1 DYNAMIC COSMOLOGICAL CONSTANT IN BRANS DICKE THEORY G P SINGH, AY KALE, J TRIPATHI 3 Department of Mathematics, Visvesvaraya National Institute of T...

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Department of Mathematics, Visvesvaraya National Institute of Technology, Nagpur - 440010, India. 2 Department of Mathematics, St. Vincent Pallotti College of Engineering and Technology, Nagpur-441108, India. 3 Department of Mathematics, SRK College of Engineering, Nagpur, India. 1 E-mail: [email protected] 2 E-mail: [email protected] 3 E-mail: [email protected] Received February 8, 2012

A new class of homogeneous and isotropic cosmological model with variable cosmological term Λ in the Brans Dicke theory has been obtained. The effect of cosmological term during accelerated expansion of the universe is investigated in the flat, open and closed FRW models of the universe. Various forms of phenomenological decay law for dynamic cosmological “constant” are considered for obtaining exact cosmological solutions. Physical behaviour of the models have also been discussed. Key words: Cosmological constant, Brans-Dicke theory, FRW models.

1. INTRODUCTION Since last few decades there is a growing interest in alternative theories of gravitation, especially scalar-tensor theories of gravity, which are very useful tools in understanding early universe models. The Brans-Dicke theory [1] of gravity is most promising one among all existing alternative theories of gravitation. It was shown that inflationary model [2], extended inflationary model [3], hyper extended inflationary model [4], chaotic inflation [5], are based on Brans-Dicke scalar tensor theory. In this theory gravitational constant is replaced by reciprocal of a massless scalar field φ . It has been suggested that large value of coupling parameter (ω ≥ 500) makes the results of BD theory practically indistinguishable from Einstein general theory of relativity [6]. A number of authors [7–23] studied cosmological models in Brans-Dicke theory to investigate various aspects of expanding models of the universe. Rom. Journ. Phys., Vol. 58, Nos. 1–2, P. 23–35, Bucharest, 2013

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G.P. Singh, A.Y. Kale, J. Tripathi

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The end of twentieth century has witness various changes in the theories of cosmic evolution of the universe. The measurements of the luminosity-redshift relations observed for 50 newly discovered type Ia supernovae with redshift z > 0.35 [24–25] and WMAP observations [26] predicted accelerated expansion of the universe [27–28]. These observations indicated that present constituent of the universe is dominated by some kind of energy with negative pressure, commonly known as dark energy, which constitutes about three fourths of the whole matter of our universe. The simplest and the most favoured candidate of dark energy is a cosmological “constant” which on one hand provide enough negative pressure to account this acceleration and on other the hand contribute an energy density of same order of magnitude than the energy density of the matter [27]. Observational data indicates that Cosmological constant Λ = 10−55 cm −2 while theoretical prediction for Λ is greater than this value by a factor of order 10120 . This discrepancy usually called cosmological “constant” problem, which is one of the puzzling problems in standard cosmology. The dynamical Λ was invoked to study the phenomenological decay of Λ so that it might be large at early epochs and reducing to a small value at the present epoch. A number of authors constructed models of more phenomenological character in which specific decay laws are postulated within the framework of general relativity. One of the very promising model among such models has the relation Λ ∝ H 2 [29–33]. The effect of cosmological ‘constant’ has been extensively studied in the literature within the framework of general relativity and its alternative theories. Singh and Singh [9] investigated a cosmological model in Brans-Dicke theory by considering cosmological “constant” as function of scalar field φ . Pimentel [10] obtained exact cosmological solutions in Brans-Dicke theory with uniform cosmological “constant”. A class of flat FRW cosmological models with cosmological “constant” in Brans-Dicke theory have also been obtained by Azar and Riazi [12]. The age of the universe from a view point of the nucleosynthesis with Λ term in Brans-Dicke theory was investigated by Etoh et al [13]. Azad and Islam [28] extended the idea of Singh and Singh [9] to study cosmological constant in Bianchi type I modified Brans-Dicke cosmology. Recently Qiang et al [34] discussed cosmic acceleration in five dimensional Brans-Dicke theory using interacting Higgs and Brans-Dicke fields. Smolyakov [35] investigated a model which provides the necessary value of effective cosmological “constant” at the classical level. Embedding general relativity with varying cosmological term in five dimensional Brans-Dicke theory of gravity in vacuum has been discussed by Reyes et al [36]. Motivated by above studies the investigation of role of dynamic cosmological “constant” has been considered in Brans-Dicke theory.

