith review of Modified gravity from Weyl connection and the $f(R,\cal{A})$ extension." pith.science (2026). https://pith.science/paper/2CDKQ5RU

@misc{pith2026241114380,
  author       = {Pith},
  title        = {Pith review of: Modified gravity from Weyl connection and the $f(R,\calA)$ extension},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2CDKQ5RU}},
  note         = {Machine review of arXiv:2411.14380}
}

abstract

We use Weyl connection and Weyl geometry in order to construct novel modified gravitational theories. In the simplest case where one uses only the Weyl-connection Ricci scalar as a Lagrangian, the theory recovers general relativity. However, by upgrading the Weyl field to a dynamical field with a general potential and/or general couplings constructed from its trace, leads to new modified gravity theories, where the extra degrees of freedom arise from the Weyl field. Additionally, since the Weyl-connection Ricci scalar differs from the Levi-Civita Ricci scalar by terms up to first derivatives of the Weyl field, the resulting field equations for both the metric and the Weyl field are of second order, and thus the theory is free from Ostrogradsky ghosts. Finally, we construct the most general theory, namely the $f(\tilde{R},\cal{A})$ gravity, which is also ghost free. Applying the above classes of theories at a cosmological framework we obtain an effective dark energy sector of geometrical origin. In the simplest class of theories we are able to obtain an effective cosmological constant, and thus we recover $\Lambda$CDM paradigm, nevertheless in more general cases we acquire a dynamical dark energy. These theories can reproduce the thermal history of the Universe, and the corresponding dark energy equation-of-state parameter presents a rich behavior.

Figures

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Forward citations

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Reference graph

Works this paper leans on

131 extracted references · 21 canonical work pages · cited by 2 Pith papers

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and ( 2) trans- forms as X → B (x)wX then its Weyl-weight is w). Nev- ertheless, note that although the Riemann tensor is anti- symmetric in the last two indices (as the Riemann tensor corresponding to the Levi-Civita connection), it lacks the antisymmetry of the ﬁrst two indices and the interchange symmetry of the index pairs. Concerning the Ricci ten- s...
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Class I Following the standard lore, a simplest choice would be S = 1 16πG ∫ d4x√−g ˜R + Sm, (12) where ˜R is the Ricci scalar corresponding to the Weyl connection given in ( 11), and Sm is the matter action. Performing variation with respect to the metric we obtain Rµ ν − 1 2 Rδµ ν + 3A2δµ ν − 6Aµ Aν = 8πGT µ ν , (13) where T µ ν is the usual energy-mome...
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Class III One can consider a further extension of the above class, by considering of theories arising from the gravitational action S = 1 16πG ∫ d4x√−g [ ˜R + h(A)Aµ Aν ˜∇µ Aν +f (A) − 1 4 Fµν F µν ] , (21) where h(A) is another arbitrary function of A. Varying the total action S + Sm with respect to the metric we obtain the ﬁeld equations ( 16), but now ...
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Class II Having the above in mind, we proceed by upgrading the Weyl ﬁeld to a dynamical one, and we include in the Lagrangian its kinetic term, namely − 1 4 Fµν F µν . Addi- tionally, using Aµ we can immediately construct a new scalar, namely its trace A ≡ Aµ Aµ , and then extend the Lagrangian by considering arbitrary functions f (A) (and thus this class...
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and ( 2) trans- forms as X → B (x)wX then its Weyl-weight is w). Nev- ertheless, note that although the Riemann tensor is anti- symmetric in the last two indices (as the Riemann tensor corresponding to the Levi-Civita connection), it lacks the antisymmetry of the ﬁrst two indices and the interchange symmetry of the index pairs. Concerning the Ricci ten- s...

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Class I Following the standard lore, a simplest choice would be S = 1 16πG ∫ d4x√−g ˜R + Sm, (12) where ˜R is the Ricci scalar corresponding to the Weyl connection given in ( 11), and Sm is the matter action. Performing variation with respect to the metric we obtain Rµ ν − 1 2 Rδµ ν + 3A2δµ ν − 6Aµ Aν = 8πGT µ ν , (13) where T µ ν is the usual energy-mome...

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Class III One can consider a further extension of the above class, by considering of theories arising from the gravitational action S = 1 16πG ∫ d4x√−g [ ˜R + h(A)Aµ Aν ˜∇µ Aν +f (A) − 1 4 Fµν F µν ] , (21) where h(A) is another arbitrary function of A. Varying the total action S + Sm with respect to the metric we obtain the ﬁeld equations ( 16), but now ...

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Class II Having the above in mind, we proceed by upgrading the Weyl ﬁeld to a dynamical one, and we include in the Lagrangian its kinetic term, namely − 1 4 Fµν F µν . Addi- tionally, using Aµ we can immediately construct a new scalar, namely its trace A ≡ Aµ Aµ , and then extend the Lagrangian by considering arbitrary functions f (A) (and thus this class...

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