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SOME EXACT BIANCHI TYPE-I COSMOLOGICAL MODELS INSCALAR-TENSOR THEORY OF GRAVITATION WITH TIME DEPENDENT

DECELERATION PARAMETER

ANIRUDH PRADHAN1, ANAND SHANKAR DUBEY2, RAJEEV KUMAR KHARE3

1Department of Mathematics,Hindu Post-graduate College,Zamania-232 331, Ghazipur, India

E-mail: [email protected]; [email protected],3Department of Mathematics, Sam Higginbottom Institute of Agriculture, Technology & Sciences

Allahabad-211 007, IndiaE-mail: [email protected]

3Department of Mathematics, Sam Higginbottom Institute of Agriculture, Technology & SciencesAllahabad-211 007, India

E-mail: [email protected]

Received October 5, 2011

A new class of a spatially homogeneous and anisotropic Bianchi type-I cos-mological models of the universe for perfect fluid distribution within the framework ofscalar-tensor theory of gravitation proposed by Saez and Ballester (Phys. Lett. 113:467,1986) is investigated by considering time dependent deceleration parameter. Two dif-ferent physically viable models of the universe are obtained. The modified Einstein’sfield equations are solved exactly and the models are in good agreement with recent ob-servations. Some physical and geometric properties of the models are also discussed.

Key words: Bianchi type-I universe, exact solution, alternative gravitation the-ory, variable decceleration parameter.

PACS: 98.80.-k.

1. INTRODUCTION

General relativity (GR) passes all present tests with flying colours, however,there are several reasons why it remains very important to consider alternative the-ories of gravity. The first one is that theoretical attempts at quantizing gravity orunifying it with other interactions generically predict deviation from Einstein’s the-ory, because gravitation is no longer mediated by a pure spin-2 field but also bypartners to the graviton. The second reason is that it is any way extremely instructiveto contrast GR’s predictions with those of alternative models, even if there were noserious theoretical motivation for them. The third reason is the existence of severalpuzzling experimental issues, which do not contradict GR in a direct way, but maynevertheless suggest that gravity does not behave at large distances exactly as New-ton and Einstein predicted. Cosmological observations notably tell us that about 96%of the total energy density of the universe is composed of unknown, non-baryonic,

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2 Exact Bianchi type-I cosmological models with time dependent deceleration parameter 1223

fluids (72% of “dark energy” and 24% of “dark matter”), and the acceleration of thetwo Pioneer spacecrafts towards the Sun happens to be larger than what is expectedfrom the 1/r2 law. Therefore, alternative theories are any way important to study.

Since the observed universe is almost homogeneous and isotropic, space-timeis usually described by a Friedman-Lemaitre-Robertson-Walker (FLRW) cosmology.But it is also believed that in the early universe the FLRW model does not give acorrect matter description. The anomalies found in the cosmic microwave back-ground (CMB) and the large structure observations stimulated a growing interest inanisotropic cosmological model of the universe. Observations by the Differential Ra-diometers on NASA’s Cosmic Background Explorer registered anisotropy in variousangle scales. It is conjectured, that these anisotropies hide in their hearts the entirehistory of the cosmic evolution down to recombination, and they are considered to beindicative of the universe geometry and the matter composing the universe. It is ex-pected, that much more will be known about anisotropy of cosmic microwave’s back-ground after the investigations of the microwave’s anisotropy probe. There is a gen-eral agreement among cosmologists that cosmic microwave’s background anisotropyin the small angle scale holds the key to the formation of the discrete structure. Thetheoretical argument [1] and the modern experimental data support the existence ofan anisotropic phase, which turns into an isotropic one.

Scalar-Tensor theories of gravitation provide the natural generalizations of gen-eral relativity and also provide a convenient set of representations for the observa-tional limits on possible deviations from general relativity. There are two categoriesof gravitational theories involving a classical scalar field φ. In first category the scalarfield φ has the dimension of the inverse of the gravitational constant G among whichthe Brans-Decke theory [2] is of considerable importance and the role of the scalarfield is confined to its effect on gravitational field equations. In the second categoryof theories involve a dimensionless scalar field. Saez and Ballester [3] developed ascalar-tensor theory in which the metric is coupled with a dimensionless scalar fieldin a simple manner. This coupling gives a satisfactory description of the weak fields.In spite of the dimensionless character of the scalar field, an anti-gravity regime ap-pears. This theory suggests a possible way to solve the missing-matter problem innon-flat FRW cosmologies. The Scalar-Tensor theories of gravitation play an im-portant role to remove the graceful exit problem in the inflation era [4]. In earlierliterature, cosmological models within the framework of Saez-Ballester scalar-tensortheory of gravitation, have been studied by Singh and Agrawal [5, 6], Ram and Ti-wari [7], Singh and Ram [8]. Mohanty and Sahu [9, 10] have studied Bianchi type-VI0 and Bianchi type-I models in Saez-Ballester theory. Recently, Tripathi et al. [11],Reddy et al. [12,13], Reddy and Naidu [14], Rao et al. [15]− [18], Adhav et al. [18],Katore et al. [19], Mohanty and Sahu [20] and Pradhan and Singh [21] have ob-tained the solutions in Sauz-Ballester Scalar-Tensor theory of gravitation in different

