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Big Rip Singularity in 5D Viscous Cosmology Send Orders for Reprints to [email protected] The Open Astronomy Journal, 2014, 7, 7-11 7 Open Access Big Rip Singularity in 5D Viscous Cosmology 1,* 2 G.S. Khadekar and N.V. Gharad 1Department of Mathematics, Rashtrasant Tukadoji Maharaj Nagpur University, Mahatma Jyotiba Phule Educational Campus, Amravati Road, Nagpur-440033, India 2Department of Physics, Jawaharlal Nehru College, Wadi, Nagpur-440023, India Abstract: Dark energy of phantom or quintessence nature with an equation of state parameter almost equal to -1 often leads to a finite future singularity. The singularities in the dark energy universe, by assuming bulk viscosity in the frame- work of Kaluza-Klein theory of gravitation have been discussed. Particularly, it is proved, that the physically natural as- sumption of letting the bulk viscosity be proportional to the scalar expansion in a spatially 5D FRW universe, can derive the fluid into the phantom region ( < -1), even if it lies in the quintessence region ( >-1) in the non viscous case. It is also shown that influence of the viscosity term acts to shorten the singularity time but it does not change the nature of sin- gularity in the framework of higher dimensional space time. Keywords: Big Rip, dark energy, future singularity, viscous cosmology. 1. INTRODUCTION including a quintessence (Wang et al. [12]) and phantom (Caldwell [13]), and (ii) interacting dark energy models, by A revolutionary development seems to have taken place considering the interaction including Chaplygin gas (Ka- in cosmology during the last few years. The latest develop- menohehik et al [14]), generalized Chaplygin gas (Bento et ments of super-string theory and super-gravitational theory al. [15]) and a linear equation of state (Babichev et al. [16]). have created interest among scientists to consider higher- The bulk viscosity in isotopic space bulk and shear viscosity dimensional space-time, for study of the early universe. A in anisotropic space are presented by (Padmanabham and number of authors [1-4] have studied physics of the universe Chitre [17]). The Planck data show that the Hubble constant in higher-dimensional space-time. Overduin and Wesson [5] H is the tension with that measure by the several direct have presented an excellent review of higher-dimensional probes on H. The Planck data favor < -1 when the "super- unified theories, in which the cosmological and astrophysical nova legacy survey" is complicated. implications of extra-dimension have been discussed. The current observations [11, 18-20] strongly suggest Kaluza-Klein theory has a long and venerable history. that the universe is dominated by a negative pressure com- However, the original Kaluza version of this theory suffered ponent, dubbed dark energy. This component can be charac- from the assumption that the 5-dimensional metric does not terized by an equation of state parameter , which is simply depend on the extra coordinate (the cylinder condition). the ratio of the pressure to the density: = p/; for example Hence these are proliferation in recent decade of various cosmological constant corresponds to =1. While it is often versions of Kaluza-Klein theory, super gravity and super assumed that -1 in accordance with weak energy condi- strings. Kaluza-Klein achievements have shown that tion, it has long been known [13] that the observations are 5-dimensional general relativity contains both Einstein's consistent with <-1, which correspond to a dark energy 4-dimensional theory of gravity and Maxwell's theory elec- density that increases with time t and the scale factor a. If the tromagnetism. In the last decade, a number of authors [6-9] energy density increases monotonically in the future, then have considered multidimensional cosmological model. the universe can undergo a future singularity, called the “Big The cosmological observations have provided increasing Rip'', for which and at a finite time. In turn, this evidence that our universe is undergoing a late time cosmic may lead to the occurrence of negative entropy [21]. One acceleration (Perlmutter et al. [10] and Riess et al. [11]). In may expect that natural effects, most likely of a quantum order to explain the acceleration expansion, cosmologist in- mechanical origin, may prevent the Big Rip, as was shown in troduces a new fluid, which posses a negative enough pres- [22, 23]. sure called dark energy. Dark energy model can be divided Viscous cosmological models have been increasingly into two categories: (i) scalar field dark energy models popular lately. From a hydrodynamics viewpoint, this is quite a natural development, as the inclusion of the viscosity *Address correspondence to this author at the Department of Mathematics, coefficients (there are two of them, shear and bulk) means Rashtrasant Tukadoji Maharaj Nagpur University, Mahatma Jyotiba Phule that one departs physically from