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Stability analysis of dilaton-inspired scalar field within the geometrical trinity of gravity

T0 review · 3 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read A dilaton-inspired scalar field with an exponential potential can drive late-time cosmic acceleration as a stable attractor in curvature and torsion gravity, while the symmetric-teleparallel counterpart with the non-coincident gauge shows…

desk verdict A competent phase-space comparison across the trinity, but the non-metricity no-attractor result is narrower than the conclusions claim. read the letter →

arxiv 2504.19245 v1 pith:KZOWNQ3A submitted 2025-04-27 gr-qc astro-ph.CO

Youri Carloni , Orlando Luongo , Andronikos Paliathanasis This is my paper · ORCID
classification gr-qcastro-ph.CO PACS 98.80.-k95.36.+x98.80.Jk04.50.kd
keywords dilatonscalarfieldgeometricaltrinityofgravityteleparallelsymmetric-teleparallelnon-metricitydynamicalsystemsstabilitydarkenergyattractorsexponentialpotentialcosmology
topics 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 asks whether a string-inspired dilaton scalar field, rewritten in the form of a Brans–Dicke Lagrangian with an exponential potential, can survive as a dynamical dark-energy model when gravity is formulated in each of the three ways that are classically equivalent at the level of actions: through curvature (general relativity), through torsion (teleparallel gravity), and through non-metricity (symmetric-teleparallel gravity). Working in a spatially curved Friedmann–Robertson–Walker universe, the authors convert each theory's field equations into an autonomous dynamical system and locate the late-time attractor points. They find that in general relativity and teleparallel gravity the field has stable attractors that drive accelerated expansion and mimic a cosmological constant at late times, with the open-universe branch better behaved. In symmetric-teleparallel gravity, however, the non-coincident gauge introduces an extra scalar field whose non-linearities prevent a complete linear stability analysis, and for the parameter values examined no attractor points exist. If correct, this means the geometrical trinity is not dynamically equivalent once a dilaton is non-minimally coupled, and only the curvature- and torsion-based versions remain viable background-level dark-energy candidates.

What carries the argument

The load-bearing mechanism is the reduction of all three theories to a single dilaton-inspired Brans–Dicke Lagrangian, $$S_{\Upsilon}=\int $d^{4}$x\,\sqrt{-g}\,$e^{{\phi}}$\left(\frac{\Upsilon}{2}-\frac{\omega_0}{2}\,$g^{{\mu\nu}}$\phi_{,\mu}\phi_{,\nu}-\hat{V}(\phi)\right),$$ where $\Upsilon$ stands for the Ricci scalar $R$, the torsion scalar $T$, or the non-metricity scalar $Q$, and $\hat{V}(\phi)=V(\phi)e^{-\phi}$. Combined with an exponential potential, which makes the auxiliary variable $\lambda=\hat{V}_{,\phi}/\hat{V}$ constant, each theory's Friedmann equations can be recast as an autonomous system in dimensionless variables such as $x=\dot{\phi}/\sqrt{H^2+|k|a^{-2}}$ and $\eta=H/\sqrt{H^2+|k|a^{-2}}$; stability is then read off from the eigenvalues of the Jacobian at the critical points. In the non-metricity case the non-coincident gauge forces $\gamma=1/\dot{\Psi}$, introducing an extra field $\Psi$ whose strong nonlinearities prevent the same complete linear analysis and, on the sampled parameter slice, produce no attractor.

What would settle it

Numerically integrate the non-metricity autonomous system on a fine grid of $(\omega_0,\lambda)$, including $\lambda\neq 0$ and $\omega_0$ outside $\{-1,0,1\}$, and search for any critical point whose Jacobian eigenvalues all have negative real part together with a negative deceleration parameter; finding even one such attractor would falsify the paper's no-attractor conclusion for the non-metricity branch.

Watch

Extended reading notes

Core claim

The paper's central discovery is a stability hierarchy across the geometrical trinity for the dilaton-inspired scalar field. After the non-minimal coupling $F(\varphi)=\varphi$, the kinetic function $\omega_0/\varphi$, and the field redefinition $\varphi=e^{\phi}$ reduce the actions to an exponential-potential Brans–Dicke form, the curvature and torsion autonomous systems admit late-time attractors at which the deceleration parameter approaches the cosmological-constant value; the preferred attractor is $P_{R,0}$ in general relativity and $P_{T,0}$ in teleparallel gravity, while other attractors are discarded because they correspond to contracting universes or unphysical regions. In the symmetric-teleparallel case the non-coincident gauge makes spatial curvature a dynamical variable and adds a second scalar degree of freedom, so the Jacobian analysis can only be carried out for the special choices $\lambda=0$ and $\omega_0\in\{-1,0,1\}$; on that slice every existing critical point is a saddle or unstable, and no attractor is found. The paper concludes that the dilaton model behaves as an effective cosmological constant in the curvature and torsion formulations, while the non-metricity formulation is dynamically disfavored for the chosen parameter sets.

