Abstract
In the present work, we study the quantum cosmology description of two Friedmann-Robertson-Walker models in the presence of a stiff matter perfect fluid and a negative cosmological constant. The models differ from each other by the constant curvature of the spatial sections, taken to be either positive or zero. We work in the Schutz’s variational formalism, quantizing the models and obtaining the appropriate Wheeler-DeWitt equations. In these models there are bound states. Therefore, we compute, for each one, the discrete energy spectrum and the corresponding eigenfunctions. After that, we use the eigenfunctions in order to construct wave packets and evaluate the time-dependent expectation values of the scale factors. Each model shows bounded oscillations for the expectation value of the scalar factor, which is never zero, which can be interpreted as an initial indication that these models may not have singularities at the quantum level.
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Acknowledgements
G. Oliveira-Neto thanks Fundação de Amparo à Pesquisa do Estado de Minas Gerais, FAPEMIG, for partial financial support. E.V. Corrêa Silva (Researcher of Conselho Nacional de Desenvolvimento Científico e Tecnológico, CNPq, Brazil), G.A. Monerat, C. Neves and L.G. Ferreira Filho thank CNPq and Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro, FAPERJ, for partial financial support. E.V. Corrêa Silva and G.A. Monerat thank Universidade do Estado do Rio de Janeiro, UERJ, for the Prociência grant, via FAPERJ.
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Appendix: Checking the Solutions
Appendix: Checking the Solutions
In general, the matricial equation (23) is not satisfied exactly by the eigenvalues and eigenvectors obtained. Actually, one expects that the difference between its r.h.s. and l.h.s. be a vector with small but non-vanishing components. We may use the largest (in absolute values) of those components as a measure of error of our approximate solution. (The same number of digits used in spectrum calculations should be used, of course.) Besides, distinct eigenvalues/eigenvectors may yield distinct errors, as shown in Fig. 7: the horizontal axis correspond to the index of the levels, and the vertical axis represents the logarithm of the corresponding error.
By limiting the number of levels used in the wave packet, we have kept this error under ≈1.6×10−11.
If N functions are used as basis functions for the wave packet (15), we obtain the N×N matricial equation Eq. (23). We have used N=250 in our paper; if we compare the eigenvalues for that situation to those obtained for, say, N=260, we observe, for the first 250 eigenvalues, the percental variations shown in Fig. 8.
Then again, the lower the eigenvalues, the better the convergence. The first 18 levels show variations of less than 1.9 %.
We have also checked the orthogonality of approximate eigenfunctions. The results are shown in Fig. 9. Once more, the lower the eigenvalues, the better the orthogonality. Similar results are obtained when we compare the 2nd, 3rd, etc. levels to the upper levels.
Finally, we have also observed the behavior of the norm of the packet. In Fig. 10 the norm as a function of time is shown for wave packets obtained for the equiprobable superposition of the 18 lowest levels, for N=250; for the case k=0, variations in the norm are smaller than 10−12, whereas for k=1 variations in the norm are smaller than 10−11. Similarly, Fig. 11 shows the norm as a function of time is shown for wave packets obtained for the equiprobable superposition of the 25 lowest levels, for N=250; for the case k=0, variations in the norm are smaller than 10−12, whereas for k=1 variations in the norm are smaller than 10−11.
The results of the checkings mentioned above give us good confidence in our results. The choice of the 18 lowest levels was made in order to keep the accuracy of our results within satisfactory values.
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Oliveira-Neto, G., Monerat, G.A., Corrêa Silva, E.V. et al. Quantization of Friedmann-Robertson-Walker Spacetimes in the Presence of a Cosmological Constant and Stiff Matter. Int J Theor Phys 52, 2991–3006 (2013). https://doi.org/10.1007/s10773-013-1590-7
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DOI: https://doi.org/10.1007/s10773-013-1590-7