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arxiv: 2604.22247 · v2 · submitted 2026-04-24 · 🌀 gr-qc · hep-th

Integrability of Conformal Killing Vectors in the Eisenhart Lift of Scalar-Field FLRW Cosmology

Takeshi Chiba , Tsuyoshi Houri This is my paper

Pith reviewed 2026-05-08 10:50 UTC · model grok-4.3

classification 🌀 gr-qc hep-th
keywords Eisenhart liftconformal Killing vectorsFLRW cosmologyscalar field potentialintegrability conditionsdifferential equations
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The pith

The potential found in prior work is the most general local one admitting a non-trivial conformal Killing vector independent of the cyclic Eisenhart coordinate in scalar-field FLRW cosmology.

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

The paper establishes that the potentials identified previously represent the complete set of local potentials that allow a non-trivial conformal Killing vector in the Eisenhart lift of scalar-field FLRW cosmology when the vector is independent of the cyclic coordinate. A reader would care because this finishes the local classification of integrable cases under these symmetries, which can guide the search for exact solutions in cosmological models. The authors derive integrability conditions from the conformal Killing equations and reduce the determinant condition of the prolonged system to a nonlinear second-order differential equation in the variable h equal to the logarithmic derivative of the potential. Solving that equation produces a regular branch that exactly matches the earlier family and a singular branch that proves incompatible with the full equations.

Core claim

We show that the potential found in our earlier work is already the most general local potential that admits a non-trivial conformal Killing vector in the sector independent of the cyclic Eisenhart coordinate. The determinant condition of the prolonged conformal Killing equations reduces to a nonlinear second-order differential equation for h=V'/V. We solve this equation locally and find two branches. The regular branch reproduces exactly the family of potentials obtained before, while the singular branch lies on the locus where the determinant equation cannot be written locally in normal form with respect to h'' and is incompatible with the full conformal Killing equations. Hence the ansatz

What carries the argument

The nonlinear second-order differential equation for h = V'/V obtained from the determinant condition on the prolonged conformal Killing equations, whose local solutions classify admissible potentials.

If this is right

  • The family of potentials from the earlier work exhausts all local cases admitting the symmetry.
  • The singular-branch solutions to the h equation do not satisfy the complete conformal Killing equations.
  • The ansatz employed in the prior analysis is exhaustive for the sector independent of the cyclic coordinate.
  • All integrable local potentials in this Eisenhart-lifted FLRW setup are captured by the regular solutions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar determinant reductions could classify potentials in non-flat FLRW or multi-field models.
  • The classification may directly yield new exact cosmological solutions by integrating the symmetry.
  • Including dependence on the cyclic coordinate could uncover additional potentials outside the current exhaustive set.

Load-bearing premise

The determinant condition on the prolonged conformal Killing equations reduces to a local nonlinear second-order differential equation for h = V'/V whose solutions classify all admissible potentials.

What would settle it

A concrete scalar potential V whose h = V'/V fails to solve the derived differential equation yet still produces a non-trivial conformal Killing vector independent of the cyclic coordinate in the full set of equations.

read the original abstract

We study the integrability conditions of the conformal Killing equations for the Eisenhart lift of a scalar field in a flat Friedmann-Lema\^\i tre-Robertson-Walker universe. We show that the potential found in our earlier work is already the most general local potential that admits a non-trivial conformal Killing vector in the sector independent of the cyclic Eisenhart coordinate. The determinant condition of the prolonged conformal Killing equations reduces to a nonlinear second-order differential equation for $h=V'/V$. We solve this equation locally and find two branches. The regular branch reproduces exactly the family of potentials obtained before, while the singular branch lies on the locus where the determinant equation cannot be written locally in normal form with respect to $h''$ and is incompatible with the full conformal Killing equations. Hence the ansatz used in our earlier work is exhaustive.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript examines the integrability conditions of the conformal Killing equations in the Eisenhart lift of a scalar field in flat FLRW cosmology. It reduces the prolonged equations (in the sector independent of the cyclic coordinate) via a determinant condition to a nonlinear second-order ODE for h = V'/V, solves this ODE locally to obtain two branches, shows that the regular branch exactly recovers the potential family from the authors' prior work, and demonstrates that the singular branch is incompatible with the full system of equations. The central claim is that the earlier ansatz is therefore exhaustive for local potentials admitting non-trivial conformal Killing vectors.

