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Quantum Bianchi Type IX Cosmology in K-Essence Theory

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Abstract

We use one of the simplest forms of the K-essence theory and apply it to the anisotropic Bianchi type IX cosmological model, with a barotropic perfect fluid modeling the usual matter content. We show that the most important contribution of the scalar field occurs during a stiff matter phase. Also, we present a canonical quantization procedure of the theory which can be simplified by reinterpreting the scalar field as an exotic part of the total matter content. The solutions to the Wheeler-DeWitt equation were found using the Bohmian formulation Bohm (Phys. Rev. 85(2):166, 1952) of quantum mechanics, employing the amplitude-real-phase approach Moncrief and Ryan (Phys. Rev. D 44:2375, 1991), where the ansatz for the wave function is of the form Ψ(ℓ μ)=χ(ϕ)W(ℓ μ)\(e^{- S(\ell ^{\mu })},\), where S is the superpotential function, which plays an important role in solving the Hamilton-Jacobi equation.

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Acknowledgments

This work was partially supported by CONACYT 167335, 179881 grants. PROMEP grants UGTO-CA-3 and UAM-I-43. A.E.G. is supported by a CONACYT graduate fellowship. This work is part of the collaboration within the Instituto Avanzado de Cosmología and Red PROMEP: Gravitation and Mathematical Physics under project Quantum aspects of gravity in cosmological models, phenomenology and geometry of space-time. Many calculations where done by Symbolic Program REDUCE 3.8.

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Authors and Affiliations

  1. Departamento de Física de la DCeI de la Universidad de Guanajuato-Campus León, A.P. 143, C.P., 37150, Guanajuato, México

    Abraham Espinoza-García & J. Socorro

  2. Departamento de Física de la Universidad Autónoma Metropolitana, Apartado Postal 55-534, 09340, México, D.F., México

    J. Socorro & Luis O. Pimentel

Authors
  1. Abraham Espinoza-García
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  2. J. Socorro
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  3. Luis O. Pimentel
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Corresponding author

Correspondence to Luis O. Pimentel.

Appendix: Energy Momentum Tensor

Appendix: Energy Momentum Tensor

The energy momentum tensor of the K-essence scalar field has the form (5)

$$ \mathrm \mathcal{T}_{\alpha \beta}=f(\phi)\left[\mathcal{G}_{X} \phi_{,\alpha}\phi_{,\beta} + \mathcal{G}(X)g_{\alpha \beta} \right], $$
(49)

the covariant derivative of which is

$$\begin{array}{rcl} \mathrm\mathcal{T}_{\alpha\beta}^{\;\; ;\beta}&=&\left[f\left(\mathcal{G}_{X}\phi_{,\alpha}\phi_{,\beta}+\mathcal{G}g_{\alpha\beta}\right)\right]^{;\beta}\\ &=&\mathrm f\left(\mathcal{G}_{X}\phi_{,\alpha}\phi_{,\beta}+\mathcal{G}g_{\alpha\beta}\right)^{;\beta}+\left(\mathcal{G}_{X}\phi_{,\alpha}\phi_{,\beta}+\mathcal{G}g_{\alpha\beta}\right)f^{;\beta}\\ &=&\mathrm f\left[\mathcal{G}_{X}\left(\phi_{,\alpha}\phi_{,\beta}\right)^{;\beta}+\phi_{,\alpha}\phi_{,\beta}\mathcal{G}_{X}^{\;\; ;\beta}+g_{\alpha\beta}\mathcal{G}^{;\beta}\right]+\left(\mathcal{G}_{X}\phi_{,\alpha}\phi_{,\beta}+\mathcal{G}g_{\alpha\beta}\right)\frac{df}{d\phi}\phi^{,\beta}\\ &=&\mathrm f\left[\mathcal{G}_{X}\left(\phi_{,\alpha}\phi^{\;\; ;\beta}_{,\beta}+\phi_{,\beta}\phi_{,\alpha}^{\;\; ;\beta}\right)+\phi_{,\alpha}\phi_{,\beta}\mathcal{G}_{XX}X^{;\beta}+\mathcal{G}_{X}X^{;\beta}g_{\alpha\beta}\right]\\ &&+\left(\mathcal{G}_{X}\phi_{,\alpha}\phi_{,\beta}+\mathcal{G}g_{\alpha\beta}\right)\frac{df}{d\phi}\phi^{,\beta}\\ &=&\mathrm \phi_{,\alpha}\left\{f\left[\mathcal{G}_{X}\phi_{,\beta}^{\;\; ;\beta}+\mathcal{G}_{XX}X^{;\beta}\phi_{,\beta}\right]+\frac{df}{d\phi}\left[\mathcal{G}-2X\mathcal{G}_{X}\right]\right\}+f\mathcal{G}_{X}\left(\phi_{,\beta}\phi_{,\alpha}^{\;\; ;\beta}+g_{\alpha\beta}X^{;\beta}\right)\\ &=&\mathrm f\mathcal{G}_{X}\left(\phi_{,\beta}\phi_{,\alpha}^{\;\; ;\beta}+g_{\alpha\beta}X^{;\beta}\right) \\ &=&0, \end{array} $$
(50)

where the term inside the curly brackets is null according to the field equation for the scalar field.

Now, making the identifications

$$ \mathrm{P(X)}=f(\phi) \mathcal{G}(X), \quad \rho(X)=f(\phi)\left[2X\mathcal{G}_{X}-\mathcal{G}(X) \right], \quad u_{\alpha}= \frac{\phi_{,\alpha}}{\sqrt{2X}} $$
(51)

it is easy to show that

$$ \left[\rho(X)+P(X)\right]u_{\alpha}u_{\beta}+P(X)g_{\alpha\beta}=f\left(\mathcal{G}_{X}\phi_{,\alpha}\phi_{,\beta}+\mathcal{G}g_{\alpha\beta}\right)=\mathcal{T}_{\alpha\beta} $$
(52)

which establishes an analogy with the energy momentum tensor of a perfect fluid (see (6)).

In the case of the SB scalar field (\(\mathrm \mathcal {G}(X)=X\)) it can be seen from the general identification (51) that P(X) = ρ(X), which means that the energy momentum tensor (6) for this restricted model has the structure of a stiff fluid.

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Espinoza-García, A., Socorro, J. & Pimentel, L.O. Quantum Bianchi Type IX Cosmology in K-Essence Theory. Int J Theor Phys 53, 3066–3077 (2014). https://doi.org/10.1007/s10773-014-2102-0

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