# Anisotropic n-dimensional quantum cosmological model with fluid

## Abstract

In the present work, we discuss the Wheeler–DeWitt quantization scheme for an n-dimensional anisotropic cosmological model with a perfect fluid in presence of a massless scalar field. We identify the time parameter using the generalization of Schutz formalism and find the wavepacket of the universe following standard Wheeler–DeWitt methodology of quantum cosmology. We also calculate the expectation values of scale factors and volume element. We show that quantized model escapes the singularity and supports bouncing universe solutions.

## 1 Introduction

Higher dimensional models had been investigated widely in the past in order to find a theory to unify gravity with other fundamental forces of physics. It started with Kaluza and Klein’s assertion that a fifth dimensions in general relativity will unify gravity with the electromagnetic field [1, 2, 3]. Further motivation of resorting to higher dimensional models came from the expectation of unifying gravity with non-Abelian gauge fields [4, 5]. Later, spacetime having more than four dimensions were motivated by 10-dimensional superstring theory and 11-dimensional supergravity theory [6, 7].

Quantum cosmology has provided a way to study the early phase of the universe. In the absence of a well accepted quantum theory of gravity, it is quite imperative to look at the quantum behaviour in individual gravitational systems. Wheeler–DeWitt equation is a widely used framework where standard quantum mechanics is used in cosmological models [8, 9, 10]. This framework of quantization suffers from various conceptual problems, which are discussed quite elaborately in the literature [11, 12, 13]. The problem of non-unitary evolution and the absence of properly oriented time are amongst them. The latter is dealt with in various ways [14, 15, 16, 17, 18, 19]. One of the ways of finding a nice time parameter is to use the evolution of the cosmic fluid following Schutz formalism [20, 21]. This method has been widely used as time parameter in quantization of cosmological models [22, 23, 24, 25]. In fact, the alleged loss of unitarity in anisotropic quantum cosmological models has been quite effectively resolved [26, 27, 28, 29, 30] using Schutz formalism. In cosmological models related to Brans–Dicke theory [31, 32], the evolution of the scalar field have been used as time parameter in order to quantize the same without adding any additional matter [33, 34, 35]. The Brans–Dicke theory of gravity has also been quantized using Schutz’s formalism [36, 37]. Similar formalism have been used by Khodadi et al. in scalar-energy dependent metric cosmology [38]. So long as a method gives rise to an oriented scalar time parameter, it is quite an effective method.

In a very recent work, Alves-Junior et al. [39] have shown the canonical quantization of n-dimensional anisotropic model coupled with a massless scalar field in the absence of any fluid where they have emphasized on natural identification of scalar field as time parameter for the evolution of quantum variables.

In the present work, we try to investigate the quantum cosmological solutions of an n-dimensional anisotropic model in which massless scalar field is minimally coupled with gravity in presence of a barotropic perfect fluid (\(P=\alpha \rho \)). The idea is to generalize the work of Alves-Junior et al. to include a fluid, as well as to generalize the recent work on anisotropic cosmologies by Pal and Banerjee [26, 27] to higher dimensions. Keeping in mind the importance of various aspects of higher dimensions, it is quite relevant to ask the questions like whether singularity-free n-dimensional quantum cosmological models realized. Also, this investigation should find if the general result, given by Pal and Banerjee [30], that it is always possible to find a self-adjoint extension of homogeneous quantum cosmologies in four dimensions.

In Sect. 2, we write Lagrangian for the gravity sector coupled with a scalar field, followed by the mention of Schutz formalism to n-dimensional models. Similar kind of generalization of Schutz fluid was done by Letelier and Pitelli [40] in the context of the quantization of an n-dimensional FLRW model. We construct the Hamiltonian using canonical transformation in Sect. 2.2. In Sect. 3, canonical quantization of the model is presented for a stiff fluid (\(\alpha =1\)). We obtain the wave packet and expectation values of scale factor and volume element. Further, we are also able to show the unitary evolution of the model with general fluid (\(\alpha \ne 1\)). This is done for an example with \(n=7\) which is mathematically tractable. In the end, in Sect. 4, we discuss our results and make some remarks.

## 2 An n-dimensional cosmological model

### 2.1 Lagrangian of the model

*P*related with density \(\rho \) by the equation of state \(P=\alpha \rho \). The metric for an n dimensional (\(n>4\)) spatially homogeneous but anisotropic cosmology is chosen as

*N*(

*t*) is the lapse function,

*a*(

*t*) is our good old usual scale factor used in flat FLRW cosmology and

*b*(

*t*) is scale factor coming from remaining \((n-4)\) dimensions. So the usual 3-space is isotropic, but the extra dimensions, though isotropic in itself, but anisotropic from the usual 3-space section.

*t*and \(V_0\) denotes \((n-1)\) dimensional volume. We have ignored the surface terms as they do not contribute to the field equations.

*S*is specific entropy, \(\zeta \) and \(\beta \) are potentials connected with rotation and \(\epsilon \) and \(\theta \) are potentials with no clear physical meaning. For n dimensional velocity vector, we can consider additional potential terms [40] like \(\zeta \) and \(\beta \) in Eq. (5) as

### 2.2 Hamiltonian of the model

*T*and \(p_T\) are related with following canonical transformation

## 3 Quantization of the model

*u*and

*v*are given as,

### 3.1 For stiff fluid \(\alpha =1\) and \(n>5\)

*k*and

*K*are constant. Here we take \(E>k+\lambda \). By superposing the function \(\psi _{\lambda ,E,k}(u,v,\phi ,T)\), the wavepacket can be formed as;

*a*,

*b*and the proper volume measure \(V= (-g)^{\frac{1}{2}}=a^3b^{n-4}\) in terms of \(\bar{u}\) and \(\bar{v}\) as,

*a*and

*b*are quite well behaved individually.

### 3.2 For stiff fluid \(\alpha =1\) and \(n=5\)

### 3.3 For fluid with \(\alpha \ne 1\)

## 4 Discussion with concluding remarks

We have found the non-singular expectation values of the scale factors and these are shown in Figs. 1, 2, 3 and 4 for different n. Expectation values of scale factors *a* and *b* coincide for \(n=7\). This is not surprising as both usual space-section and the extra 3-dimensional space, although anisotropic between each other to start with, are 3-dimensional isotropic space like sections in themselves.

Non-zero minima of the expectation values of volume element (Fig. 5) clearly show the contraction of the universe followed by expansion avoiding any singularity indicating a bouncing universe. Similar results were obtained in quantization of anisotropic models in absence of fluid in the past [39]. This result is obtained with quite reasonable boundary conditions, \(\varPsi \rightarrow 0\) for infinite values of its arguments, \(\bar{u}\) and \(\bar{v}\). For a discussion, we refer to the work of Vilenkin and Yamada [43] and Tuccil and Lehners [44].

For a general fluid with \(\alpha \ne 1\), we are able to show the self-adjoint extension and thus unitary evolution of the model for \(n=7\) despite the mathematical difficulty of the model. However, the result of it is in accordance with four-dimensional anisotropic model Bianchi-I as expected [26].

Our results may also be considered as the higher dimensional generalization of unitary evolution of the anisotropic models as shown in [26, 27, 28, 29].

## Notes

### Acknowledgements

The author would like to thank Narayan Banerjee for insightful discussions and comments in preparing the manuscript. The author was financially supported by CSIR, India (Grant no. 09/921(0106)/2014-EMR-I).

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