Cosmological Models Filled with Perfect Fluid & Dark Energy in f(r,t) Theory of Gravity

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1 August 05 Volume 6 Issue 8 pp Cosmological Models Filled with Perfect Fluid & Dark Energy in f(r,t) Theory of Gravity 79 Article D. D. Pawar * & S. S. Pawar School of Mathematical Sciences, Swami Ramanand Teerth Marathwada University, Vishnupuri, Nanded-606, India MGM s College of Engineering Near Airport, Hingoli Road Nanded, India Abstract In this paper we have investigated the physical behaviour of homogeneous and anisotropic f R, T theory of gravity in the presence of Kaluza-Klein cosmological model in the frame of perfect fluid and dark energy. Where R is the Ricci scalar and T is the stress energy tensor of the matter. Here we consider the case when the perfect fluid obeys the following euations of state P 0,. Also we studied the cases when dark energy is given by a with uintessence or Chaplygin gas. Some physical properties of the models are also discussed. Keywords: Dark energy, perfect fluid, theory of gravity, Kaluza-Klein field euation.. Introduction Now a day s different observation gives the suggestion that our universe is almost homogeneous and isotropic space time it is first describe by Friedman-Lemaitre-Robertson-Walker (FLRW) cosmology. Since, in early universe the FLRW statement about matter description is not correct. The cosmological view is that the universe might be anisotropic and also inhomogeneous in the vary early era and that in the course of its evaluation these characteristics has been take out by some process or mechanism and finally we get the homogeneous and isotropic universe. In anisotropic cosmological model of the universe the large scale observation intersects with cosmic microwave background (CMB) therefore it is important to study the cosmological model within the framework of different anisotropic space time. According to the history of the expansion of the universe the cosmological observation shows that the current universe not only expanding but also accelerating. So in a late time accelerated expansion of the universe has been confirmed by high red shift Ia supernovae experiment (SNe.Ia) [] also some observations like cosmic background radiation (CMBR) data [] and large scale structure [] combination shows that the late time acceleration expansion of the universe. All such results strongly imply that existence of an extra component in a universe with negative pressure i.e. the dark energy, which * Correspondence Author: School of Mathematical Sciences, Swami Ramanand Teerth Marathwada University, Vishnupuri, Nanded-606, (India). dypawar@yahoo.com ISSN: 5-80

2 August 05 Volume 6 Issue 8 pp is about 70% in our universe. The dark energy is described by an euation of state EoS where p d and are pressure and energy density of dark energy and is a parameter of EoS. d pd This is effectively describing cosmic acceleration. The universe expand with acceleration gives the conformation of existence of dark energy and the simplest candidate for dark energy is the vacuum energy w is mathematically eual to cosmological constant term in the system [] which is first investigated by Einstein as a sure and sustainable cosmological solution to the gravitational field euation. Now in addition with cosmological constant for dark energy scenario and the proposal given by dynamical dark energy is also reliable by some scalar field mechanism which suggests that energy form with negative pressure is provided by scalar field evolved down a proper potential. Now some more examples of dark energy models are uintessence [5].Quintessence is a hypothetical form of dark energy and is thought to be the fifth fundamental force. Since cosmological constant stays constant throughout time and uintessence changes over time due to its dynamic character which is obtained by the euation of state EoS. This is the most common type of dark energy [6]. Chaplygin gas models [7] are used to combine the two different concepts as a dark energy and dark matter and thus reduce the two physical parameters in one by exotic euation of state EoS p C. Since the original Chaplygin gas C was introduce into Aerodynamics [8]. On chaplygin model there are some good number of works [9-6], Quintom field [8], Phantom field [9],K-essence[0],Tachyon field [], Holographic[-] and agegraphic [] from these candidates to play the role of dark energy the uintessence and the Chaplygin gas have more concerned a lot in recent years. Since uintessence behave like a cosmological constant by combining positive energy density and negative pressure. Which obeys euation of state EoS pd d. Many researchers are working on dark energy in different theories of gravity like [5-]. In a theory of gravity the development is going on whose results is that the new theories of gravity f T and f R are recently developed. f T Theory has interesting features.that it may explain the current acceleration without involving dark energy and is also the generalized version of teleparallel gravity in which Weitzenbock connection is used instead of Levi-Civita connection. Many considerable works has been done in this f T theory so far like [-]. In f R theory of gravity the Ricci scalar is used in standard Einstein Hilbert Lagrangian. In different ways this f R is investigated by many researchers like [-5]. For f R the mostly studied solution is that of spherically symmetric solution because of its closeness to the nature. Felice A.D., Tsuikawa and Noiri et al. present details reviews of a number of popular models of modified f R gravity. f, theory of gravity developed by Harko et al.[6] where gravitational Lagrangian is given by an arbitrary function of Ricci scalar R and of trace T of stress energy tensor and obtained several models corresponding to some explicit form of the function f R, T. Currently, Samanta and Dhala constructed higher dimensional cosmological model filled with The generalized R T d ISSN: 5-80

