2-A Cosmological Model with Varying G and ∧ in General Relativity

doi:10.4236/ojapps.2013.31B1015

Paper Menu >>

Journal Menu >>

Open Journal of Applied Sciences, 2013, 3, 74-78

doi:10.4236/ojapps.2013.31B1015 Published Online April 2013 (http://www.scirp.org/journal/ojapps)

2-A Cosmological Model with Varying G and 

in General Relativity

Harpreet1, R. K.Tiwari2, H. S. Sahota1

1Department of Applied Sciences , Sant Baba Bhag Singh Institute of Engineering & Technology, Jalandhar, India

2Goverment Engineering College, Reva, M.P., India

Email: [email protected]

Received 2013

ABSTRACT

Spatially homogeneous and anisotropic Cosmological models play a significant role in the description of the early

stages of evolution of the universe. The problem of the cosmological constant is still unsettled. The authors recently

considered time dependent G and  with Bianchi type–I Cosmological model .We considered in this paper homogene-

ous Bianchi type -I space-time with variable G and  containing matter in the form of a perfect fluid assuming the

cosmological term proportional to R-2 (where R is scale factor). Initially the model has a point type singularity, gravita-

tional constant G (t) is decreasing and cosmological constant  is infinite at this time. When time increases  decreases.

Unlike in some earlier works we have neither assumed equation of state nor particular form of G. The model does not

approach isotropy, if ‘t’ is small .The model is quasi-isotropic for large value of ‘t’.

Keywords: Bianchi Type-I Universe; Varying G and ; Cosmology

1. Introduction

Cosmology is the scientific study of large scale proper-

ties of the universe as a whole. Cosmology is study of mo-

tion of crystalline objects. The origin of universe is great-

est cosmological mystery even today. As we are aware

that the expansion of the universe is undergoing time

acceleration [Perlmutter et al., (1997,1998,1999), Riess

et al., (1998,2004), Allen et al., (2004), Peebles et al.,

(2003), Padmanabhan. (2003) &Lima. (2004)]. Present

universe is suitably represented by Fried-

mann-Robertson-Walker model which is isotropic and

homogeneous in nature. To resolve the problem of a

huge difference between the effective cosmological con-

stant observed today and the vacuum energy density pre-

dicted by the quantum field theory, several mechanisms

have been proposed by Weinberg (1989). A possible way

is to consider a varying cosmological term due to the

coupling of dynamic degree of freedom with the matter

fields of the universe. The cosmological constant is small

because the universe is old. Models with dynamically

decaying cosmological term representing the energy den-

sity of vacuum have been studied by R. G. Vishwakarma,

(2000,2001,2005), A. I. Arbab, (1998) and Berman

(1991,1991b). Cosmological scenarios with a time vary-

ing cosmological constant were proposed by several re-

searchers. A number of models with different decay laws

for the variation of cosmological term were investigated

during the last two decades[ Chen & Wu (1990); Pavan

(1991); Carvalho et al.,(1992); Lima & Maia(1994);

Lima & Trodden (1996); Arbab & Abdel-Rahman(1994);

Cunha & Santos (2004); Carneiro & Lima(2005)].

A lot of work has been done by Saha (2005a, 2005b,

2006a, 2006b), in studying the anisotropic Bianchi type-I

Cosmological Model in general relativity with varying G

and . In this paper we study homogeneous Bianchi type

-I space-time with variable G and  containing matter in

the form of a perfect fluid. We obtain solution of the

Einstein field equations assuming the cosmological term

proportional to R-2 (where R is scale factor).

2. The Metric and Field Equations

We consider the Bianchi type - I metric in the orthogonal

form

(1)

The non-zero components of the Ricci tensor Rij

We assume that cosmic matter is taken to be perfect

fluid given by the energy- momentum tensor

(2)

where p,



are the isotropic pressure and energy density

of the fluid. We take equation of state

HARPREET ET AL. 75

vi is four velocity vector of the fluid satisfying

(3)

Einstein's field equations with time dependent G and 

are

(4)

For the metric (1) and energy-momentum tensor (2) in

commoving system of coordinates, the field equation (4)

yields.

(5)

(6)

(7)

(8)

In view of vanishing of the divergence of Einstein

tensor, we have

(9)

The usual energy conservation equation of general

relativity quantities is

(10)

Equation (9) together with (10) puts G and  in some

sort of coupled field given by

(11)

implying that  is a constant whenever G is constant.

Using equation(21) in equation (10) and then integrating,

we get k > 0

(11

We define, R as the average scale factor of Bianchi type-

I universe.

