G. A. Monerat

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

The quantization of the Friedmann-Robertson-Walker spacetime in the presence of a negative cosmological constant was used in a recent paper to conclude that there are solutions that avoid singularities (big bang-big crunch) at the quantum level. We show that a proper study of their model does not indicate that it prevents the occurrence of singularities at the quantum level, in fact the quantum probability of such event is larger than the classical one. Our numerical simulations based on the powerful variational sinc collocation method (VSCM) also show that the precision of the results of that paper is much lower than the 20 significant digits reported by the authors.

Tunneling probability for the birth of an asymptotically de Sitter universe

In the present work, we quantize a closed Friedmann-Robertson-Walker model in the presence of a positive cosmological constant and radiation. It gives rise to a Wheeler-DeWitt equation for the scale factor which has the form of a Schrodinger equation for a potential with a barrier. We solve it numerically and determine the tunneling probability for the birth of a asymptotically DeSitter, inflationary universe, initially, as a function of the mean energy of the initial wave-function. Then, we verify that the tunneling probability increases with the cosmological constant, for a fixed value of the mean energy of the initial wave-function.

Symplectic method in quantum cosmology

E. V. Corrêa Silva∗,1 G. A. Monerat†,1 G. Oliveira-Neto‡,1 C. Neves§,1 and L. G. Ferreira Filho¶2 Departamento de Matemática e Computação, Faculdade de Tecnologia, Universidade do Estado do Rio de Janeiro, Rodovia Presidente Dutra, Km 298, Pólo Industrial, CEP 27537-000, Resende-RJ, Brazil. Departamento de Mecânica e Energia, Faculdade de Tecnologia, Universidade do Estado do Rio de Janeiro, Rodovia Presidente Dutra, Km 298, Pólo Industrial, CEP 27537-000, Resende-RJ, Brazil. (Dated: March 24, 2009)

AN EARLY UNIVERSE MODEL WITH STIFF MATTER AND A COSMOLOGICAL CONSTANT

In the present work, we study the quantum cosmology description of a Friedmann-Robertson-Walker model in the presence of a stiff matter perfect fluid and a negative cosmological constant. We work in the Schutzs variational formalism and the spatial sections have constant negative curvature. We quantize the model and obtain the appropriate Wheeler-DeWitt equation. In this model the states are bounded therefore we compute the discrete energy spectrum and the corresponding eigenfunctions. In the present work, we consider only the negative eigenvalues and their corresponding eigenfunctions. This choice implies that the energy density of the perfect fluid is negative. A stiff matter perfect fluid with this property produces a model with a bouncing solution, at the classical level, free from an initial singularity. After that, we use the eigenfunctions in order to construct wave packets and evaluate the time-dependent expectation value of the scale factor. We find that it oscillates between maximum and minimum values. Since the expectation value of the scale factor never vanishes, we confirm that this model is free from an initial singularity, also, at the quantum level.

Can noncommutativity affect the whole history of the universe

We study a classical, noncommutative (NC), Friedmann–Robertson–Walker (FRW) cosmological model. The spatial sections may have positive, negative or zero constant curvatures. The matter content is a generic perfect fluid. The initial noncommutativity between some canonical variables is rewritten, such that, we end up with commutative variables and a NC parameter. Initially, we derive the scale factor dynamic equations for the general situation, without specifying the perfect fluid or the curvature of the spatial sections. Next, we consider two concrete situations: a radiation perfect fluid and dust. We study all possible scale factor behaviors, for both cases. We compare them with the corresponding commutative cases and one with the other. We obtain, some cases, where the NC model predicts a scale factor expansion which may describe the present expansion of our universe. Those cases are not present in the corresponding commutative models. Finally, we compare our model with another NC model, where the noncommutativity is between different canonical variables. We show that, in general, it leads to a scale factor behavior that is different from our model.

