1 August 05 Volume 6 Issue 8 pp Bali,. & Swati, Bianchi Type VIII Inflationary Universe with Massless Scalar Field in General elativity Bianchi Ty...

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Bianchi Type VIII Inflationary Universe with Massless Scalar Field in General Relativity Raj Bali* & Swati Department of Mathematics, University of Rajasthan, Jaipur-302004, India

Abstract Bianchi Type VIII inflationary universe in the presence of massless scalar field with flat potential is investigated. To get the deterministic model of the universe, we assume that the shear () is proportional to expansion () as considered by Thorne [20] and Collins et al.[21]. The model represents decelerating and accelerating phases of universe and spatial volume increases exponentially with time representing inflationary scenario. The model initially represents anisotropic space-time. However, it isotropizes in special case. The Higgs field evolves slowly but the universe expands. Keywords: Bianchi VIII, inflationary, massless scalar field.

1. Introduction Anisotropic cosmological models including the so called Bianchi cosmologies (Ellis et al.[1], Gron and Hervik[2]) are of great theoretical importance and have been object of close studies (Hewitt et al.[3]), Coley and Hervik[4]). Bianchi Type VIII cosmological models are the most general ever-expanding Bianchi cosmologies and are therefore of special interest. The late time behaviour of Bianchi Type VIII space-time in presence of diffusion is studied by Shogin and Hervik[5]. The state of accelerated expansion of the universe is termed as Inflation. It was first proposed in the beginning of the 1980s and now a days receives a great deal of attention. The inflationary epoch comprises the first part of the electro weak epoch following the grand unification epoch. It lasted from 10-33 to 10-32 seconds. Following the inflationary period, the universe continues to expand. Inflationary universes provide a potential solution to the formation of structure problem in Big-Bang cosmology like Horizon, Flatness and Magnetic monopole problems. Guth[6] introduced the idea of early inflationary phase in the context of grand unified field theories (Zel’dovich and Khlopov[7]) in which symmetry breaking phase transitions occur with the decrease of temperature at the very early stages of evolution of universe (Linde[8]). Rothman and Ellis[9] explained that we can have isotropic space-time if we work with anisotropic metrics and these metrics can be isotropized in a very general circumstances. Stein-Schabes[10] has shown that inflation will take place if effective potential V() has flat region when Higgs field () evolves slowly but the universe expands in an exponential way due to vacuum field energy. It has been assumed that the scalar field takes sufficient time to cross the flat region so that *

Correspondence: Raj Bali, CSIR Emeritus Scientist, Department of Mathematics, University of Rajasthan, Jaipur-302004, India. E-mail: [email protected] ISSN: 2153-8301

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Prespacetime Journal| August 2015 | Volume 6 | Issue 8 | pp. 679-683 Bali, R. & Swati, Bianchi Type VIII Inflationary Universe with Massless Scalar Field in General Relativity

680

universe expands slowly to become homogeneous and isotropic. Anninos et al.[11] has discussed the significance of inflation for isotropization of the universe. Many authors viz. Bali et al.[12,13,14,15], Singh and Kumar[16], Reddy et al.[17,18], Katore et al.[19] studied inflationary cosmological models in different Bianchi space-times with self interacting scalar field in General Relativity. Motivated by the above mentioned studies, we investigate inflationary scenario in Bianchi Type VIII space-time for a massless scalar field with flat potential using the condition that shear () is proportional to expansion (). We find that the model represents accelerating universe and spatial volume increases exponentially representing inflationary scenario. The model represents anisotropic space-time initially but isotropizes in special case.

2. Metric and Field Equations We consider Bianchi Type VIII metric in the form ds 2 dt 2 R 2 ( t)[d 2 cosh2d 2 ] S 2 ( t)[d sinh d ]2 (1) where R and S are functions of t-alone. The Lagrangian is that of gravity minimally coupled to Higgs scalar field () with effective potential V() given by Stein-Schabes[10]:

L

1 g R g ij i j V( d 4 x 2

(2)

The variation of L with respect to the dynamical field, leads to the Einstein field equation

1 R ij R g ij Tij 2

(3)

(in geometrical unit 8G = 1, c = 1) where

1 Tij i j V( g ij 2

(4)

and

1 dV i g i d g

(5)

where is the Higgs field, V the effective potential and gij the metric tensor. Einstein’s field equation (3) with (4) and (5) for the metric (1) leads to

S S 2 2 R S R V( RS R S 4R 4 2 R 2 3S 2 2 2R 1 2 4 2 V( R R 4R R 2 2 2 2 R 2R S 1 S V( R 2 RS R 2 4R 4 2

(6)

(7)

(8)

The equation for scalar field leads to ISSN: 2153-8301

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Prespacetime Journal| August 2015 | Volume 6 | Issue 8 | pp. 679-683 Bali, R. & Swati, Bianchi Type VIII Inflationary Universe with Massless Scalar Field in General Relativity

681

2R S dV R S d

(9)

To get the deterministic solution in terms of cosmic time t, we assume that shear () is proportional to expansion () as considered by Thorne[20] and Collins et al.[21]. Thus, we have

S Rn

(10)

[13]

As considered before , we consider that effective potential V() is flat and to get the inflationary scenario. Thus equation (9) leads to

V() = constant

2R S 0 R S

(11)

V() = k (constant).