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1224 Anirudh Pradhan, Anand Shankar Dubey, Rajeev Kumar Khare 3

context.Recently, Kumar and Singh [22] obtained exact Bianchi type-I cosmological

models in Saez and Ballester Scalar-Tensor theory of gravitation by assuming theconstant deceleration parameter. In literature it is common to use a constant decel-eration parameter, as it duly gives a power law for metric function or correspondingquantity. In 1998, published observations of Type Ia supernovae by the High-z Su-pernova Search Team (Riess et al. [23]) followed in 1999 by Supernova CosmologyProject (Perlmutter et al. [24]) suggested that the expansion of the universe is acceler-ating. Recent observations of SNe Ia of high confidence level (Tonry et al. [25]; Riesset al. [26]; Clocchiatti et al. [27]) have further confirmed this. Also, the transitionredshift from deceleration expansion to accelerated expansion is about 0.5. Now for aUniverse which was decelerating in past and accelerating at the present time, the DPmust show signature flipping (see the Refs. Padmanabhan and Roychowdhury [28],Amendola [29], Riess et al. [30]). So, there is no scope for a constant DP at presentepoch. So, in general, the DP is not a constant but time variable. Motivated by thediscussions, in this paper, we have obtained a new class of exact solutions of fieldequations given by Saez and Ballester [3] in Bianchi type-I space-time by consider-ing a time dependent deceleration parameter. The out line of the paper is as follows:In Section 2, the metric and the field equations are described. Section 3 deals withthe solutions of the field equations and its physical significance. Section 4 deals withan other solutions of the field equations and physical and geometric behaviour of themodel are also discussed. Finally, conclusions are summarized in the last Section 5.

2. THE METRIC AND FIELD EQUATIONS

We consider spatially homogeneous and anisotropic Bianchi type-I space-timegiven by

ds2 =−dt2+A2dx2+B2dy2+C2dz2, (1)where the metric potentials A, B and C are functions of cosmic time t alone. Thisensures that the model is spatially homogeneous.

The field equations given by Saez and Ballester [3] for combined scalar andtensor fields are

Gij−ωφm(φ,iφ,j−

1

2gijφ,kφ

,k

)=−8πTij , (2)

where the scalar field φ satisfies the equation

2φmφ,i;i+mφ,kφ,k = 0, (3)

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where Gij = Rij − 12Rgij is the Einstein tensor; ω and m are constants; Tij is the

stress-energy tensor of the matter; comma and semicolon denotes partial and covari-ant differentiation, respectively.

The energy-momentum tensor Tij for a perfect fluid has the form

Tij = (p+ρ)uiuj−pgij , (4)

where p is the thermodynamical pressure, ρ the energy density, ui the four-velocityof the fluid satisfying

gijuiuj = 1. (5)

In co-moving system of coordinates, we have ui=(0,0,0,1). For the energy momen-tum tensor (4) and Bianchi type-I space-time (1), Einstein’s modified field equations(2) and (3) yield the following five independent equations

A

A+B

B+AB

AB=−8πp+ 1

2ωφmφ2, (6)

A

A+C

C+AC

AC=−8πp+ 1

2ωφmφ2, (7)

B

B+C

C+BC

BC=−8πp+ 1

2ωφmφ2, (8)

AB

AB+AC

AC+BC

BC= 8πρ− 1

2ωφmφ2, (9)

φ+ φ

(A

A+B

B+C

C

)+mφ2

2φ= 0, (10)

where an over dot denotes derivative with respect to cosmic time t. The law ofenergy-conservation equation (T ij

;j = 0) gives

ρ+(ρ+p)

(A

A+B

B+C

C

)= 0. (11)

It is worth noting here that our approach suffers from a lack of Lagrangian approach.There is no known way to present a consistent Lagrangian model satisfying the nec-essary conditions discussed in this paper.