3 August 05 Volume 6 Issue 8 pp f, theory of gravity. Shri Ram et al. [7] have examined anisotropic cosmological models in f(r,t) theory of gravitation. Pawar and Solanke [8] have investigated the physical behaviour of LRS Bianchi type I cosmological model in f(r,t) theory of gravity. Dark energy cosmological models in f(r,t) theory of gravity studied by Pawar and Agrawal [9].Therefore it is hoped that gravity may explain the present phase of cosmic acceleration of our universe. perfect fluid in R T Now a day there is more work on binary mixture of perfect fluid and dark energy in that Ban Saha [0] worked on self-consistent system of Binachi Type I. T Singh and R Chaubey [] has considered Binachi type V model of the Universe with a binary mixture of perfect fluid and dark energy. Katore [] have considered cosmological model of the universe with a binary mixture of perfect fluid and dark energy Akarsu and Kilinc [] worked on specially homogeneous but totally anisotropic and non-flat Binachi type II cosmological models with general relativity and in the presence of two minimally interacting fluids i.e. Perfect fluid as the Matter fluid and hypothetical anisotropic fluid as the dark energy fluid. Cataldo et al [] described that a simplest non trivial cosmological scenario for an interacting mixture of two cosmic fluids by Power law scale factors i.e. expansion (contraction) as a power law in time. Bainchi type IX two fluids cosmological models in General Relativity have obtained by Pawar and Dagwal [5]. Singh et al.[6] have obtained two-fluid cosmological model of Bianchi type- V with negative constant deceleration parameter. Two fluid cosmological models in Kaluza- Klein space-time examined by Samanta [7]. Recently, two fluids tilted cosmological model in General Relativity and Axially Bianchi type-i mesonic cosmological models with two fluid sources in Lyra Geometry presented by Pawar and Dagwal [8-9] Since the five dimensional theory introduced by Kaluza-Klein by Maxwell s theory of electromagnetism and Einstein s gravity theory. The candidate of fundamental theory is a Kaluza-Klein due to its potential function to unite the fundamental interaction. Kaluza-Klein cosmological model have been studied with different type of matter and cosmological constant with considerable inflection. In this paper we have investigated the physical behaviour of homogeneous and anisotropic f R, T theory of gravity in the presence of Kaluza-Klein cosmological model in the frame of perfect fluid and dark energy. Where R is the Ricci scalar and T is the stress energy tensor of the matter. The dark energy is given by Chaplygin gas or uintessence and also the perfect fluid obeys the euation of state 0,.This work is an extension of G.C. Samanta, S.N.Dhala [50]. p with. Field Euations and Solutions Here we consider specially Kaluza-Klein space time field euation in the form of ds = dt - a ( dx dy dz ) b d (.) where a and b represents the functions of t only. ISSN: 5-80