Load-bearing premise

The claim that the non-metricity branch has no stable late-time attractors rests on testing only a narrow set of parameter values and on one particular way of rewriting the extra field; if a wider scan or another rewriting turns up a stable accelerating solution, the claim collapses.

Editorial extensions

If this is right

  • The dilaton-inspired scalar field is a viable late-time dark-energy candidate in general relativity and teleparallel gravity, reaching a state that reproduces a cosmological constant.
  • In the open-universe branch the model is more robust: for the parameter choices tested, the deceleration parameter stays in the physically allowed region across the whole cosmic history.
  • In teleparallel gravity most attractor configurations require $\omega_0<0$, meaning the dilaton behaves as a phantom-like field there, while viable quintessence-like behavior is also possible at $P_{T,0}$ for $\omega_0>0$.
  • In symmetric-teleparallel gravity no attractor point exists for $\lambda=0$ and $\omega_0\in\{-1,0,1\}$, so that formulation cannot support late-time accelerated expansion as a stable background solution on this slice.
  • The three geometric formulations, though action-level equivalent, are not dynamically equivalent once the non-minimal dilaton coupling is present.

Reading between the lines

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

  • If the no-attractor result for the non-metricity branch persists under a broad scan of $(\omega_0,\lambda)$, symmetric-teleparallel geometry would be ruled out as a stable background for this dilaton dark-energy model, demoting the trinity equivalence to a purely kinematic statement.
  • A natural extension is to test non-exponential potentials or add the vector-field coupling the authors mention; a potential with a stable minimum could reintroduce attractors in the non-metricity branch, marking the no-attractor finding as specific to the exponential-potential slice.
  • The stability comparison could be tied to observations by computing the predicted $w_0w_a$ parameters in each attractor region and matching them against current baryon-acoustic-oscillation datasets, turning the phase-space classification into a model-selection test.
  • Because the non-metricity analysis treats curvature as a dynamical variable, a fuller numerical search for stable spirals or limit cycles, rather than only fixed points, would clarify whether the system has any late-time attractor at all.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper studies a dilaton-inspired scalar field, recast as a Brans-Dicke-like theory, in the three gravitational frameworks of the geometrical trinity: general relativity (curvature), teleparallel gravity (torsion), and symmetric teleparallel gravity (non-metricity). Using a non-flat FRW metric and an exponential potential, the authors construct autonomous dynamical systems, find critical points, and perform linear stability analyses for each framework. For GR and teleparallel gravity they report attractor points that can yield late-time acceleration and mimic a cosmological constant, with stability regions summarized in parameter-space plots. For the non-metricity case they work in the non-coincident gauge, introduce an auxiliary scalar field, and report that for the sampled parameters there are no attractor points, concluding that the non-metricity dilaton model is dynamically disfavored. The central comparison across the trinity, and especially the negative Q-sector result, is the main claim of the paper.

Significance. If the conclusions were fully established, the paper would provide a useful dynamical-systems comparison of the same dilaton-inspired scalar action across curvature, torsion, and non-metricity formulations in non-flat cosmologies. The GR and teleparallel analyses are internally coherent: eigenvalues are tabulated, phase portraits are provided, and the parameter regions are discussed. The paper also makes a concrete, falsifiable statement about which geometric formulation supports stable late-time acceleration for this model. However, the key negative result for symmetric teleparallel gravity rests on a small parameter sample and on a gauge reparameterization whose coverage of the phase space is not demonstrated; consequently the trinity hierarchy claimed in the conclusions is currently stronger than the evidence supports. The authors build on their own prior phase-space work, and the manuscript does not include machine-readable code or numerical data, which limits independent verification of the region plots.