(12)

The Hubble parameter H, volume expansion



, shear



and deceleration parameter q are given by

Einstein's field equations (5)-(8) can be also written in

terms of Hubble parameter H, shear



and deceleration

parameter q as

(13)

(14)

On integrating (5) - (8), we obtain

(15)

and

(16)

where k1 and k2 are constants of integration. From (14),

we obtain

(17)

Implying that  0

Thus the presence of positive  lowers the upper limit

of anisotropy whereas a negative  contributes to the

anisotropy.

Equation (17) can also be written as

(18)

where





 is the critical density and 8









is the vacuum density.

From (13) and (14), we get,

(19)

Thus the universe will be in decelerating phase for

negative  and for positive  universe will slow down

the rate of decrease,showing that the rate of volume ex-

pansion decreases during time evolution and presence of

positive , slows down the rate of this decrease whereas

a negative  would promote it.

3. Solution of the Field Equation

The system of equations (5)-(8) and (11) supply only five

equations in seven unknown parameters (A, B, C,, p, ,

HARPREET ET AL.

and G). Two extra equations are needed to solve the sys-

tem completely. For this purpose we take cosmological

term is proportional to R-2 , where ‘a’ is a positive con-

stant.i.e we take the decaying vacuum energy density

(20)

This variation law was proposed by Olson et al.,

(1987), Pavon (1991), Maia et al., (1994); Silveira et al.,

(1994,1997) and Torres et al.,(1996).Because observa-

tions suggest that  is very small in the present universe,

a decreasing functional form permits  to be large in

early universe.

Using equation (111) and equation (20) in eq (11) we

get

(21)

From equations (13), (14), (20) and (201) we get

(22)

Find the time evolution of Hubble parameter, integrate

(22), we get

(23)

where t0 is a constant of integration. The integration con-

stant is related to the choice of origin of time.

From eq (23),we obtain the scale factor

(24)

By using equation (24) in (15) and (16) in the metric

(1), we get

(25)

where m1,m2 and m3 are constants.

For the model (25), the spatial V, density , gravita-

tional constant G and cosmological constant  are

(26)

(27)

(28)

(29)

Expansion scalar  and shear  are

(30)

(31)

(32)

4. Observations and Conclusion

1) Thus we observe that as spatial volume V0 at t =

= and expansion scalar  is infinite,

which shows that universe starts evolving with zero

volume at t = infinite rate of expansion.

2) The scale factors also vanish at t = and hence

the model has a point type singularity at initial epoch.

Initially at t = the energy density ‘



‘, pressures ‘p’,

shear , cosmological term  tend all infinite but G is

finite.

3) As t increases the spatial volume increases but the

expansion rate decreases. Thus the rate of expansion

slows down with increase in time.

4) As t the spatial volume V becomes infinitely

large. All parameters





, p,



,   0 asymptotically

but G is decreasing. Therefore at large value of t model

gives empty universe. The cosmic scenario starts from a

big bang at t = and continues until t =.If t =

then gravitation constant is zero and as t increase G also

increases.

The possibility of G increasing with time, at least in

some stages of the development of the universe, has been

investigated by Abdel-Rahman (1990), Chow (1981),

Levit (1980) and Milne (1935). 2



 Include Berman

(1990), Berman and Som (1990), Berman et al., (1989),

and Bertolami (1986b,1986a). This form of  is physi-

cally reasonable as observations suggest that  is very

small in the present universe. A decreasing functional

form permits  to be large in the early universe.

5) The ratio /0 as t∞ .So the model approach

isotropy for large value of t.

In summary, we have investigated the Bianchi type-I

cosmological model with variable G and  in presence of

perfect fluid with cosmological term proportional to R-2

(R is scale factor) suggested by Silveira et al., (1994,

1997) and others. Initially the model has a point type

singularity, gravitational constant G (t) is decreasing and

cosmological constant  is infinite at this time when time

increases  decrease.

It is interesting that Beesham (1994) , Lima and Car-

valho (1994), Kalligas, et al., (1995) and Lima (1996)

have also derived the Bianchi type I cosmological mod-

els with variable G and  assuming a particular from of

G and by taking equation of state .But we have neither

assumed equation of state nor particular form of G. The

model approach isotropy for large value of t, the model is

quasi-isotropic i.e. 0.







HARPREET ET AL. 77

REFERENCES

[1] B. Saha, “Anisotropic Cosmological Models with a Per-

fect Fluid and a Term,” Astrophysics and Space Science ,

Vol. 302, No. 1-4, 2006a, pp. 83-91.

[2] B. Saha, “Anisotropic Cosmological Models with a Per-

fect Fluid and Dark Energy Reexamined,” International

Journal of Theoretical Physics, Vol. 45, No. 5, 2006b, pp.

952-964.