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Dynamic cosmological “constant” in Brans Dicke theory

25

2. FIELD EQUATIONS

The field equation of Brans Dicke theory in presence of cosmological constant may be written as Rij − 1 Rgij − Λgij + ω2 2 φ

φ φ − 1 g φ φ;k + 1 φ − g , φ = 8π T , ij φ ij ;i ; j 2 ij ;k φ ;i ; j , φ = φ;;ii =

8π T i ,i 2ω + 3

(1) (2)

where φ is the scalar field. The energy momentum tensor Tij of cosmic fluid can be defined as Tij = (ρ + p )ui u j − g ij p

.

(3)

Let us consider a homogeneous and isotropic universe represented by FRW spacetime metric 2 ds 2 = dt 2 − R 2 (t ) dr 2 + r 2 (d θ2 + sin 2 θ d φ2 ) , 1 − kr

(4)

where R (t ) is the scale factor, k = 1, 0, −1 for spaces of positive, vanishing and negative curvature which represents closed, flat and open model of the universe respectively. The FRW metric (4) and energy momentum tensor (3) along with Brans-Dicke field equations yield the following equations 2 3R 2 + 3 R φ − ωφ + 3 k = 8π ρ + Λ, R φ 2φ 2 φ R2 R2

(5)

2 φ φ φ 2 2 R + R 2 + + ω 2 + 2 R + k2 = −8π p + Λ, R R φ 2φ Rφ R φ

(6)

φ ρ − 3p φ + 3 R = 8π + 2Λ . φ R φ φ 3 + 2ω 3 + 2ω

(7)

The geometrical quantities of observational interest Hubble parameter H and deceleration parameter q are defined by

H=R, R

(8)

( H + H ) q=− 2

H2

(9)

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G.P. Singh, A.Y. Kale, J. Tripathi

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In order to find exact solutions of basic field equations (5)-(7), one must ensures that set of equations should be closed. Thus two more physically reasonable relations are required amongst the variables. Now considering a well accepted power law relation [10–12] between scale factor R (t ) and scalar field φ of the form φ = φ0 R α ,

(10)

the set of field equations (5)-(7), may be written as 6 + 6α − ωα 2 2

R 2 3k 8π ρ + Λ, 2 + 2 = φ0 R α R R

(11)

2 2 2 (2 + α) R + 2 + 2α + 2α + ωα R 2 + k2 = −8πα p + Λ, R 2 φ0 R R R

(12)

R 8π R 2 α R + α(α + 2) R 2 (3 + 2ω) = φ R α (ρ − 3 p) + 2Λ, 0

(13)

A combination of equations (11)-(13) leads to 2 2(3 − ωα) R + ( 6 − 4ωα − ωα 2 ) R 2 + 6 k2 = 2Λ. R R R

(14)

This equation is playing an important role in obtaining various cosmological solutions. 3. COSMOLOGICAL MODELS FOR FLAT FRW SPACE-TIME

It has been presented in the literature [15] that the findings of BOOMERANG experiment [37] strongly suggest the possibility of a flat universe. For flat model of the universe represented by FRW space-time (k = 0), equation (14) reduces to 2 2(3 − ωα) R + ( 6 − 4ωα − ωα 2 ) R 2 = 2Λ. R R

(15)

Now various phenomenological models of the dynamical cosmological “constant” Λ will be considered. In the absence of Λ term equation (15) reduces to the case already discussed by Johri and Kalyani [11]. 3.1. CASE I: MODEL WITH Λ ∝ H 2 Considering the commonly used relation between the cosmological “constant” and the Hubble parameter (H) ([33] and references there in) as

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Dynamic cosmological “constant” in Brans Dicke theory

27

Λ = βH 2 ,

(16)

2(3 − ωα) H + (12 − 6ωα − ωα 2 − 2β ) H 2 = 0 ,

(17)

equation (15) assumes the form

12 − 6ωα − ωα 2 − 2β ) ( H which may be written as − 2 = =a 2(3 − ωα) H

(say),

(18)

where a is a constant. On integration (18) yields the solution 1

R = ( at + b ) a for a ≠ 0, R = c1e

H 0t

for a = 0.

(19) (20)

Here c1, b and H0 are constants of integration. 3.1.1. Subcase I: model with power law solution

In order to obtain nonsingular cosmological model of the universe using power law relation (19) between the scale factor and cosmic time t. That gives relations for the scalar field and cosmological term respectively α

φ = k1 ( at + b ) a , Λ=

β

( at + b )

.

2

(21) (22)

Using these relations one can obtain following expressions for energy density and pressure φ (6 − ωα 2 + 6α − 2β) (23) ρ= 0 , 2− α 8π ( at + b ) a p=−

φ0 6 + 4α − 2β − 2a (2 + α) + α 2 (2 + ω) . 2− α 16π at + b a

(

)

(24)

In this case the geometrical quantities of observational interest take the form 1 , at + b

(25)

q = −1 + a .