4 August 05 Volume 6 Issue 8 pp Kaluza-Klein universe is generalization of the open universe in FRW cosmology. Therefore its study is important in the presence of dark energy models in a universe with non-zero curvature []. For the matter source the energy momentum tensor is given by where, Ti uiu i u is the flow vector satisfy the euation, i u u By using euation (.) we find, ( P ) P (.) 0 T 0 in a co-moving system of co-ordinates. Now varying the action, i g (.) T, T T T P S = 6 f T) g d 5 x + L m g d 5 x (.5) Of the gravitational field with concerned to the metric tensor components g, We get the field euation of f T ) gravity model as (Harko et al [6]) i f R T ) R f T ) g ( g i ) f R T ) 8T ft T ) T ft T ) (.6) ( g ) where T L m and, g g (.) T Pg (.7) Now we define the covariant derivative as, f T ) f T ) f R T ), f T T ) and i. R T Here f T ) denote arbitrary function of Ricci scalar R and the trace T of the stress energy tensor of mattert L. m Represent the matter Lagrangian density in the present study. Here we have assumed that stress energy tensor of matter is given by, T ( P ) u u Pg (.8) i And also assuming that the function f T ) given by (Harko et al. [6]) f T ) R f ( T ) (.9) Where f (T ) denote arbitrary function of trace of the stress energy tensor of matter. Now use the euations (.7) and (.8), in an euation (.6) we get the new form, R Rg 8T f ( T ) T (Pf ( T ) f ( T )) g (.0) ISSN: 5-80

5 August 05 Volume 6 Issue 8 pp Where f( T) is differentiation f( T ) with respect to the argumentt. We take, f ( T ) T (.) Where, is constant (Harko et al[6]) Now in a co-moving co-ordinate systems use euations (.), (.), (.) and (.) in a field euation (.0),for a metric (.) we get, a a ab a b ab a b a a a a P P 8 8, (.), (.) a ab 8 P. (.) a ab Where overhead prime denotes differentiation with respect to cosmic time t only. Now subtract euation (.) from (.) we get, a a ab b 0 (.5) a a ab b Rearranging euation (.5) we get, d a b a bv 0 (.6) dt a b a b V dy Since above euation has form of P( t) y Q( t) which is the linear differential euation of dt first order. Solution of euation (.6) is, a b V l, a b where l integrating constant. a b l a b V a l d log b V On integrating above euation both side we get, a l log dt m, b V here m integrating constant. a dt m exp l, here m e m 0 (.7) b V Now euation (.7) becomes, dt a KV exp d, (.8) V ISSN: 5-80

6 August 05 Volume 6 Issue 8 pp l Where K m, d Now from euation (.7) and V a b we get, dt b KV expd (.9) V l Where K m, d K i i, and d i ( i,) satisfy the relation, K K and d d 0 Where Put K K therefore K K (.0) And, Put d D therefore d D (.) Euation (.8) and (.9) becomes, dt a KV exp D (.) V dt b K V exp D (.) V Where K 0 and D are constant of integration.. Universe filled with perfect fluid and dark energy We consider the evaluation of Kaluza-Klein space-time universe filled with perfect fluid and dark energy in f(r,t) frame in details. For the perfect fluid and dark energy, is the energy density and P is the pressure. i.e., P P P (.) The energy momentum tensor can be decomposed as, Ti P P u iu P P i (.) In above euation is the dark energy density, P its pressure. We also use the notation of and P to denote the energy density and the pressure of the perfect fluid respectively. Here we consider the case when the perfect fluid obeys the following euations of state. P (.) Where is the constant and lies in the interval 0, depending on its numerical value we define types of universe as following, 0 (Dust universe), (Radiation universe), (Hard universe), (Zeldovich universe or stiff matter) ISSN: 5-80

7 August 05 Volume 6 Issue 8 pp In a co-moving frame the balance of the energy density is due to conservation law of energy momentum tensor. V P P (.) V The dark energy is supposed to interact with itself only and is minimally coupled to the gravitational field. As a result the evaluation euation for the energy density decouples from that of the perfect fluid and from euation (.) we obtain two balance euations. V P 0 (.5) V And V P 0 (.6) V From euation (.) and (.5) 0 0, P V V (.7) Where 0 is integrating constant.. Case with a Quintessence Let us consider the case when dark energy is given by a uintessence which obeys the euation of state, (.8) P Where the constant varies between From euation (.8) and (.6), we get 0, V where is integration constant. 0. Case with Chaplygin Gas and zero, i.e.,,0 P 0 V. Let us consider the case when dark energy is represented by Chaplygin gas. P C (.0) C With being a positive constant. From euation (.0) and (.6), We get, With 0C 0C C, V being an integration constant. P C V 0C (.9) (.) ISSN: 5-80