[3] B. Saha, “Bianchi Type I Universe With Viscous Fluid,”

Modern Physics Letters A, Vol. 20, No. 28, 2005, p. 2127.

doi：10.1142/S021773230501830X

[4] B. Saha, “Anisotropic Cosmological Models with Perfect

Fluid and Dark Energy,”Chinese, 2005.

[5] R. G. Vishwakarma, “A Model to Explain Varying, G and

Simultaneously,” General Relativity and Gravitation,

Vol. 37, No. 7, 2005, pp. 1305-1311.

doi：10.1007/s10714-005-0113-0

[6] S. Carneiro and J. A. S. Lima, “Time Dependent Cosmo-

logical Term and Holography,” Modern International

Journal of Physics A, Vol. 20, No. 11, 2005, p. 2465.

doi:10.1142/S0217751X0502478X

[7] J. V. Cunha, R. C. Santos, “The Existence Of An Old

Quasar At Z = 3.91 And Its Implications For Λ(T) Defla-

tionary Cosmologies Read,” Modern International Jour-

nal of Physics D, Vol. 13, No. 7, 2004, p. 1321.

doi.org/10.1142/S0218271804005481

[8] A. G. Riess, et al., Astro nomical Journal, pp. 607- 665.

[9] S. W. Allen, et al., “Probing Dark Energy with Constella-

tion-X,” Monthly Notices of the Royal Astronomical So-

ciety, Vol. 353, No. 2, 2004, pp. 457-467.

doi:10.1111/j.1365-2966.2004.08080.x

[10] J. A. S. Lima “Alternative Dark Energy Models: An

Overview,” Brazilian Journal of Physics, Vol. 34, No. 1a,

2004, pp. 194-200.

doi：10.1590/S0103-97332004000200009

[11] T. Padmanabhan, “Cosmological Constant the Weight of

the Vacuum,” Physics Report, Vol. 380, No. 5-6, 2003,

pp. 235-320. doi:10.1016/S0370-1573(03)00120-0

[12] P. J. E. Peebles and B. Ratra, “The Cosmological Con-

stant and Dark Energy,” Reviews of Modern Physics, Vol.

75, No. 2, 2003, pp. 559-606.

doi：10.1103/RevModPhys.75.559

[13] R. G. Vishwakarma, “Study of the Magnitude-Redshift

Relation for Type Ia Supernovae in a Model Resulting

from a Ricci-Symmetry,” General Relativivity and Grav-

ity, Vol. 33, 2001, p. 1973.

[14] R. A. Vishwakarma, “A Study of Angular Size-redshift

Relation for Models in Which ᴧ Decays as the Energy

Density,” Class Quantum Gravity, Vol. 17, 2000, pp.

38-33. doi：10.1088/0264-9381/17/18/317

[15] S. Perlmutter, et al., “Measurements of Ω and Λ from 42

High-Redshift Supernovae,” Astrophysical Journal, 1999,

pp. 517- 565.

[16] I. Arbab, “Bianchi Type I Universe with Variable G and

ᴧ,” General Relativity and Gravitation, Vol. 30, No.

9,1998, pp. 1401-1405. doi:0.1023/A:1018856625508

[17] S. Perlmutter, et al., “Discovery of a Supernova Explo-

sion at Half the Age of the Universe,” Nature, Vol. 391,

1998.

[18] A. G. Riess, et al., “Observational Evidence from Super-

novae for an Accelerating Universe and a Cosmological

Constant,” the Astronphysical Journal, Vol. 116 , 1998, p.

1009.

[19] S. Perlmutter, et al., “Measurements of the Cosmological

Parameters Ω and Λ from the First Seven Supernovae at z

≥ 0.35,” the Astrophysical Journal, Vol. 483, No. 2,1997,

p. 565.doi：10.1086/304265.

[20] V. Silveira and J. Waga, “Cosmological Properties of a

Class of ᴧ Decaying Cosmologies,” Physical Review D,

Vol. 56, 1997, p. 4625.

[21] J. A. S. Lima and M. Trodden, “Decaying Vacuum En-

ergy and Deflationary Cosmology in Open and Closed

Unverses,” Physical Review D, Vol. 53, No. 8, 1996, pp.

4280-4286. doi: 10.1103/PhysRevD.53.4280

[22] J. A. S. Lima, “Thermodynamics of Decaying Vacuum

Cosmologies,” Physical Review D, Vol. 54, p. 2571.