The spatial volume for the model (1) is given by

V 3 =ABC. (12)

We define a= (ABC)13 as the average scale factor so that the Hubble’s parameter is

anisotropic and may be defined as

H =a

a=

1

3

(A

A+B

B+C

C

). (13)

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We also have

H =1

3(Hx+Hy+Hz) , (14)

where Hx =AA , Hy =

BB and Hz =

CC .

The deceleration parameter q, the scalar expansion θ, shear scalar σ2 and theaverage anisotropy parameter Am are defined by

q =−aaa2, (15)

θ =A

A+B

B+C

C, (16)

σ2 =1

2

(3∑

i=1

H2i −

1

3θ2

), (17)

Am =1

3

3∑i=1

(4Hi

H

)2

, (18)

where4Hi =Hi−H(i= x,y,z).

3. SOLUTIONS OF THE FIELD EQUATIONS

The field equations (6)-(10) are five equations necessitating six unknowns A,B, C, p, ρ and φ. One additional constraint relating these parameters are requiredto obtain explicit solutions of the system. We consider the deceleration parameter astime dependent to get the deterministic solution.

We define the deceleration parameter q as

q =−aaa2

=−

(H+H2

H2

)= b(t) say. (19)

As discussed in previous section of introduction, the deceleration parameter is con-sidered as time dependent. The equation (19) may be rewritten as

a

a+ b

a2

a2= 0. (20)

In order to solve the Eq. (20), we assume b = b(a). It is important to note here thatone can assume b= b(t) = b(a(t)), as a is also a time dependent function. It can bedone only if there is a one to one correspondences between t and a. But this is onlypossible when one avoid singularity like big bang or big rip because both t and a areincreasing function.

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The general solution of Eq. (20) with assumption b= b(a), is given by∫e∫

bada = t+m, (21)

where m is an integrating constant.One can not solve Eq. (21) in general as b is variable. So, in order to solve

the problem completely, we have to choose∫

bada in such a manner that Eq. (21) be

integrable without any loss of generality. Hence we consider∫b

ada= lnL(a). (22)

which does not affect the nature of generality of solution. Hence from Eqs. (21) and(22), we obtain ∫

L(a)da= t+m. (23)

Of course the choice of L(a), in Eq. (23), is quite arbitrary but, since we are look-ing for physically viable models of the universe consistent with observations, weconsider

L(a) =1

α√1+a2

, (24)

where α is an arbitrary constant. In this case, on integrating, Eq. (23) gives the exactsolution

a(t) = sinh(αT ). (25)where T = t+m. Recentlly, relation (25) is also used by Amirhashchi et al. [31] tostudy the evolution of dark energy models in a spatially homogeneous and isotropicFRW space-time filled with barotropic fluid and dark energy by considering a timedependent deceleration parameter. Substituting (25) into (15), we obtain

q =−tanh2 (αT ), (26)

which is a variable deceleration parameter.Now subtracting (6) from (7), (6) from (8), (7) from (8) and taking second

integral of each expression, we obtain the following three relations, respectively

A

B= d1 exp

(x1

∫a−3dt

), (27)

A

C= d2 exp

(x2

∫a−3dt

), (28)

B

C= d3 exp

(x3

∫a−3dt

), (29)

where d1, d2, d3, x1, x2 and x3 are integrating constants.

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From above Eqs. (27)-(29), the metric functions A(t), B(t) and C(t) are ex-plicitly obtained as

A(t) = a1aexp

(b1

∫a−3dt

), (30)

B(t) = a2aexp

(b2

∫a−3dt

), (31)

C(t) = a3aexp

(b3

∫a−3dt

), (32)

where

a1 = (d1d2)13 , a2 =

(d3d1

) 13

, a3 = (d2d3)− 1

3 , b1 =1

3(x1+x2),

b2 =1

3(x3−x1), b3 =−

1

3(x2+x3).

It is worth mentioned here that these constants also satisfy the following two condi-tions

a1a2a3 = 1, b1+ b2+ b3 = 0. (33)

The second integral of (10) leads to

φ(t) =

[k(m+2)

2

∫a−3dT

] 2(m+2

, (34)

where k is a constant due to first integral while the constant of second integral istaken as zero for simplicity without any loss of generality.