8 August 05 Volume 6 Issue 8 pp Analysis and Discussion Since we have the field euations (.) to (.) are highly non-linear. Thus extra conditions are needed to solve the system completely. For that we have used two different volumetric expansion laws. lt V (.) And V (.) Where Cl, and n are positive constant. In this way all possible expansion histories, the exponential expansion (.) power law expansion (.) have be covered. The model with exponential expansion and power law totally depends on n. As n exhibit accelerating volumetric expansion while for n Exhibits volumetric expansion with constant velocity, the model for n exhibits decelerating volumetric expansion. 5. Some Physical Parameters Since the isotropy of the expansion can be parameterized after defining the directional Hubble s parameters and the average Hubble s parameters of the expansion.the directional Hubble s parameters in the direction of x, y, z and for the Kaluza-Klein metric define in (.) Is defined as, a b H x H y H z And H (5.) a b Average Hubble s parameter is given by, V H where V a b is the volume of the universe (5.) V a b H (5.) a b The Anisotropy parameter of the expansion is defined as, Hi H (5.) i H H i i,,, represent the directional Hubble s parameters in the direction of x, y, z Where and respectively. 0 Corresponds to isotropic expansion. The space approaches isotropy,in a case of diagonal 0 energy momentum tensor T i 0, i,,, if 0, V andt as t (See [70] for details) Let us introduce the dynamical scalars such as expansion parameters and shear as usual. u i i H (5.5) ; ISSN: 5-80

9 August 05 Volume 6 Issue 8 pp (5.6) where ui; P u ; Pi P (5.7) Here the proection tensor P has the form P g u u (5.8) For the Kaluza-Klein metric (.) these dynamical scalars have the form, a b H (5.9) a b 5. Model for Exponential Expansion i a b (5.0) a b After solving the field euations (.) to (.) for the exponential volumetric expansion (.) by considering the euation (.) and (.) we obtain the scale factor as follows, lt D lt a KC e exp e (5.) lc lt D lt b K C e exp e (5.) lc The metric (.) with the help of (5.) and (5.) can now be written as, lt D lt 6 lt D lt ds dt K C e exp e dx dy dz K C e exp e d (5.) lc l The directional Hubble s parameters H x, H y, H z and H have values given by, a D lt H x H y H z e l a C (5.) D lt H e l (5.5) C Mean Hubble s parameter H by using euation (5.) we get H l. Anisotropy parameter of the expansion from euation (5.) is, D 8lt e (5.6) lc Here we observe that this behaviour of is euivalent to the ones obtained for the model that correspond to the exponential expansion in Bianchi type III cosmological model with anisotropic dark energy []. From euation (5.9) and (5.0) the dynamical scalars are given by, l (5.7) ISSN: 5-80

10 August 05 Volume 6 Issue 8 pp D 8lt e (5.8) C To study the non-singular behaviour of the model (5.), we consider the Riemann-curvature invariant R for the metric (.) given by, D 8lt R 0l e (5.9) C R 0l as t.hence the model (5.) is not a singular in the finite past and in the infinite future. Now use euation (.) in (.7), (.9) and (.), We get, 0, P 0 lt (5.0) lt 0, lt P 0 (5.) lt 0C C, 0C lt PC (5.) lt aa d The deceleration parameter (5.) a dt H 5. Model for Power Law Expansion After solving the field euations (.) to (.) for the Power Law expansion (.) by considering the euation (.) and (.) we obtain the scale factor as follows n D t a KC t exp (5.) C n D t b K C t exp (5.5) C The metric by using euation (5.) and (5.5) we get, n D t 6 n 6D t ds dt K C t exp dx dy dz K C t exp d C C (5.6) The directional Hubble s parameters H x, H y, H z and H are, a D n H x H y H z, b D n H (5.7) a t b t n Mean Hubble s parameter H by using euation (.) we get H t. Anisotropy parameter of the expansion from euation (5.) is, ISSN: 5-80