(26)

H=

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G.P. Singh, A.Y. Kale, J. Tripathi

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The SNIa observations [38] suggest the range for deceleration parameter as – 0.715 to ± 0.045 which yields 0.24 ≤ a ≤ 0.33. 18

2.

16

2.

a = 0.24

14

a=0.33

12 10

R(t)

2.

a = 0.24

2.

a = 0.33 2

φ(t)

8

1.

6

1.

4

1.

2

1.

0

0

2

4

6

8

1

1 0

2

4

Time

6

8

1

Time

Figs. 1 and 2 show the variation of Scale factor and scalar field against cosmic time t. 700

0

600

-50

a=0.24

500

-100

a=0.33

400

-150 Λ

300

-200

200

-250

100

-300

ρ

0

0

2

4

Time

6

8

10

-350

a=0.24 a=0.33

0

2

4

Time

6

8

10

Figs. 3 and 4 show the variation of energy density and cosmological constant against cosmic time t .

φ0 = 1 a = 0.24 and 0.33 is considered. For 8π these values of constants, β can be obtained from equation (18) as β= –337.92 and – 327.39 resp. This shows that for all values of a in the above range, β< 0 which clearly shows that Λ < 0 . It can be easily seen that energy density, pressure, expansion scalar are decreasing with evolution of the universe. The deceleration parameter suggests that for all a < 1 the model presents accelerating expansion of the universe.

In this case b = 1, ω = 600, α = 0.2 ,

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Dynamic cosmological “constant” in Brans Dicke theory

29

3.1.2. Subcase II: model with exponential solution

Considering expression of scale factor as in (20), the scalar function assumes the form H t φ = c1e 0 (27) In this case the energy density and pressure assumes uniform value. 3.2. CASE II: MODEL WITH Λ ∝ R − n

Several authors ([33] and references there in) considered a familier relation between the scale factor and cosmological constant as Λ ∝ R − n . Λ = Λ0 R−n .

Therefore

(28)

By use of equation (28), one can write equation (15) as 2 2Λ 2(3 − ωα ) R + ( 6 − 4ωα − ωα 2 ) R 2 = n0 . R R R

(29)

With the change of variable u = R 2 , equation (29) may be written as du + ( 6 − 4ωα − ωα (3 − ωα) dR

2

) 1u= R

2Λ 0 R1− n (3 − ωα )

(30)

On solving this Leibnitz linear differential equation one can have expression for scale factor as

R = R0 t 1

2Λ

2

n

(31)

0 where R0 = 2 12 6 3 n n − ωα − ωα − + ωα Using this value of scale factor, the scalar field and cosmological “constant” assumes the form n

φ = φ0 R0 α t Λ=

2α n

Λ0 R0 n t 2

(32) (33)

This value of Λ is similar to the result obtained by authors [39 and references there in]. In this case energy density and pressure are expressed as φ0 R0 α (12 + 12α − 2ωα 2 − n 2 Λ 0 R0 − n ) 2 α −2 t n ρ= 8πn 2

(34)

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G.P. Singh, A.Y. Kale, J. Tripathi

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−φ0 R0 α (12 − 4n + 8α − 2αn + 4α 2 + 2ωα 2 − n 2 Λ 0 R0 − n ) 2 α t n p= 8πn 2

−2

(35)

Here the geometrical quantities of observational interest are H= 2, nt

(36)

q = −1 + n . 2

(37)

The range for deceleration parameter suggested by the SNIa observation [38] as –0.715 to ± 0.045 gives 0.48≤ n ≤ 0.66. 180

15000

160

n=0.48 n=0.66

10000

n=0.48

140

n=0.66

120

Φ(t)

R(t)

100

80 60

5000

40 20 0 0

2

4

Time

6

8

0 0

10

2

4 6 Time

8

10

Figs. 5 and 6 show plot of Scale factor and scalar field against cosmic time t. 4

9000

0

8000

-0.5

n=0.48

7000

n=0.66

-1.5 -2

5000

Λ -2.5

4000

-3

3000

-3.5

2000

-4

1000

-4.5

0 0

n=0.48

-1

n=0.66

6000

ρ

x 10

1

2

3

Time

4

5

-5 0

1

2

3

4

5

Time

Figs. 7 and 8 show plot of energy density and cosmological constant against cosmic time t.

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Dynamic cosmological “constant” in Brans Dicke theory

31

φ0 = 1 and n = 0.48 and 0.66 is considered. 8π These assumed values, gives Λ0 = – 477.8189 and – 414.4102 which shows similar behaviour of Λ as in previous case. It can be easily seen that energy density, pressure are decreasing with evolution of the universe. The deceleration parameter suggests that for all n < 2 the model presents accelerating expansion of the universe.

