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arXiv:2009.05187 · Physics Letters B

Wheeler-DeWitt equation rejects quantum effects of grown-up universes as a candidate for dark energy

Dongshan He, Qing-yu Cai

Published Aug 31, 2020• 5 citations
arXivHTMLPDF

Abstract

In this paper, we study the changes of quantum effects of a growing universe by using Wheeler-DeWitt equation (WDWE) together with de Broglie-Bohm quantum trajectory approach. From WDWE, we obtain the quantum modified Friedmann equations which have additional terms called quantum potential compared to standard Friedmann equations. The quantum potential governs the behavior of the early universe, providing energy for inflation, while it decreases rapidly as the universe grows. The quantum potential of the grown-up universe is much smaller than that required for accelerating expansion. This indicates that quantum effects of our universe cannot be treated as a candidate for dark energy.

DOI
10.1016/j.physletb.2020.135747

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In this paper, we study the changes of quantum effects of a growing universe by using Wheeler-DeWitt equation (WDWE) together with de Broglie-Bohm quantum trajectory approach. From WDWE, we obtain the quantum modified Friedmann equations which have additional terms called quantum potential compared to standard Friedmann equations. The quantum potential governs the behavior of the early universe, providing energy for inflation, while it decreases rapidly as the universe grows. The quantum potential of the grown-up universe is much smaller than that required for accelerating expansion. This indicates that quantum effects of our universe cannot be treated as a candidate for dark energy.
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We construct a Bohmian quantum cosmological model for a spatially flat Friedmann-Robertson-Walker universe filled with a single scalar field whose potential provides a unified description of cold dark matter and dark energy at the background level. Starting from the Einstein-Hilbert action supplemented by a scalar field, we derive the minisuperspace Lagrangian and the associated canonical Hamiltonian formulation. By means of a nontrivial canonical transformation, the minisuperspace dynamics is mapped into that of a two-dimensional hyperbolic oscillator with a fixed frequency ratio, rendering the Wheeler-DeWitt equation exactly solvable by separation of variables. The resulting Wheeler-DeWitt solutions are expressed in terms of parabolic cylinder functions and are parametrised by a continuous separation constant, reflecting the constrained nature of the theory and the absence of a standard Schrödinger time parameter. Adopting the de Broglie-Bohm formulation, we derive deterministic guidance equations in minisuperspace and construct a well-defined Bohmian Hubble parameter directly in terms of the pilot-wave phase. Finally, we present a Wheeler-DeWitt-derived toy wave function for which the Bohmian trajectories and the associated cosmological expansion history can be obtained analytically, reproducing the late-time ΛCDM behaviour while exhibiting quantum modifications at earlier epochs.
The Einstein's general relativity is formulated in the Hamiltonian form for a spatially Bat, isotropic and homogeneous universe. Subsequently, we perform the canonical quantization procedure to the Hamiltonian to obtain the Wheeler—DeWitt equation. Solving the Wheeler—DeWitt equation and employing the de Broglie—Bohm interpretation to the wave function of the universe, we obtain a new version of spatially Bat Friedmann equation for the early universe where the scale factor of the universe is taken to be sufficiently small.
Friedmann equations play a central role in cosmology for describing the evolution of the universe. Initially, the Friedmann equations were derived by Alexander Friedmann in 1922. He derived the equations from Einstein’s field equations for a universe which is spatially homogeneous and isotropic. The equations were again derived from Newtonian mechanics by Milne. However in this paper, we derive the spatially flat Friedmann equation from Wheeler-DeWitt equation. We apply the Bohm-de Broglie interpretation to the wave function of universe. In addition, we also set a condition to the wave function, where a large scale factor of the universe is needed.   Key words: Canonical quantum cosmology, Friedmann equations, Wheeler-DeWitt equation.
We argue that the Copenhagen interpretation of quantum mechanics cannot be applied to quantum cosmology. Among the alternative interpretations, we choose to apply the Bohm-de Broglie interpretation of quantum mechanics to canonical quantum cosmology. For minisuperspace models, we show that there is no problem of time in this interpretation, and that quantum effects can avoid the initial singularity, create inflation and isotropize the Universe. For the general case, it is shown that, irrespective of any regularization or choice of factor ordering of the Wheeler-DeWitt equation, the unique relevant quantum effect which does not break spacetime is the change of its signature from lorentzian to euclidean. The other quantum effects are either trivial or break the four-geometry of spacetime. A Bohm-de Broglie picture of a quantum geometrodynamics is constructed, which allows the investigation of these latter structures.