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arxiv: 2606.20894 · v1 · pith:WSMD3DQXnew · submitted 2026-06-18 · 🌀 gr-qc · hep-th

Exact solutions using power law scalar potential in the Saez-Ballester-K-essence like theory

J. Socorro , A. Gil-Ocaranza , Ximena L\'opez-Mujica , Cesar Aar\'on Pacheco-V\'azquez This is my paper

Pith reviewed 2026-06-26 16:07 UTC · model grok-4.3

classification 🌀 gr-qc hep-th
keywords K-essence cosmologySaez-Ballester potentialexact solutionsHamiltonian formalismWheeler-DeWitt equationde Sitter expansionFLRW modelaccelerated expansion
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The pith

A field redefinition maps a K-essence model with power-law potential to an exactly solvable FLRW cosmology, producing classical and quantum solutions with late-time de Sitter expansion.

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

The paper shows that redefining the scalar field from ϕ to varphi transforms the equations of a K-essence model built on a negative power-law Saez-Ballester potential into a mathematical structure identical to a previously solved FLRW cosmological model. This mapping permits exact classical solutions for the scale factor and scalar field to be derived in the Hamiltonian formalism. The resulting dynamics describe late-time accelerated expansion in which the deceleration parameter approaches -1, matching a de Sitter phase. Exact solutions of the associated Wheeler-DeWitt equation are also obtained, furnishing a consistent quantum description in which the scalar field serves as a cosmic background.

Core claim

By means of a suitable field redefinition from ϕ to varphi, the resulting field equations acquire a mathematical structure analogous to that of a previously solved FLRW cosmological model. This correspondence allows us to obtain exact classical solutions for both the scale factor and the scalar field within the Hamiltonian formalism. The resulting cosmological dynamics exhibits a late-time accelerated expansion, with the deceleration parameter approaching the asymptotic value q→ -1, characteristic of a de Sitter phase. At the quantum level, the corresponding Wheeler-DeWitt equation is derived and exact quantum solutions are obtained.

What carries the argument

The field redefinition from ϕ to varphi that maps the K-essence equations onto the structure of a previously solved FLRW model, enabling exact solvability in the Hamiltonian formalism.

If this is right

  • Exact classical expressions exist for both the scale factor and the scalar field.
  • The deceleration parameter q approaches -1 asymptotically, producing de Sitter-like expansion at late times.
  • Exact solutions to the Wheeler-DeWitt equation exist and describe the quantum evolution of the model.
  • The scalar field functions as a cosmic background in the quantum description.

Where Pith is reading between the lines

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

  • The same redefinition technique could be tested on other K-essence potentials to check whether exact solvability appears more generally.
  • The classical solutions could be matched to observational Hubble data to constrain the power-law index in the potential.
  • The quantum solutions might be used to extract expectation values for the scale factor at early times.

Load-bearing premise

The field redefinition from ϕ to varphi maps the K-essence equations to an analogous structure of a previously solved FLRW model while preserving physical equivalence and allowing exact solvability in the Hamiltonian formalism without introducing inconsistencies.

What would settle it

Substituting the derived exact solutions back into the original unredefined field equations and checking whether they satisfy the equations identically would falsify the claimed correspondence if they do not.

Figures

Figures reproduced from arXiv: 2606.20894 by A. Gil-Ocaranza, Cesar Aar\'on Pacheco-V\'azquez, J. Socorro, Ximena L\'opez-Mujica.

Figure 1
Figure 1. Figure 1: FIG. 1: Huge inflation in the volume function, eq. (28), and the corresponding scalar field [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Deceleration parameter using the scale factor function, eq. (28), and all the values that [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Moderate inflation in the volume function, eq. (34), and the corresponding scalar field [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: In this subcase, the evolution of the volume of the universe is very slow; there is no [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: There is no real growth of the scale factor, since evolution occurs in positive chunks in [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Deceleration parameter using the scale factor function, eq. ( [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: the probability density of the equation (57), where we choose the values Ψ [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: the probability density of the equation (57), where we choose the values Ψ [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: the probability density of the equation (60), where we choose the values [PITH_FULL_IMAGE:figures/full_fig_p016_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: the probability density of the equation (60), where we choose the values [PITH_FULL_IMAGE:figures/full_fig_p016_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: the probability density of the equation (63), where we choose the values [PITH_FULL_IMAGE:figures/full_fig_p017_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12: the probability density of the equation (69), where we choose the values [PITH_FULL_IMAGE:figures/full_fig_p018_12.png] view at source ↗
read the original abstract