[23] L. F. B. Torres and I. Waga, “Decaying Lambda Cos-

mologies and Statistical Properties of Gravitational Lens-

es,” Monthly Noices of the Royal Astronomical Society,

Vol. 279, 1996, pp. 712-726.

doi: 10.1093/mnras/279.3.712

[24] D. Kalligas, P. S. Wesson and C. W. F. Everitt, “Bianchi

Type I Cosmological Models with VariableG and Λ: A

Comment, General Relativity and Gravitivity, Vol. 27,

1995, pp. 645-650. doi：10.1007/BF02108066

[25] A. I. Arbab and A. M. M. Abdel-Rahaman, “Nonsingular

Cosmology with a Time-dependent Cosmological Term,”

Physical Review D, Vol. 50, 1994, pp. 7725-7728.

doi：10.1103/PhysRevD.50.7725

[26] A. Beesham, General Relativity and Gravitivity, Vol. 26

1599.

[27] J. A. S. Lima and J. M. F. Maia, Physical Review D, Vol.

49, 1994, p. 5579.

[28] J. A. S.Lima and J. C. Carvalho, “Dirac's Cosmology with

Varying Cosmological Constant,” General Relativity and

Gravitivity, Vol. 26, 1994, pp. 909-916.

doi10.1007/BF02107147

[29] M. D. Maia and G. S. Silva, “Geometrical Constraints on

the Cosmological Constant,” Physical Review D, Vol. 50 ,

1994, pp. 7233-7238. doi：10.1103/PhysRevD.50.7233

[30] V. Silveira and J. Waga, “Decaying ᴧ Cosmologies and

Power Spectrum,” Physical Review D, Vol. 50, 1994, pp.

4890-4894. doi：10.1103/PhysRevD.50.4890

[31] J. C. Carvalho, J. A. S. Lima and I. Waga, “Cosmological

Consequences of a Time-dependent Term,” Physical

Review D, Vol. 46, No. 6, 1992, pp. 2404-2407.

doi：10.1103/PhysRevD.46.2404

[32] M. S. Berman, “Cosmological Models with Variable

Gravitational and Cosmological Constants,” General

Relativity and Gravitation, Vol. 23, 1991a, pp. 465-469.

doi：10.1007/BF00756609

[33] M. S. Berman, “Cosmological Models with Variable-

HARPREET ET AL.

Cosmological Term,” Physical Review D, Vol. 43, 1991b,

pp. 1075-1078.

[40] M. S. Berman, M. M. Som and F. M. Gomide,

“Brans-Dicke Static Universes,” General Relativity and

Gravitation, Vol. 21, 1989, pp. 287-292.

doi：10.1007/BF00764101

[34] D. Pavon, “Nonequilibrium Fluctions in Cosmic Vacuum

Decay,” Physical Review D, Vol. 43, 1991, pp. 375-378.

doi：10.1103/PhysRevD.43.375 [41] S. Weinberg, “The Cosmological Constant Problem,”

Review Modern Physics, Vol. 61, 1989, pp. 1-23.

doi：10.1103/RevModPhys.61.1

[35] A. M. M. Abdel - Rahaman, “A Critical Density Cosmo-

logical Model with Varying Gravitation and Cosmologi-

cal Constants,” General Relativity and Gravitation, Vol.

22, No. 6, 1990, pp. 655-663. doi：10.1007/BF00755985

[42] T. S. Olson and T. F. Jordan, “Ages of the Universe for

Decreasing Cosmological Constants,” Physical Review D,

Vol. 35, 1987, pp. 3258-3260.

doi:10.1103/PhysRevD.35.3258

[36] M. Berman, “Static Universe in a Modified Brans- dicke

Cosmology,” International Journal of Theoretical Phys-

ics, Vol. 29, 1990, pp. 567-570.

doi：10.1007/BF00672031

[43] O. Bertolami Brans-Dicke, “Cosmology with a Scalar

Field Dependent Cosmological Term,” Fortsch Physics,

Vol. 34, 1986b, p. 829.

[37] M. S. Berman and M. M. Som, “Brans-Dicke Models

with Time-dependent Cosmological Term,” International

Journal of Theory Physics, Vol. 29, 1990, pp. 1411-1414.

doi：10.1007/BF00674120

[44] O. Bertolami, “Time-dependence Cosmological Term,”

Nuovo Cimento B, Vol. 93, 1986a, pp. 36-42.

doi:10.1007/BF02728301

[38] W. Chen and Y. S. Wu, “Implication of a Cosmological

Constant Varying as ,” Physical Review D, Vol. 41,

1990, p. 695.

[45] T. L. Chow, “The Variability of the Gravitational Con-

stant,” Nuovo Cimento Lettere, Vol. 31, 1981, pp.

119-120. doi:10.1007/BF02822409

[46] L. S. Levitt, “The Gravitational Constraint at Time Zero,”

Nuovo Cimento,Lettere, Serie 2, Vol. 29 , 1980, p. 23.

[39] E. A. Milne, “Relativity, Gravitation and World struc-

ture,” Oxford University Press, Oxford, 1935.