Using Eq. (25) in (30)-(32), we get the following expressions for metric coef-ficients

A= a1 sinh(αT )exp

(b1

∫cosech3(αT)dT

), (35)

B = a2 sinh(αT )exp

(b2

∫cosech3(αT)dT

), (36)

C = a3 sinh(αT )exp

(b3

∫cosech3(αT)dT

). (37)

Again substituting (25) in (34), the scalar field is obtained as

φ=

[k(m+2)

2

∫cosech3(αT)dT

] 2(m+2)

, (38)

Eq. (38) implies that

φmφ2 = k2cosech6(αT) (39)

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Substituting (35)-(39) in (8) and (9), the expressions for isotropic pressure (p) andenergy density (ρ) for the model are obtained as

p=

(1

2ωk2−β3

)cosech6(αT)−α2

{2+coth2 (αT)

}, (40)

ρ= 3α2 coth2 (αT )+

(1

2ωk2+β2

)cosech6(αT), (41)

whereβ1 = b21+ b

22+ b

23,

β2 = b1b2+ b2b3+ b3b1,

β3 = b22+ b23+ b1b2. (42)

In view of (33), it is observed that the solutions given by (35)-(42) satisfy the energyconservation equation (11) identically and hence represent exact solutions of the Ein-stein’s modified field equations (6)-(10). It is evident that the energy conditions ρ≥ 0and p≥ 0 are satisfied under the appropriate choice of constants. Due to exponentialnature of scale factor we are not in position to derive analytical conditions exactlybut apparent that such feature are there in present study. From Eq. (41), it is ob-served that ρ is a positive decreasing function of time and it approaches to zero asT →∞. This behaviour is clearly depicted in Figure 1 as a representative case withappropriate choice of constants of integrations and other physical parameters usingreasonably well known situations.

The rate of expansion Hi in the direction of x, y, and z read as

Hx = αcoth(αT )+ b1cosech3(αT), (43)

Hy = αcoth(αT )+ b2cosech3(αT), (44)

Hz = αcoth(αT )+ b3cosech3(αT). (45)

The Hubble parameter, expansion scalar and shear of the model are, respectivelygiven by

H = αcoth(αT ), (46)θ = 3αcoth(αT ), (47)

σ2 =1

2β1cosech

6(αT). (48)

The spatial volume (V ) and anisotropy parameter (Am) are found to be

V = sinh3 (αT ), (49)

Am =1

3β1

cosech6(αT)

α2 coth2 (αT ). (50)

From the above results, it can be seen that the spatial volume is zero at T = 0, andit increases with the cosmic time. The parameter Hi, H , θ, and σ diverge at the

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Fig. 1 – The plot of energy density ρ versus T . Here α= 1, k = β2 = 1.

Fig. 2 – The plot of anisotropic parameter Am versus T . Here α= 1.

initial singularity. There is a Point Type singularity (MacCallum [31]) at T = 0 in

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the model. The dynamics of the mean anisotropic parameter depends on the constantβ1 = b21+b

22+b

23. From Eq. (50), we observe that at late time when T →∞,Am→ 0.

Thus, our model has transition from initial anisotropy to isotropy at present epochwhich is in good harmony with current observations. Figure 2 depicts the variationof anisotropic parameter (Am) versus cosmic time T. From the figure, we observe thatAm decreases with time and tends to zero as T →∞. Thus, the observed isotropyof the universe can be achieved in our model at present epoch. From Eq. (26), itcan be seen that for T →∞, q = −1 and for T → 0, q = 0. Hence our universe isin accelerating phase during the evolution of the universe which is consistent withrecent observations of Type Ia supernovae (Riess et al. [23]; Perlmutter et al. [24];Tonry et al. [25]; Riess et al. [26]; Clocchiatti et al. [27]).

4. OTHER EXACT SOLUTION OF THE FIELD EQUATIONS AND THEIR PHYSICALASPECTS

For a suitable choice of L(a), Eq. (23) gives after integration the exact solutionfor the scale factor, where the increase in terms of time evolution is

a(t) = tet. (51)

We define the deceleration parameter q as usual,

q =− aaa2

=− a

aH2. (52)

Using (51) into (52), we find

q =−1+ 1

(1+ t2). (53)

Using Eq. (25) in (30)-(32), we get the following expressions for metric coefficients

A= a1(tet)exp

(b1

∫(tet)−3dt

), (54)

B = a2(tet)exp

(b2

∫(tet)−3dt

), (55)

C = a3(tet)exp

(b3

∫(tet)−3dt

). (56)

Again substituting (25) in (34), the scalar field is obtained as

φ=

[k(m+2)

2

∫(tet)−3dt

] 2(m+2)

, (57)

Eq. (57) implies thatφmφ2 = k2(tet)−6 (58)

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Substituting (54)-(58) in (8) and (9), the expressions for isotropic pressure (p) andenergy density (ρ) for the model are obtained as

p=2−3(t+1)2

t2+ωk2−2β32(tet)6

, (59)

ρ= 3

(t+1

t

)+ωk2+2β22(tet)6

. (60)

where β2 and β3 are already defined in previous section.