major comments (3)
  1. [Sec. IV.C, Table VI, and Sec. V] The statement in Sec. V that 'we found no attractor solutions' for the non-metricity framework is not supported by the evidence in Table VI. Table III defines critical points P_Q,2 through P_Q,5 whose existence conditions allow general lambda and omega0, but Table VI computes eigenvalues only for lambda = 0 and omega0 in {-1, 0, 1}; moreover P_Q,4 and P_Q,5 do not exist at lambda = 0 and are therefore never tested. Since the paper's claimed hierarchy among the three geometries depends on the Q-sector having no attractors, the conclusion should be restricted to the sampled parameter set, as the abstract correctly does, or the analysis should be extended to the lambda != 0 existence regions of Table III.
  2. [Sec. II.C, Eqs. (19)-(20)] The reduction of the non-metricity scalar to the Lagrangian in Eq. (20) uses the substitution gamma = 1 / dot-Psi, but the manuscript does not demonstrate that this transformation is invertible on the phase space of interest. If dot-Psi vanishes on relevant trajectories, or if the map from gamma to Psi is not one-to-one, the autonomous system studied in Sec. IV.C may not represent all non-coincident-gauge Q cosmologies, and the negative stability result could be an artifact of this ansatz. This issue should be addressed by proving the invertibility over the relevant domain or by explicitly stating the substitution as a restricted gauge ansatz with its domain of validity.
  3. [Sec. IV.C, Eqs. (32)-(34)] The reduction to the three-dimensional Q system uses the square-root constraint in Eq. (34), which fixes the sign of y and requires division by z. This branch choice restricts the phase space and excludes z = 0, yet the no-attractor conclusion is drawn from this reduced system. The authors should justify that the chosen branch is representative, or show that the excluded regions cannot contain stable critical points; otherwise the Q-sector conclusion remains conditional on this additional modeling assumption.
minor comments (5)
  1. [Table VI] The parameter is typeset as w0 in Table VI but as omega0 elsewhere; please use consistent notation throughout.
  2. [References] References [42] and [89] appear to be the same paper (Carloni and Luongo, Class. Quant. Grav. 42, 075014, 2025); please consolidate the duplicate citation.
  3. [Sec. III.A and Sec. V] There is a typo 'verly early stages' in Sec. III.A, and the conclusion that an open universe 'appeared more robust' is based on a small number of selected parameter sets; please qualify this statement accordingly.
  4. [Figs. 2 and 4] The deceleration-parameter plots would be easier to verify if each curve were labeled with its parameter set and with the attractor point toward which the chosen initial condition converges.
  5. [Appendix A] The stability regions plotted in Figs. 5 and 6 are presented without analytic boundary curves; providing the explicit conditions or a reproducibility statement would strengthen the reliability of the classification.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the stability analysis is derived from explicit actions and displayed autonomous systems; free parameters are inputs, not fitted outputs.

full rationale

The paper's derivation chain is a parameter-space stability analysis, not a prediction of an externally fitted quantity. The actions (1)-(6), the field equations (10), (15), (21), the dimensionless variables (22)-(23) and (30)-(31), and the autonomous systems (24), (27), and (32) are all written out explicitly; the critical points (Tables I-III) and eigenvalues (Tables IV-VI) are computed from those displayed systems. The free parameters λ and ω0 are model inputs, and the 'cosmological constant at late times' statements are read off from the deceleration parameter at the computed attractors for selected parameter values, not fitted to data. The non-metricity no-attractor result is explicitly qualified in the abstract ('for the chosen set of free parameters') and is obtained by evaluating the eigenvalues in Table VI for ω0 in {-1, 0, 1} and λ = 0; the broader wording in Section V is a generalization of that computation, and any concern about the three-point scan or the γ = 1/Ψdot reparameterization is a modeling or scope risk, not a circular reduction. The self-citations to the authors' earlier phase-space papers provide conventions and the non-coincident-gauge expression for Q (Eq. 19), all of which are reproduced in the text; the central conclusion does not reduce to an unverified self-citation. No step was found in which an output quantity is identical by construction to an input parameter, a fitted value is renamed a prediction, or an ansatz is adopted solely through a self-citation.