11 August 05 Volume 6 Issue 8 pp D (5.8) 8n C n t Form euation (5.9) and (5.0) the dynamical scalars are given by, H (5.9) t D (5.0) 8n Now to study the non-singular behaviour of the model (5.6).We consider the Riemanncurvature invariant R for the metric (.) given by, 0n D 8n R (5.) 8n t C t t Now use euation (.) in (.7), (.9) and (.), We get, 0, P 0 (5.) C 0, 0C The deceleration parameter P 0 (5.), 0C PC (5.) d n (5.5) dt H 6. Some particular cases Case-I For 0, 0the euations (5.0),(5.),(5.) and (5.) becomes, 0, lt 0 0, lt 0 0, 0 0, 0 P (6.) P (6.) P (6.) P (6.) ISSN: 5-80

12 August 05 Volume 6 Issue 8 pp Case-II For, 0the euations (5.0),(5.),(5.) and (5.) becomes, 0 P 8lt (6.5) 0, P 0 lt (6.6) 0 P 8n (6.7) 0, P 0 (6.8) Case-III For,,the euations (5.0),(5.),(5.) and (5.) becomes, 0 P 8lt (6.9) 0, P 0 (6.0) 0 P 8n (6.) Case-IV For, the euations (5.0),(5.),(5.) and (5.) becomes, 0 0, P 5 (6.) 5 5lt 5lt 0 0, P (6.) lt lt 0 0, P 5 (6.) 5 5n 5n 0 0, P (6.5) n n Case-V For (Hard Universe), the euations (5.0),(5.),(5.) and (5.) becomes, 0 0, P 6lt (6.6) 6lt ISSN: 5-80

13 August 05 Volume 6 Issue 8 pp , P (6.7) lt lt 0 0, P (6.8) 6n 6n 0 0, P (6.9) n n 7. Conclusion We consider the Kaluza-Klein space-time with binary mixture of dark energy and perfect fluid in the frame of f(r,t) theory. To obtain the complete solution of field euations we assume the two expansion laws namely: exponential law and power law. We have been discussed the solution for both volumetric laws of expansion for constant deceleration parameter. An accelerated expansion of the model is due to the presence of the dark energy into the system. As a conseuence the initial anisotropy of the model rapidly passes away. When n, anisotropy goes to decays and it becomes null monotonically in both volumetric laws of expansion, if n 0,, we studied five particular cases in then the universe tends to isotropy. Also for details. It is note that our cosmological models bear a resemblance to [50]. References [] S. Perlmutter et al. (999) Astrophys. J [] Caldwell, R.R.Doran, M.:Phys.Rev.D69, 057(00). [] Daniel, S.F., Caldwell, R.R., Cooray, A., Melchiorri, A.:Phys.Rev.D77, 05(008). [] A.Einstein, Kosomologische Betrachtungen Zurallge-Meinen.Relativ;tii atstheorie, Sitzungsber.Preuss.Acad.wiss.,-5(97). [5] B. Ratra, J. Peebles (988) Phys. Rev. D 7. [6] R.R.Caldwell,R.Dave,and P.J.Steinhardt, Cosmological imprint of an Energy component with general euation of state, Phys.Rev.Lett. 80(8),58-585(998). [7] Kamenshchik et al. Phys.Lett. B5 (00) [8] S.A.Chaplygin, On gas et, in Scientific notes of the Department od Physico Math.Science of Moscow University,90,Issue,pp -. [9] R.Bean and O. Dore, Are Chaplygin gases serious contenders to the dark energy throne? Phys. Rev. D 68,055(00). [0] M.C.Bento,O.Bertolami,and A.A.Sen, Generalized Chaplygin gas and CMBR constraints, Phys.Rev.D 67,0600(00). [] M.Biesiada, W.Godlowski and M.Szydlowski, Generalized Chaplygin Gas models tested with SNIa, Astrophys.J.6,8-8(005), astroph/0005. [] J.C.Fabris,S.V.Goncalvez,and P.E.de Souza, Density perturbations in Universe dominated by the Chaplygin gas, Gen. Reativ.Gravitation,5-6(00). [] V.Gorini,A.Kamenshchik and U.Moschella and V.Pasuier, The Chaplygin gas as a model for dark energy, gr-c/ ISSN: 5-80

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