Fig. 3 – The plot of energy density ρ versus t. Here k = β2 = 1.

In view of (33), it is observed that the solutions given by (54)-(60) satisfy theenergy conservation equation (11) identically and hence represent exact solutions ofthe Einstein’s modified field equations (6)-(10). It is evident that the energy condi-tions ρ ≥ 0 and p ≥ 0 are satisfied under the appropriate choice of constants. FromEq. (60), it is observed that ρ is a positive decreasing function of time and it ap-proaches to zero as t→ 0. Figure 3 shows this behaviour of ρ.

The rate of expansion Hi in the direction of x, y, and z read as

Hx =

(t+1

t

)+ b1(te

t)−3, (61)

Hy =

(t+1

t

)+ b2(te

t)−3, (62)

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Fig. 4 – The plot of anisotropic parameter Am versus t. Here β1 = 1.

Hz =

(t+1

t

)+ b3(te

t)−3. (63)

The Hubble parameter, expansion scalar and shear of the model are, respectivelygiven by

H =

(t+1

t

)(64)

θ = 3

(t+1

t

), (65)

σ2 =1

2β1(te

t)−6 (66)

The spatial volume (V ) and anisotropy parameter (Am) are found to be

V = (tet)3, (67)

Am =1

3β1

(t+1

t

)−2

(tet)−6, (68)

where β1 is already defined in previous section.

From the above results, it can be seen that the spatial volume is zero at t = 0,and it increases with the cosmic time. The parameter Hi, H , θ, and σ diverge at theinitial singularity. There is a Point Type singularity (MacCallum [32]) at t = 0 inthe model. The mean anisotropic parameter is a decreasing function of time. Figure

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4 plots the variation of anisotropic parameter (Am) versus cosmic time t. From thefigure, we observe that Am decreases with time and tends to zero as t→∞. Thus,the observed isotropy of the universe can be achieved in our model at present epoch.

5. CONCLUDING REMARKS

In this paper we have studied a spatially homogeneous and anisotropic Bianchitype-I space-time within the framework of the scalar-tensor theory of gravitation pro-posed by Saez and Ballester [3]. To find the deterministic solution, we have consid-ered a time dependent deceleration parameter. In the first case, it is observed thatas T →∞, q = −1. This is the case of de Sitter universe. For T → 0, q = 0. Thisshows that in the early stage the universe was decelerating where as the universe isaccelerating at present epoch which is corroborated from the recent supernovae Ia ob-servation (Riess et al. [23]; Perlmutter et al. [24]; Tonry et al. [25]; Riess et al. [26];Clocchiatti et al. [27]). The parameter Hi, H , θ, and σ diverge at the initial singu-larity. There is a Point Type singularity (MacCallum [32]) at T = 0 in the model.The rate of expansion slows down and finally tends to zero as T → 0. The pres-sure, energy density and scalar field become negligible where as the scale factors andspatial volume become infinitely large as T →∞, which would give essentially anempty universe. We also obtain similar type of characteristic of the model describedin second case (Section 4).

Both the models represent expanding, shearing and non-rotating universe, whichapproach to isotropy for large value of t. This is consistent with the behaviour of thepresent universe as already discussed in introduction. The scalar field decreases tozero as t→∞ in both models. If we set k = 0, then the solutions reduce to the so-lution in general relativity and the Saez-Ballester scalar-tensor theory of gravitationtends to standard general theory of relativity in every respect. In literature we canget the solutions of the field equations of Saez-Ballester scalar-tensor theory of grav-itation by using a constant deceleration parameter. So the solutions presented in thispaper are new and different from other author’s solutions. Our solutions may be use-ful for better understanding of the evolution of the universe in Bianchi-I space-timewithin the framework of Saez-Ballester scalar-tensor theory of gravitation.

Acknowledgments. Author (AP) would like to thank the Inter-University Centre for Astronomyand Astrophysics (IUCAA), Pune, India for providing facility and support where part of this work wascarried out.

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