Noncommutativity in the early universe

In the present work, we study the noncommutative version of a quantum cosmology model. The model has a Friedmann–Robertson–Walker (FRW) geometry, the matter content is a radiative perfect fluid and the spatial sections have zero constant curvature. In this model, the scale factor takes values in a bounded domain. Therefore, its quantum mechanical version has a discrete energy spectrum. We compute the discrete energy spectrum and the corresponding eigenfunctions. The energies depend on a noncommutative parameter β. We compute the scale factor expected value (〈a〉) for several values of β. For all of them, 〈a〉 oscillates between maxima and minima values and never vanishes. It gives an initial indication that those models are free from singularities, at the quantum level. We improve this result by showing that if we subtract a quantity proportional to the standard deviation of a from 〈a〉, this quantity is still positive. The 〈a〉 behavior, for the present model, is a drastic modification of the 〈a〉 behavior in the corresponding commutative version of the present model. There, 〈a〉 grows without limits with the time variable. Therefore, if the present model may represent the early stages of the universe, the results of the present paper give an indication that 〈a〉 may have been, initially, bounded due to noncommutativity. We also compute the Bohmian trajectories for a, which are in accordance with 〈a〉, and the quantum potential Q. From Q, we may understand why that model is free from singularities, at the quantum level.

Spectral: Solving Schroedinger and Wheeler–DeWitt equations in the positive semi-axis by the spectral method

Abstract The Galerkin spectral method can be used for approximate calculation of eigenvalues and eigenfunctions of unidimensional Schroedinger-like equations such as the Wheeler–DeWitt equation. The criteria most commonly employed for checking the accuracy of results is the conservation of norm of the wave function, but some other criteria might be used, such as the orthogonality of eigenfunctions and the variation of the spectrum with varying computational parameters, e.g. the number of basis functions used in the approximation. The package Spectra, which implements the spectral method in Maple language together with a number of testing tools, is presented. Alternatively, Maple may interact with the Octave numerical system without the need of Octave programming by the user. Program summary Program title: Spectral Catalogue identifier: AEQQ_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEQQ_v1_0.html Program obtainable from: CPC Program Library, Queen’s University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 20417 No. of bytes in distributed program, including test data, etc.: 2149904 Distribution format: tar.gz Programming language: Maple, GNU Octave 3.2.4 Computer: Any supporting Maple Operating system: Any supporting Maple RAM: About 4 Gbytes Classification: 1.9, 4.3, 4.6. Nature of problem: Numerical solution of Schrodinger-like eigenvalue equations (especially the Wheeler–DeWitt equation) in the positive semi-axis Solution method: The unknown wave function is approximated as a linear combination of a suitable set of functions, and the continuous eigenvalue problem is mapped into a discrete (matricial) eigenvalue problem Restrictions: Limitations are due to memory usage only Unusual features: The package may not work properly in older versions of Maple, due to a bug in that CAS; for that reason an interface with the GNU Octave system is provided, requiring no user intervention or Octave programming during calculations Running time: Seconds to hours, depending on the number of basis functions used and on the complexity of the potential used

Reply to "Comment on 'Quantization of FRW spacetimes in the presence of a cosmological constant and radiation' "

[gr-qc/0611029] contains a valid criticism of the numerical precisionof the results reported in a recent paper of ours [Phys. Rev. D 73, 044022 (2006)], as well as freshideas on how to characterize a quantum cosmological singularity. However, we argue that, contraryto what is suggested in the Comment, the quantum cosmological models we studied show hardlyany sign of singular behavior.

Probing singularities in quantum cosmology with curvature scalars

Abstract We provide further evidence that the canonical quantization of cosmological models eliminates the classical Big Bang singularity, using the de Broglie–Bohm interpretation of quantum mechanics. We compute the ‘local expectation value’ of the Ricci and Kretschmann scalars, for some quantum FRW models. We show that they are finite for all time.

Explorando sistemas hamiltonianos: estudo analítico

A double pendulum submitted to external torques is employed to introduce some basic fundamentals of dynamical systems theory to physics undergraduate courses, soon after the student takes the analytical mechanics discipline. This system is a good example for the introduction of such techniques. Hamiltons equations of motion indicate the existence of stationary solutions (equilibrium points) in the phase space of the model. The identification of the nature of these points allows the description of the system dynamics around their linear neighborhood. Moreover, qualitative results obtained in the linear neighborhood of a fixed point are not changed by the introduction of non-vanishing constant external torques. This work emphasizes the analysis of the linear neighborhood of the equilibrium points.