Significance. If the reduction and branch analysis hold, the result establishes a complete local classification of admissible potentials, confirming that no other families exist in the considered sector. This strengthens the prior work by replacing an ansatz with a proof of generality and provides a concrete example of using prolongation techniques and determinant conditions to classify symmetries in cosmological models. Such exhaustiveness results are useful for identifying integrable cases and exact solutions in scalar-field cosmology.

minor comments (3)
  1. The abstract states that the singular branch 'lies on the locus where the determinant equation cannot be written locally in normal form with respect to h'''; a brief explicit statement of this locus (e.g., the condition on the coefficients that makes the leading term vanish) would improve readability without lengthening the abstract.
  2. In the derivation of the determinant condition, the transition from the prolonged conformal Killing system to the single ODE for h should be cross-referenced to the specific equations being eliminated; this would make the reduction steps easier to follow for readers unfamiliar with the prolongation procedure.
  3. The manuscript should include a short remark confirming that the regular-branch solutions satisfy the original conformal Killing equations identically (not just the reduced ODE), perhaps by direct substitution or by noting that the eliminated equations are automatically satisfied on this branch.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful summary of our manuscript and for the positive assessment of its significance. The referee correctly identifies that our work establishes the exhaustiveness of the potential family from our prior paper by solving the integrability conditions via the determinant condition on the prolonged conformal Killing equations. We appreciate the recommendation for minor revision; however, no specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation from CK equations is independent

full rationale

The paper derives a nonlinear second-order ODE for h = V'/V directly from the determinant condition on the prolonged conformal Killing equations in the cyclic-coordinate-independent sector. Local solutions are classified into regular and singular branches; the regular branch recovers the prior family while the singular branch is shown incompatible with the full system. This exhaustiveness argument is self-contained and does not reduce to a fit, self-definition, or load-bearing self-citation. The reference to earlier work merely identifies the recovered family after the independent derivation is complete.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on the standard Eisenhart lift construction for a scalar field in flat FLRW, the definition of conformal Killing vectors, and local solvability of the resulting nonlinear ODE; no new entities or fitted parameters are introduced.

axioms (2)
  • domain assumption The Eisenhart lift metric for the scalar-field FLRW system is well-defined and the conformal Killing equations can be prolonged in the usual way.
    Invoked in the setup of the integrability conditions.
  • domain assumption Local analysis in the sector independent of the cyclic coordinate is sufficient to classify all relevant potentials.
    Stated as the sector under study.

pith-pipeline@v0.9.0 · 5446 in / 1317 out tokens · 25695 ms · 2026-05-08T10:50:27.739718+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

15 extracted references · 14 canonical work pages

  1. [1]

    L. P. Eisenhart, Dynamical trajectories and geodesics, Ann. Math.30(1928) 591 [doi:10.2307/1968307]

    work page doi:10.2307/1968307 1928

  2. [2]

    Duval , author G

    C. Duval, G. Burdet, H. P. K¨ unzle and M. Perrin, “Bargmann structures and Newton-Cartan theory,” Phys. Rev. D31, 1841–1853 (1985) doi:10.1103/PhysRevD.31.1841

    work page doi:10.1103/physrevd.31.1841 1985

  3. [3]

    org/10.1103/PhysRevD.43.3907

    C. Duval, G. W. Gibbons and P. A. Horvathy, “Celestial mechanics, conformal structures, and gravitational waves,” Phys. Rev. D43, 3907–3922 (1991) doi:10.1103/PhysRevD.43.3907 [arXiv:hep-th/0512188]

    work page doi:10.1103/physrevd.43.3907 1991

  4. [4]

