Analytical Solutions to the Wheeler-DeWitt Equation in Rosen-Lagrangian Cosmology via the Eisenhart Lift

Published 31 May 2026 in gr-qc | (2606.01500v1)

Abstract: The Rosen Lagrangian framework promotes the cosmological constant to a scale-factor-dependent quantity, $Λ(a)=Λ{0}a^λ$, thereby providing a dynamical dark energy scenario for $λ\neq 0$. In the special case $λ=0$, the model naturally reduces to the standard $Λ$CDM cosmology. Within this framework, the conformal Killing equations are employed to determine the conformal factor $\mathcal{F}(a)$, which is expressed in terms of the effective potential $V{\rm eff}$ and its derivative $V'_{\rm eff}$. Furthermore, the Eisenhart lift formalism introduces an additional field $χ$, allowing the cosmological dynamics to be reformulated through a purely kinetic lifted action. This geometrical construction provides a powerful approach to quantum cosmology by transforming the Wheeler-DeWitt equation into a tractable form that admits analytic solutions. Such solutions are particularly relevant in cosmological epochs dominated by the cosmological constant, including both the inflationary era of the early Universe and the late-time accelerated expansion. Consequently, this framework offers a promising avenue for connecting geometrical methods, quantum cosmology, and dynamical dark energy within a unified description.

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Summary

The paper develops analytic solutions for the Wheeler-DeWitt equation by applying the Eisenhart lift to a Rosen-Lagrangian cosmological model.
It employs a geometric reformulation that extends the minisuperspace and uses Bessel functions to solve the quantum cosmology equations.
The results predict cyclic, bouncing universe scenarios and impose constraints on the dynamical cosmological constant parameter (λ > -2).

Analytical Solutions to the Wheeler-DeWitt Equation in Rosen-Lagrangian Cosmology via the Eisenhart Lift

Introduction

This paper undertakes a geometrical reformulation of quantum cosmology, focusing on the Rosen-Lagrangian cosmological framework where the cosmological constant is parameterized as a scale-factor-dependent field, $\Lambda(a) = \Lambda_0 a^{\lambda}$ . The central methodological advance is the employment of the Eisenhart lift, elevating the dynamical system to a 2D minisuperspace via an auxiliary cyclic field $\chi$ . This extension renders the cosmological Lagrangian purely kinetic, facilitating analytical solutions for the Wheeler-DeWitt (WDW) quantum cosmological equation. The work analyzes both the classical and quantum consequences, the structure of the conformal factor via conformal Killing equations, and the emergence of cyclic or bouncing universes dependent on the parameter choice for $\lambda$ . Additionally, explicit analytic solutions in terms of Bessel functions for the WDW equation are derived, elucidating the quantum structure of such cosmologies.

Rosen-Lagrangian Formalism and the Eisenhart Lift

The Rosen-Lagrangian framework is characterized by promoting the cosmological constant $\Lambda$ to a dynamical quantity with explicit scale factor dependence, $\Lambda(a) = \Lambda_0 a^{\lambda}$ . This approach naturally interpolates between standard $\Lambda$ CDM cosmology ( $\lambda=0$ ) and a broader class of dynamical dark energy models, with associated matter degrees of freedom strictly in the energy density sector. The effective potential and the resulting Hamiltonian reproduce the canonical Friedmann and acceleration equations, while permitting broader phenomenology for $\lambda \neq 0$ .

The Eisenhart lift transforms the 1D minisuperspace trajectory (in $a$ ) onto a 2D configuration space $(a,\chi)$ , where the Rosen Lagrangian is recast into a purely kinetic term:

$\chi$ 0

Here, $\chi$ 1 denotes the effective potential and $\chi$ 2 is an introduced mass scale. The auxiliary field $\chi$ 3 is cyclic, directly encoding a conserved quantity into the formalism.

The configuration space is manifestly a 2D Riemannian manifold with the lift metric:

$\chi$ 4

Geodesic analysis and the explicit calculation of Christoffel symbols show that the dynamical system is governed by coupled equations where the Eisenhart condition $\chi$ 5 ensures consistency with the Friedmann framework.