the case of an ideal fluid Educational Campus, Amravati Road, Nagpur-440033, India; and incorporate the deviations from the thermal equilibrium Tel: 91-9011323123; E-mail: [email protected] to first order. The case of an ideal (non-viscous) fluid is, 1874-3811/14 2014 Bentham Open 8 The Open Astronomy Journal, 2014, Volume 7 Khadekar and Gharad after all an idealized model, useful in practice for many sit- The five dimensional Einstein field equations are: uations but not for all, especially not when fluid motion near 2= , (3) solid boundaries is involved. In cosmological context, as the cosmic fluid is assumed to be spatially isotropic, the shear viscosity is usually left out. Anisotropic deviations such as + = - (4) those encountered in the Kasner universe are quickly smoothed out being the only coefficient left , the bulk vis- The effective pressure on the thermodynam- cosity. Some of the early treatises on viscous cosmology are ical grounds, in conventional physics has to be positive described [24-28]. A brief review of the singularities in four quantity. dimensions and five dimensions in viscous cosmology near The energy conservation equation leads to: future is discussed by [29]. Higher dimensional FRW cos- mological model with traversable static wormhole and Big . (5) Rip is discussed in detail by Brevik et al. [30]. Recently, Gorbunova and Diego Saez-Gomez [31] described the influ- From the above equations, the differential equation for ence of Casimir effects near the Big Rip singularity in vis- the scalar expansion, with and as a free parameter can be cous cosmology. Also Padmanabhan and Chitre [17] studied derived as: a constant and an oscillating dark energy fluid that crosses + (6) the phantom barrier (<-1) and derives the universe evolu- tion to a Big rip singularity. This equation can be written as a differential equation for The present paper is the generalization of the work [32] in the view of the relation: in the framework of Kaluza-Klein cosmology. The role of 64G • bulk viscosity in a universe having Big Rip singularity will = . We will be interested in the region around 3 be studied. For this, it has been considered that Kaluza-Klein cosmology with cosmological constant is equal to zero. The = -1, so it is convenient to introduce the EoS: shear viscosity will be put equal to zero, in conformity with (7) usual practice. We present the formulation for Kaluza-Klein Then the above Eq. (6) takes the form: cosmology when the Equation of State (EoS) is of general form, =() and examine the consequences of cosmic fluid (8) with bulk viscosity. We discuss some spatial cases. In all the cases, the presence of bulk viscosity tends to promote the big we denote the starting value of by . From Eq. (8) we rip phenomenon. Then it is observed that the barrier ( = -1) obtain: between the quintessence region (> -1) and the phantom 3 d region (<-1) can be crossed, as a consequence of the bulk . (9) viscosity. The case , means that we go a step further; it 64G 0 64G = f () + f ( ) means that the momentum transfer involves second order 3 quantities in deviation from reversibility, still maintaining This is the general relation between the cosmological the scalar property of . The general case concerning the time t (>0) and the density . dependencies of both the function f() and the bulk viscosity in terms of the density is discussed in detail. In the following section, we solve Eq. (9) for different values, a bulk viscosity function 2. BASIC FORMALISM: KALUZA-KELIN COSMOL- OGY 3. THE CASE WHEN = CONSTANT Let us put We assume the Kaluza-Klein type cosmological model in the form: , (10) , (1) where is a constant. This means that . We investigate the following with some different choice for . where (t) is a scale factor and the spatial curvature k, as well as cosmological constant is equal to zero. The Hubble 3.1. parameter H =, and scalar expansion = This is the non viscous case, Eq. (9) gives with 5 being the velocity of the fluid. We assume the equation of state (EoS) in the form: . (11) =(), (2) If > 0 for , the Big Rip singularity takes place i.e. where the thermodynamical variable () is an arbitrary >0 a finite value of the time t is compatible with a Big Rip function of . Similarly, we assume the bulk viscosity is ar- singularity (= ). This is the conventional phantom case bitrary = (). corresponds to (<-1). Big Rip Singularity in 5D Viscous Cosmology The Open Astronomy Journal, 2014, Volume 7 9 By using (11) we can also express density with time in , (21) the form: there occurs a Big Rip; all the quantities (t), (t), diverges. , (12) where the initial scalar expansion is 3.3. In this case, we assumed that the constant has to be . (13) positive as is positive. From (3) the bulk viscosity as a function of energy density : Similarly the scalar expansion is . (22) , (14) Again from Eq. (9) with the help of Eq. (22) we get: and the scale factor is .
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