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Quantum Einsteinian Cubic Cosmology

Nephtalí Eliceo Martínez Pérez, Cupatitzio Ramírez Romero

Abstract

We study Cosmological Einsteinian Cubic Gravity (CECG) arXiv:1810.08166v3 in the context of minisuperspace quantum cosmology. CECG is a modification of Einstein's gravity by cubic curvature terms that yield a nontrivial contribution to the dynamics of FRW backgrounds while keeping the Friedmann equations at second order. First, we study the Hamiltonian formulation of the effective one-dimensional FRW CECG action using Ostrogradski's canonical variables and Dirac's algorithm for constrained systems. Since the momentum $p_a$ conjugate to the scale factor is a polynomial of degree five in $\dot{a}$, we implement canonical transformations $(a,p_a)\to (A,P)$ that enable us to write the Hamiltonian constraint explicitly. Second, we perform the Wheeler-DeWitt quantization using the new canonical variables. Although FRW CECG has no extra degree of freedom besides the scale factor, its non-standard Hamiltonian yields a higher-derivative Wheeler-DeWitt equation. We obtain exact solutions for the spatially flat case, and WKB-type solutions for the spatially closed case. Finally, we consider a homogeneous scalar field $φ$ with inflationary potential and obtain WKB wave functions leading to strong correlations between coordinates and momenta.

Quantum Einsteinian Cubic Cosmology

Abstract

We study Cosmological Einsteinian Cubic Gravity (CECG) arXiv:1810.08166v3 in the context of minisuperspace quantum cosmology. CECG is a modification of Einstein's gravity by cubic curvature terms that yield a nontrivial contribution to the dynamics of FRW backgrounds while keeping the Friedmann equations at second order. First, we study the Hamiltonian formulation of the effective one-dimensional FRW CECG action using Ostrogradski's canonical variables and Dirac's algorithm for constrained systems. Since the momentum conjugate to the scale factor is a polynomial of degree five in , we implement canonical transformations that enable us to write the Hamiltonian constraint explicitly. Second, we perform the Wheeler-DeWitt quantization using the new canonical variables. Although FRW CECG has no extra degree of freedom besides the scale factor, its non-standard Hamiltonian yields a higher-derivative Wheeler-DeWitt equation. We obtain exact solutions for the spatially flat case, and WKB-type solutions for the spatially closed case. Finally, we consider a homogeneous scalar field with inflationary potential and obtain WKB wave functions leading to strong correlations between coordinates and momenta.

Paper Structure

This paper contains 9 sections, 72 equations, 11 figures.

Table of Contents

  1. Introduction
  2. FRW action
  3. Hamiltonian formulation
  4. Wheeler-DeWitt equation
  5. Pure quantum states
  6. Spatially closed universe
  7. Scalar field
  8. Quantization
  9. Conclusions

Figures (11)

  • Figure 1: The left-hand side of (\ref{['friedcanon']}) for different ranges of $\beta$.
  • Figure 2: Plot of $\Lambda^\beta(\beta)$ according to (\ref{['effla']}). Blue/purple segments correspond to $\alpha_2/\alpha_3$ in (\ref{['alphaminus']}), respectively.
  • Figure 3: Hessian determinant (\ref{['hess']}) evaluated on the two different solutions (\ref{['alphaminus']}).
  • Figure 4: $\psi(A)=\cos(R A/\hbar)$ constructed with the real roots of (\ref{['six']}). With negative $\beta$, we restrict $A\in \mathbb{R}^+$ according to the sign of the factor $(1+48 \beta \kappa^2 H^4)$ in (\ref{['AP']}) (see Fig. \ref{['gig5']}).
  • Figure 5: Exact solutions (\ref{['analytic']}) of the WDW equation with positive $\beta$. For large $A$, the amplitude decreases as $A^{-1/4}$.
  • ...and 6 more figures