Conformal Killing Structure and Conservation Laws

A distinctive feature of the Eisenhart-lifted framework is the transparent extraction of underlying symmetries via conformal Killing equations. The conformal factor is determined as

$\chi$ 6

This structure directly relates the evolution of the scale factor to the underlying geometry of configuration space. The explicit analytic form of the Killing vectors and the corresponding conserved charges elucidate integrable sub-classes and enable construction of hidden symmetries, potentially relating the Eisenhart-lifted system to broader classes characterized by Noether symmetries.

The behavior of the universe (cyclic, re-collapsing, or bouncing) is dictated by $\chi$ 7, with the maximum scale factor given by $\chi$ 8, and the cyclic period connected to the Eisenhart lift constant of motion. For $\chi$ 9, the universe undergoes cyclic evolution with an oscillation period compatible with observationally inferred upper bounds from quantum memory universe considerations.

Quantum Cosmology: Wheeler-DeWitt Equation and Analytic Solutions

The Eisenhart lift significantly simplifies the canonical quantization of the theory. In the lifted space, the Wheeler-DeWitt equation assumes the form (with the Laplace-Beltrami operator and vanishing quantum correction due to $\lambda$ 0):

$\lambda$ 1

Separation of variables yields an ODE for $\lambda$ 2, which upon a variable transformation reduces to a Bessel equation; the general solution is expressed as:

$\lambda$ 3

where $\lambda$ 4, $\lambda$ 5 encodes the quantum number associated with the cyclic field $\lambda$ 6, and $\lambda$ 7, $\lambda$ 8 are elementary Bessel functions.

The parameter $\lambda$ 9 thus governs the quantum (semiclassical) structure of cosmological evolution, affecting oscillation frequency and the semiclassical limit. For increasingly positive $\Lambda$ 0 (or $\Lambda$ 1), more rapid oscillations in the WDW wavefunction are observed, indicating a robust classical correspondence at large $\Lambda$ 2. In contrast, for negative $\Lambda$ 3, quantum gravitational effects dominate at small scale factor.

Figure 1: Oscillatory behavior of the Wheeler-DeWitt wavefunction solution as a function of the scale factor $\Lambda$ 4 for several values of $\Lambda$ 5 ( $\Lambda$ 6), with increasing $\Lambda$ 7 driving more rapid oscillations.

This structure underlies the assertion that the existence, normalization, and physical relevance of quantum cosmological states are contingent on $\Lambda$ 8, i.e., $\Lambda$ 9.

Physical Implications and Prospective Developments

The formalism provides a precise analytical mapping between geometry, symmetry, and quantum cosmological states in models with dynamical dark energy. The Eisenhart lift completes the geometric structure of the Rosen-Lagrangian cosmology, indicating that cyclic and oscillatory solutions can be generically realized with suitable parameter choices. The WDW equation's analytic tractability in this setting exposes the impact of dynamical cosmological terms on quantum gravitational boundary conditions and the emergence of semiclassical epochs.

The results directly inform questions on cosmic recurrence times, the quantization of periodic cosmological models, and scenarios where the universe undergoes nonsingular bounces. The construction is extendable: coupling to additional fields (e.g., scalar-tensor gravity, non-minimal coupling) or modified gravities would generalize the structure of $\Lambda(a) = \Lambda_0 a^{\lambda}$ 0 and might produce richer symmetry algebras and more diverse quantum structures.

From an integrability perspective, interplay between Eisenhart and Noether symmetries is suggested as a key route for classifying exact cosmological solutions. Direct analytic correspondence between conserved charges in the Eisenhart-lifted theory and observable cosmological parameters remains an avenue for exploration, with potential to constrain dynamical $\Lambda(a) = \Lambda_0 a^{\lambda}$ 1 models via cosmological surveys.

Conclusion

This work establishes that the Eisenhart lift facilitates a transparent and tractable analytic approach to quantum cosmology in Rosen-Lagrangian models with a dynamical cosmological term. The formalism unifies geometric, symmetry, and quantization methods, yielding cyclic and bouncing cosmologies, and explicit Wheeler-DeWitt wavefunctions. Physical and mathematical consistency demands $\Lambda(a) = \Lambda_0 a^{\lambda}$ 2, placing significant constraints on models with varying dark energy. Future research should address extended symmetry structures, matter couplings, and observational consequences in more general dynamical gravity scenarios.