    Eisenhart lifts and symmetries of time-dependent systems,

    M. Cariglia, C. Duval, G. W. Gibbons and P. A. Horvathy, “Eisenhart lifts and symmetries of time-dependent systems,” Annals Phys.373, 631–654 (2016) doi:10.1016/j.aop.2016.07.033 [arXiv:1605.01932 [hep-th]]

    work page doi:10.1016/j.aop.2016.07.033 2016

  5. [5]

    Cosmological aspects of the Eisenhart–Duval lift,

    M. Cariglia, A. Galajinsky, G. W. Gibbons and P. A. Horvathy, “Cosmological aspects of the Eisenhart–Duval lift,” Eur. Phys. J. C78, no.4, 314 (2018) doi:10.1140/epjc/s10052-018-5789-x [arXiv:1802.03370 [gr-qc]]

    work page doi:10.1140/epjc/s10052-018-5789-x 2018

  6. [6]

    N. Kan, T. Aoyama, T. Hasegawa and K. Shiraishi, Eisenhart–Duval lift for minisuperspace quantum cosmology, Phys. Rev. D104(2021) 086001 [arXiv:2105.09514] [doi:10.1103/PhysRevD.104.086001]

    work page doi:10.1103/physrevd.104.086001 2021

  7. [7]

    N. Kan, T. Aoyama, T. Hasegawa and K. Shiraishi, Third quantization for scalar and spinor wave functions of the Universe in an extended minisuperspace, Class. Quant. Grav.39(2022) 165010 [arXiv:2110.06469]

    work page arXiv 2022

  8. [8]

    N. Kan, T. Aoyama and K. Shiraishi, Spinorial Wheeler–DeWitt wave functions inside black hole horizons, Class. Quant. Grav.40(2023) 165006 [arXiv

    2023

  9. [9]

    Chiba and T

    T. Chiba and T. Houri, Hidden symmetries of power-law inflation, Class. Quant. Grav.41(2024) 19LT01 [doi:10.1088/1361-6382/ad7185]

    work page doi:10.1088/1361-6382/ad7185 2024

  10. [10]

    Chiba and T

    T. Chiba and T. Houri, Eisenhart lift for scalar fields in the FLRW universe, Class. Quant. Grav.42(2025) 055003 [arXiv:2409.16325] [doi:10.1088/1361-6382/ada90d]

    work page doi:10.1088/1361-6382/ada90d 2025

  11. [11]

    L. J. Garay, J. J. Halliwell and G. A. Mena Marug´ an, Path-integral quantum cosmology: A class of exactly soluble scalar-field minisuperspace models with exponential potentials, Phys. Rev. D43(1991) 2572 [doi:10.1103/PhysRevD.43.2572]

    work page doi:10.1103/physrevd.43.2572 1991

  12. [12]

    de Ritis, G

    R. de Ritis, G. Marmo, G. Platania, C. Rubano, P. Scudellaro and C. Stornaiolo, New approach to find exact solutions for cosmological models with a scalar field, Phys. Rev. D42(1990) 1091 [doi:10.1103/PhysRevD.42.1091]. – 10 –

    work page doi:10.1103/physrevd.42.1091 1990

  13. [13]

    Cariglia, Hidden symmetries of Eisenhart-Duval lift metrics and the Dirac equation with flux, Phys

    M. Cariglia, Hidden symmetries of Eisenhart-Duval lift metrics and the Dirac equation with flux, Phys. Rev. D86(2012) 084050 [doi:10.1103/PhysRevD.86.084050]

    work page doi:10.1103/physrevd.86.084050 2012

  14. [14]

    Cariglia, Hidden symmetries of dynamics in classical and quantum physics, Rev

    M. Cariglia, Hidden symmetries of dynamics in classical and quantum physics, Rev. Mod. Phys.86(2014) 1283 [doi:10.1103/RevModPhys.86.1283]

    work page doi:10.1103/revmodphys.86.1283 2014

  15. [15]

    K. Finn, S. Karamitsos and A. Pilaftsis, Eisenhart lift for field theories, Phys. Rev. D98(2018) 016015 [doi:10.1103/PhysRevD.98.016015]. – 11 –

    work page doi:10.1103/physrevd.98.016015 2018