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International Journal of the Physical Sciences

Derivation of the spatially flat Friedmann equation from Bohm-De Broglie interpretation in canonical quantum cosmology

Chrsquo, Ch’ng Han Siong, Shahidan Radiman

Published Feb 9, 2013• 4 citations
Publisher

Abstract

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.

DOI
10.5897/ijps2013.3819

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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.
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.
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.
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.
Abstract In quantum cosmology, one has to select a specific wave function solution of the quantum state equations under consideration in order to obtain concrete results. The simplest choices have been already explored, in different frameworks, yielding, in many cases, quantum bounces. As there is no consensually established boundary condition proposal in quantum cosmology, we investigate the consequences of enlarging known sets of initial wave functions of the Universe, in the specific framework of the Wheeler–DeWitt equation interpreted along the lines of the de Broglie–Bohm quantum theory, on the possible quantum bounce solutions which emerge from them. In particular, we show that many asymmetric quantum bounces are obtained, which may incorporate non-trivial back-reaction mechanisms, as quantum particle production around the bounce, in the quantum background itself. In particular, the old hypothesis that our expanding Universe might have arisen from quantum fluctuations of a fundamental quantum flat space-time is recovered, within a different and yet unexplored perspective.
We summarize a fully worked-out consistent histories formulation of a quantum cosmological model, and illustrate with an example calculation of the probability that the quantum universe becomes singular. Specifically, we take a complete Wheeler-DeWitt canonical quantization of a flat Friedmann-Robertson- Walker cosmological model sourced with a free massless scalar field, and construct the decoherence functional which determines which quantum histories may be consistently assigned probabilities. As an example, we exhibit the probability that the quantum universe is singular in the sense that it assumes zero volume. The classical solutions of this model are all singular; we show that the quantum solutions are as well.We comment on the crucial role played by quantum decoherence in arriving at this conclusion. © 2010 American Institute of Physics.