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

Technical Summary: Exact Solutions in Saez-Ballester-K-essence Like Theory with Power Law Potentials

Problem Statement
The paper addresses the lack of exact classical solutions for cosmological models involving scalar fields with power-law potentials, specifically $V(\phi) = V_0 \phi^{\pm \lambda}$, within the context of K-essence and Saez-Ballester formalisms. While dynamic system methods exist, they often yield only specific or approximate solutions. The authors aim to construct a "mixed formalism" combining K-essence and Saez-Ballester theories with a scalar potential to derive exact classical and quantum solutions for a flat Friedmann-Lemaître-Robertson-Walker (FLRW) universe. The study focuses on two scenarios: quintessence ($\epsilon = 1$) and phantom fields ($\epsilon = -1$), utilizing a negative power-law potential $V(\phi) = V_0 \phi^{-\lambda}$.

Methodology
The authors employ a field redefinition strategy to transform the problem into a solvable form. By introducing the transformation $\phi = e^{\varphi}$, the original negative power-law potential $V(\phi) = V_0 \phi^{-\lambda}$ is converted into an exponential potential $V(\varphi) = V_0 e^{-\lambda \varphi}$. This transformation maps the complex Saez-Ballester-K-essence system to a known problem with established exact solutions.

The analysis proceeds in two distinct regimes:

1. Classical Regime: The authors construct the Lagrangian and Hamiltonian densities for the FLRW metric. Using the Hamiltonian formalism, they derive the canonical momenta and the Hamiltonian constraint. By solving the resulting system of differential equations without imposing specific ansätze for the scale factor or scalar field, they obtain exact time-dependent solutions for the scale factor $A(t)$ and the scalar field $\phi(t)$. The analysis is bifurcated based on the parameter $\lambda$ relative to $\sqrt{3}$ and the sign of the kinetic term $\epsilon$.
2. Quantum Regime: The Wheeler-DeWitt (WDW) equation is derived by promoting the canonical momenta to operators ($\Pi_q \to -i\hbar \partial_q$). The authors address the factor-ordering ambiguity using a semi-general ordering parameter $Q$. Through a canonical transformation of variables ($x, y$), the WDW equation is reduced to an ordinary differential equation solvable via special functions (exponential, Bessel functions $J_\nu$ and $K_\nu$).

Key Results

• Classical Solutions (Quintessence, $\epsilon=1$):

  • Case $\lambda = \sqrt{3}$: The model yields a solution exhibiting "huge inflation" where the scale factor grows exponentially. The deceleration parameter $q(t)$ asymptotically approaches $-1$, characteristic of a de Sitter phase.
  • Case $\lambda < \sqrt{3}$: The evolution is less explosive, allowing for parameter adjustments to achieve a moderate inflationary stage with appropriate e-folds, similar to standard inflationary models.
  • Case $\lambda > \sqrt{3}$: The evolution of the universe volume is extremely slow, failing to produce real inflation. The authors suggest this case is not viable for standard inflationary scenarios.

• Classical Solutions (Phantom, $\epsilon=-1$):

  • For any value of $\lambda$, the scale factor exhibits periodic growth rather than continuous expansion. The scalar field acts complementarily to the scale factor (reaching minima when the scale factor reaches maxima). The authors note the absence of singularities but highlight that the periodic nature and lack of sustained growth render this scenario potentially non-viable for describing our universe's current accelerated expansion.

• Quantum Solutions:

  • Quintessence: The wave function solutions involve exponential and Bessel functions depending on $\lambda$. For $\lambda = \sqrt{3}$, the probability density shows a spontaneous collapse direction for the scale factor $A$ while increasing along the scalar field $\phi$. This behavior is interpreted as the scalar field remaining as a "cosmic background" while the universe unfolds.
  • Phantom: The probability density is highly sensitive to constants, exhibiting "fuzzy subuniverses" of lower probability. The authors note that for certain parameter ranges, the order of the Bessel function becomes imaginary, requiring a wave packet approach.

Significance and Claims
The paper claims to provide a consistent classical and quantum description of cosmological evolution generated by this specific class of K-essence models. The primary contribution is the demonstration that a mixed Saez-Ballester-K-essence formalism with a power-law potential can be exactly solved by mapping it to an exponential potential problem.

The authors assert that in this formalism, the scalar field persists as a cosmic background throughout the universe's evolution, a feature glimpsed from the quantum solution perspective where the probability density decays with the scale factor. They conclude that the $\lambda = \sqrt{3}$ case in the quintessence scenario offers a viable model for late-time accelerated expansion (de Sitter phase), whereas the phantom scenario and the $\lambda > \sqrt{3}$ quintessence case appear less viable for describing standard cosmological inflation due to periodic or slow evolution, respectively. The work suggests that specific subcases in the quantum solution space (dictated by parameters like $Q$ and $\lambda$) determine the likelihood of producing an inflationary universe.