We investigate a K-essence like cosmological model whose scalar-field potential is constructed from a negative power-law S\'aez--Ballester potential. By means of a suitable field redefinition from $\phi$ to $\varphi$, we show that the resulting field equations acquire a mathematical structure analogous to that of a previously solved Friedmann-Lema\^itre-Robertson-Walker (FLRW) cosmological model. This correspondence allows us to obtain exact classical solutions for both the scale factor and the scalar field within the Hamiltonian formalism. The resulting cosmological dynamics exhibits a late-time accelerated expansion, with the deceleration parameter approaching the asymptotic value $q\rightarrow -1$, characteristic of a de Sitter phase. At the quantum level, the corresponding Wheeler-DeWitt (WDW) equation is derived and exact quantum solutions are obtained. These results provide a consistent classical and quantum description of the cosmological evolution generated by this class of K-essence models. In this formalism, the scalar field remains as a cosmic background where the universe unfolds, which is glimpsed from the quantum solution perspective.

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

1 major / 2 minor

Summary. The manuscript investigates a K-essence cosmological model whose scalar potential derives from a negative power-law Saez-Ballester term. A field redefinition ϕ → varphi is used to map the field equations onto the structure of a previously solved FLRW model, permitting exact classical solutions for the scale factor and scalar field within the Hamiltonian formalism. The resulting dynamics shows late-time acceleration with q → −1 (de Sitter asymptote). The corresponding Wheeler-DeWitt equation is derived and solved exactly at the quantum level.

Significance. If the redefinition preserves the Hamiltonian structure, the work supplies rare exact classical and quantum solutions for this class of models. Such explicit solutions can serve as benchmarks and the mapping technique may extend to related scalar-tensor or K-essence systems.

major comments (1)
  1. [Field redefinition and Hamiltonian formalism] Field redefinition (abstract and Hamiltonian section): the mapping ϕ → varphi must be shown to be a canonical transformation in minisuperspace, i.e., the symplectic form p_ϕ dϕ equals p_ϕ dϕ plus a total derivative so that the Hamiltonian constraint and its quantization remain equivalent. The manuscript provides no explicit verification of Poisson brackets or the symplectic structure; without it the claimed exact solutions and WDW wave functions are not guaranteed to describe the original theory.
minor comments (2)
  1. Add the specific reference for the 'previously solved FLRW cosmological model' to which the equations are mapped.
  2. Clarify the explicit form of the redefinition and the resulting Hamiltonian constraint with numbered equations so that the correspondence can be checked directly.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for the constructive comment regarding the field redefinition. We address the major point below and will revise the manuscript accordingly to strengthen the presentation.

read point-by-point responses

  1. Referee: [Field redefinition and Hamiltonian formalism] Field redefinition (abstract and Hamiltonian section): the mapping ϕ → varphi must be shown to be a canonical transformation in minisuperspace, i.e., the symplectic form p_ϕ dϕ equals p_ϕ dϕ plus a total derivative so that the Hamiltonian constraint and its quantization remain equivalent. The manuscript provides no explicit verification of Poisson brackets or the symplectic structure; without it the claimed exact solutions and WDW wave functions are not guaranteed to describe the original theory.

    Authors: We agree that an explicit verification of the canonical character of the transformation is required for rigor. The redefinition ϕ → varphi is a point transformation in the configuration space of the minisuperspace variables. Because the original Lagrangian density is rewritten in terms of the new field varphi while preserving the form of the kinetic term (up to a redefinition of the potential), the transformation induces a corresponding redefinition of the conjugate momentum such that the symplectic form is preserved up to an exact differential. Consequently the Hamiltonian constraint remains equivalent. Nevertheless, the manuscript does not contain the explicit computation of the Poisson brackets or the pull-back of the symplectic two-form. We will add a short subsection (or appendix) in the Hamiltonian formalism section that (i) computes the new momenta, (ii) verifies {varphi, p_varphi} = 1, and (iii) shows that the transformed constraint is identical to the original one. With this addition the subsequent classical solutions and the Wheeler-DeWitt quantization are guaranteed to apply to the original theory. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on explicit field redefinition to an independent prior model

full rationale

The central step is a field redefinition ϕ → varphi that maps the K-essence + Saez-Ballester equations onto the structure of a previously solved FLRW model, after which exact solutions are imported. This is a standard reduction technique rather than a self-definitional loop or fitted-input prediction. The abstract and description present the prior FLRW model as external; no load-bearing self-citation chain, uniqueness theorem, or ansatz smuggling is quoted. The Hamiltonian and WDW steps follow from the mapped equations, not by renaming the input. The result is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on the assumption that a field redefinition produces an exactly solvable analogous system; no free parameters or invented entities are identifiable from the abstract.

axioms (1)
  • domain assumption The field redefinition from ϕ to varphi maps the K-essence equations to those of a standard FLRW model without loss of physical content.
    Invoked in the abstract as the step that enables exact classical and quantum solutions.

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discussion (0)

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Reference graph

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