(12)

where Equations (7) and (8) lead to

2R 2 2R S 2 S 2 2R 2 2k R RS R 2 R 4 R

(13)

From equations (13) and (10), we have

2n 2)R 2 2kR 2 R 2n 2 2 2R R R f(R). Thus equation (14) leads to To get the solution, we assume that R 2f f '

(14)

2n 2 2 2kR 2 R 2n 2 2 f R R

(15)

which leads to

k R 2n 2 2 2 f R R 2n 2 n2 4n 2n 2 2

(16)

where

f'

df dR

(17)

To get the solution in terms of cosmic time t, we assume n = 2 and = 0. Thus equation (16) leads to 2

dR k 1 2 1 R 3 dt 4 8

(18)

which leads to

R a sinh ( t

(19)

2k 1 2 1 a 2 8 3

(20)

where

2 Thus ISSN: 2153-8301

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Prespacetime Journal| August 2015 | Volume 6 | Issue 8 | pp. 679-683 Bali, R. & Swati, Bianchi Type VIII Inflationary Universe with Massless Scalar Field in General Relativity

S R 2 a 2sinh 2 t

682

(21)

After suitable transformation of coordinates, the metric (1) leads to the form

ds 2

dT 2

2

a 2sinh 2 T [d 2 cosh 2 d 2 a 4sinh 4 Td sinh d 2

(22)

3. Physical and Geometrical Aspects Using the assumed condition V() = constant = k, the equation (11) leads to

4 cosech 4 T a

(23)

which leads to coth3 T N cothT (24) 4 3 3 a where ℓ and N are constants. Higgs field () decreases slowly and for large value of T, it tends to a finite value. The spatial volume (V3) is given by

V 3 R 2S a 4 sinh 4 T

(25)

The directional Hubble parameters Hx, Hy, Hz are given by

R cothT R S 2R Hz 2 coth T S R

Hx Hy

(26) (27)

The mean Hubble parameter (H) is given by

1 4 H H x H y H z cothT 3 3

(28)

The expansion (), the shear () and deceleration parameter (q) are given by

3H 4 coth T

(29)

3 2 1 2 1 2 H i coth 2 T 2 i1 3 3 2

Thus, we have

coth T 3 /V coth 2 T 3 V tanh 2 T q 2 2 V 4 V 4

(30) (31)

Discussion and Conclusion ISSN: 2153-8301

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Prespacetime Journal| August 2015 | Volume 6 | Issue 8 | pp. 679-683 Bali, R. & Swati, Bianchi Type VIII Inflationary Universe with Massless Scalar Field in General Relativity

683

The Higgs field decreases slowly but the universe expands and at late time, it has finite value. The spatial volume increases exponentially representing inflationary scenario. Since

0

hence the model represents anisotropic space-time in general. However, it represents isotropic space-time for n = 1. The Hubble parameter (H) is initially large but decreases with time. The model describes an unified expansion history of the universe which starts with decelerating expansion and the expansion accelerates at the late time. The decelerating expansion at initial epoch provides obvious provision for the formation of large structure in the universe. The formation of structure is better supported by decelerating expansion. Thus the model is astrophysically relevant. Also late time acceleration is in agreement with the observations of 16 Type Ia supernovae made by Hubble space Telescope (Riess et al.[23]). The model has Point Type singularity at T = 0 (MacCallum[22]).

References [1] Ellis G F R Maartens R and MacCallum M A H 2012 Text Book on Relativistic Cosmology (Cambridge University Press) [2] Gron O and Hervik S 2007 Text Book on Einstein’s General Theory of Relativity with Modern Applications in Cosmology (Springer) [3] Hewitt C G Brisdon R Wainwright J A 2001 Gen. Relativ. Gravit. 33 65 [4] Coley A and Hervik S 2008 Class. Quant. Gravity 25 198001 [5] Shogin D and Hervik S 2014 arxiv:1402.6864 v1(gr-qc) [6] Guth AH 1981 Phys. Rev. D 23 347 [7] Zel’dovich Ya B and Khlopov Yu M 1978 Phys Lett B 79 239 [8] LInde A D 1982 Phys. Lett. B 108 389 [9] Rothman T and Ellis G F R 1986 Phys. Lett. B 180 19 [10] Stein-Schabes J A 1987 Phys. Rev. D 35 2345 [11] Anninos P Matzner R A Rothman T and Ryan M P Jr 1991 Phys. Rev. D 43 3821 [12] Bali R Jain V C 2002 Pramana – J. Phys. 59 1 [13] Bali R 2011 Int. J. Theor. Phys. 50 3043 [14] Bali R and Swati 2012 Chin. Phys. Lett. 29 080404-1 [15] Bali R and Singh S 2014 Canad. J. Phys. 92 1 [16] Singh C P and Kumar S 2007 Pramana – J. Phys. 68 707 [17] Reddy D R K Naidu R L and Rao AS 2008 Int. J. Theor. Phys. 47 1016 [18] Reddy D R K Naidu R L and Rao AS 2009 Astrophys. Space Sci. 319 89 [19] Katore SD Rane RS Wankhade KS and Sarkate NK 2010 Ramana – J. Phys. 74 669 [20] Thorne K S 1967 Astrophys. J. 148 51 [21] Collins C B Glass EN and Wilkinson DA 1980 Gen. Relativ. Gravit. 12 805 [22] MacCallum M A H 1971 Comm. Math. Phys. 20 57 [23] Riess A G et al. 2004 Astrophys. J. 607 665

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