Notes on the quantization of FRW model in the presence of a cosmological constant and radiation

Abstract

In the present work, we use the formalism of quantum general relativity in order to quantize a FriedmannRobertson-Walker model in the presence of a negative cosmological constant and radiation. The model has spatial sections with positive constant curvature. The wave-function of the model satisfies a Wheeler-DeWitt equation, for the scale factor, which has the form of the Schrödinger's equation for the quartic anharmonic oscillator. We find the eigenvalues and eigenfunctions by using a method first developed by Chhajlany and Malnev. After that, we use the eigenfunctions in order to construct wave-packets for evaluating the time-dependent, expected value of the scale factor. We find that, the expected value of the scale factor oscillates between maximum and minimum values. Since the scale factor never vanishes, we conclude that the model does not have a singularity.One of the motivations for the quantization of cosmological models was that of avoiding the initial Big Bang singularity. Since the pioneering work in quantum cosmology due to DeWitt [1], workers in this field have been attempting to prove that quantum cosmological models entail only regular space-times. An important contribution to this issue was given by Hartle and Hawking [2], who proposed the no-boundary boundary condition, which selects only regular space-times to contribute to the wave-function of the Universe, derived in the path integral formalism. Therefore, by construction, the no-boundary wave-functions are everywhere regular and predict a non-singular initial state for the Universe. Using that boundary condition, in certain particular cases the noboundary wave-function can be explicitly computed [2][3][4]. Another way by which one may compute the wave-function of the Universe is by directly solving the Wheeler-DeWitt equation [1]. The wave-function of the Universe for some important models have been computed using this approach [5][6][7][8][9][10].Several important theoretical results and predictions in quantum cosmology have been obtained with a negative cosmological constant [11], [12] and [3]. Besides that, we think it is important to understand more about such models which represent bound Universes (analogous to uni-dimensional atoms, in the present situation).In the present paper, we use the formalism of quantum cosmology in order to quantize a Friedmann-Robertson-Walker model in the presence of a negative cosmological constant and radiation. The radiation is treated by means of the variational formalism developed by Schutz [13]. The model has spatial sections with positive constant curvature. The wavefunction of the model satisfies a Wheeler-DeWitt equation, for the scale factor, which has the form of the Schrödinger's equation for the quartic anharmonic oscillator. We find the eigenvalues and eigenfunctions by using a method first developed by Chhajlany and Malnev. After that, we use the eigenfunctions in order to construct wave-packets for evaluating the time-dependent, expected value of the scale factor. We find that the expectatio...

Quantum Cosmology with Many Fluids and the Choice of Cosmological Time

“…3(d) is similar to the behavior of the expected value in the case of a FLRW type cosmological model with positive curvature, negative cosmological constant and radiation, studied in Ref. [27].…”

Comment on “Quantization of Friedmann-Robertson-Walker spacetimes in the presence of a negative cosmological constant and radiation”

“…(8) is not yet fully specified, since the eigenfunctions φ n (a) are known up to a phase factor which can be chosen arbitrarily. By looking at the coefficients in Table II of [1] and of Tables II-VII of [2] we see that these wave functions have been chosen by Monerat and collaborators to be positive in a small neighborhood of a = 0 + . The wave packet obtained with this prescription for k = 1 corresponds at t = 0 to a state sharply localized around a = 0.1, which quickly spreads at later times according to the Heisenberg principle.…”

Quantização de sistemas hamiltonianos via método de diferenças finitas

“…Recentemente utilizamos o método de diferenças finitas no esquema de CrankNicolson [11] para obter a quantização de modelos cosmológicos homogêneos e isotrópicos com diferentes fontes de matéria [12][13][14][15]. Métodos espectrais [16] também têm se mostrado importantes ferramentas na determinação do espectro de energia de sistemas físicos com soluções analíticas desconhecidas.…”