Assumptions & free parameters 2 free parameters · 4 assumptions · 1 invented entities

The central stability results rest on a handful of modeling choices: an exponential potential, constant lambda, the Brans-Dicke representation of the dilaton, and for Q a specific non-coincident gauge parameterization. None of these are benchmarked against external data, so the ledger reflects the parameter dependence and auxiliary field introduced in the Q sector.

free parameters (2)
  • omega0 (Brans-Dicke coupling) = chosen values: -8, -6, -4, -3, -1, 0, 1, 4, 8 depending on scenario
    The kinetic coupling of the scalar field; the paper varies it to find attractor regions. For Q, only -1, 0, 1 with lambda = 0 are used.
  • lambda (exponential potential slope) = chosen values: 0, 1, 2, -1, -2 depending on scenario
    Defines V(phi) proportional to e^(lambda phi) after field redefinition; constant lambda simplifies the autonomous system. The no-attractor Q result uses only lambda = 0.
assumptions (4)
  • standard math The FRW metric with k = +/-1 and the cosmological field equations are derived from the given actions using the standard variational procedure.
    Section II uses the minisuperspace Lagrangian approach; no nonstandard assumptions.
  • domain assumption The exponential potential makes lambda = V_phi / V constant, which is required to close the autonomous system.
    The stability analysis is only performed for this single potential family; the conclusions do not extend to general potentials.
  • domain assumption The non-coincident gauge is represented by the auxiliary field Psi through gamma = 1/Psi_dot (Eq. 20).
    This is a gauge/parameterization choice for Q; the paper notes the coincident gauge has ghosts, but the completeness of this parameterization is not proven.
  • standard math Linear stability classification via eigenvalues of the Jacobian assumes the Hartman-Grobman theorem applies at the critical points (no purely imaginary or zero eigenvalues except where stated).
    Standard dynamical systems theory; the paper follows Copeland et al. [36].
invented entities (1)
  • Auxiliary scalar field Psi
    purpose: To express the non-coincident gauge connection component gamma in the non-metricity scalar Q as gamma = 1/Psi_dot, yielding a local Lagrangian L_Q in Eq. (20).
    It is a mathematical reparameterization of the connection, not a physical particle; no independent observational or experimental handle is suggested.

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Pith. "Pith review of Stability analysis of dilaton-inspired scalar field within the geometrical trinity of gravity." pith.science (2026). https://pith.science/paper/KZOWNQ3A

@misc{pith2026250419245,
  author       = {Pith},
  title        = {Pith review of: Stability analysis of dilaton-inspired scalar field within the geometrical trinity of gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KZOWNQ3A}},
  note         = {Machine review of arXiv:2504.19245}
}
read the original abstract

We investigate the dynamics of the dilaton-inspired scalar field, formally rewritten by means of a Brans-Dicke Lagrangian, within the framework of \emph{geometrical trinity of gravity}. In this respect, we perform a stability analysis by adopting a non-flat Friedmann-Robertson-Walker (FRW) metric and considering the well-established exponential potential in three distinct gravitational frameworks: general relativity, teleparallel gravity, and symmetric-teleparallel gravity. By comparing the scalar field behaviors across these theories, we highlight the role of curvature, torsion, and non-metricity in shaping cosmic evolution. Our analysis reveals that, both in general relativity and teleparallel gravity, the dilaton-inspired field can drive the accelerated expansion of the universe, effectively behaving as cosmological constant at late times. In contrast, within the symmetric teleparallel gravity scenario, performing a complete linear stability analysis is prevented by the use of the non-coincident gauge. Nevertheless, the latter paradigm introduces complexity into the autonomous system, resulting in a structurally different analysis. For general relativity and teleparallel scenarios, we remark the regions of attractor solutions and unphysical domains in which we do not expect the viability of our dilaton-inspired Lagrangian. However, within the framework of symmetric-teleparallel gravity, the stability analysis reveals no attractor points for the chosen set of free parameters. In support of these findings, physical conclusions, kinematical studies, and consequences on Friedmann dynamics are thus explored.

Figures

Figures reproduced from arXiv: 2504.19245 by the authors.

Figure 1
Figure 1. FIG. 1: Phase-space portrait graphs for scalar field with the deceleration parameter in general relativity framework. [PITH_FULL_IMAGE:figures/full_fig_p014_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: The deceleration parameters are computed under the assumption of a closed or an open universe in general relativity [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Phase-space portrait graphs for scalar field with the deceleration parameter in teleparallel gravity framework. [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4: The deceleration parameters are computed under the assumption of a closed or an open universe in teleparallel [PITH_FULL_IMAGE:figures/full_fig_p017_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Free parameter regions to establish scalar field stability in general relativity for both closed and open universes. [PITH_FULL_IMAGE:figures/full_fig_p024_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Free parameter regions to establish scalar field stability in teleparallel gravity for both closed and open universes. [PITH_FULL_IMAGE:figures/full_fig_p025_6.png]

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Reviewed August 16, 2026 · model on record in the stance chip above.