In this case R0 = 1, ω = 600, α = 0.2 ,

4. COSMOLOGICAL MODELS FOR NON-FLAT FRW SPACE-TIME 4.1. CASE I: MODEL WITH Λ ∝ H 2

In this case a cosmological model is obtained by considering the expression for cosmological constant as in equation (16). Equation (14) along with equation (16) gives 2 2(3 − ωα) R + ( 6 − 4ωα − ωα 2 − 2β ) R 2 − = −62k R R R

(38)

With the change of variable u = R 2 , equation (38) takes the form 2 du + ( 6 − 4ωα − ωα − 2β ) 1 u = −6k 1 dR (3 − ωα) R (3 − ωα) R

(39)

Further on integration equation (39), yield

R = R1t , where R1 =

( ωα

2

(40)

6k . − 6 + 4ωα + 2β )

This value of scale factor suggests φ = φ0 R1α t α ,

Λ=

β , t2

(41) (42)

Here the expressions for energy density and pressure are ρ=

p=−

φ0 R1α R1α (6 + 6α − ωα 2 ) + 6k − 2 R12 β) α− 2 , t 8π 2 R12

φ0 R1α R1α (2 + 2α 2 + 2α + ωα 2 ) + 2k − 2 R12 β) α− 2 t . 8π 2 R12

(43)

(44)

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G.P. Singh, A.Y. Kale, J. Tripathi

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The geometrical quantities of observational interest are H =1, t

(45)

q =0.

(46)

It can easily seen that energy density, pressure are decreasing with evolution of the universe. The deceleration parameter suggests that the model presents uniform expansion of the universe. 4.2. CASE II: MODEL WITH Λ ∝ R − n

Assuming expression for cosmological constant as in equation (28), equation (14) becomes 2 2Λ 2(3 − ωα) R + ( 6 − 4ωα − ωα 2 ) R 2 + 6k2 = n0 . R R R R

(47)

Using u = R 2 , equation (47) takes the form du + ( 6 − 4ωα − ωα dR (3 − ωα)

2

) 1u= R

2Λ 0 −6k R1− n . + R(3 − ωα) (3 − ωα)

(48)

This Liebnitz linear differential equation has solution u=

2Λ 0 −6 k + R 2− n . (6 − 4ωα − ωα 2 ) 6 − 4ωα − ωα 2 − 3n + 6 + ω nα − 2ωα

(49)

Here in order to obtained solutions two particular cases are considered for n that are n = 1 and n = 2. Model For n = 1: In this case the expression for scale factor is obtained as R = a1t 2 + b1t + c1 ,

(50)

which gives the expression for the scalar function and cosmological constant as φ = φ0 ( a1t 2 + b1t + c1 )α , Λ=

Λ0 (a1t + b1t + c1 ) . 2

(51) (52)

Here the energy density and pressure take the following form ρ=

φ0 (a1t 2 + b1t + c1 )α 8π

(6 + 6α − ωα 2 )(2a1t + b1 ) 2 + 6k Λ0 − , 2 2 2 2(a1t + b1t + c1 ) (a1t + b1t + c1 )

(53)

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Dynamic cosmological “constant” in Brans Dicke theory

p=

33

−φ0 (a1t 2 + b1t + c1 )α (2 + α)2a1 − Λ 0 (2 + 2α 2 + ωα 2 + 2α)(2a1t + b1 )2 + 2k (54) + . 2 8π 2(a1t 2 + b1t + c1 )2 (a1t + b1t + c1 )

The geometrical quantities of observational interest are 2(a1t + b1 ) , a1t 2 + b1t + c1

(55)

2a1 (a1t 2 + b1t + c1 ) . (2a1t + b1 ) 2

(56)

H=

q=−

Equation (56) suggests that deceleration parameter is dynamical. Variable deceleration parameter has already been presented by Singh and Kale [41]. Considering a variable deceleration parameter Singh et al. [42] and Pradhan et al. [43] have presented cosmological models in Lyra’s manifold. Model For n = 2: Here the scalar factor assumes the form

R = R2 e a2t .

(57)

Again using this, one can easily obtained the expressions for the scale factor and cosmological constant as

φ = φ0 R2 α e a2 α t Λ = Λ 0 R2 −2 e −2 a2

(58)

, t

.

(59)

In this case the energy density and pressure have the following form ρ=

p=−

φ0 R2 α e a2 8π

φ0 R2 α e a2 8π

α t

α t

(6 + 6α − ωα 2 ) a2 2 3k − Λ + 2 2 a0 t , 2 R2 e 2

(2 + 2α + 2α 2 + ωα 2 ) a2 2 k−Λ + 2 2 0a t . 2 R e 2 2

(60) (61)

The Hubble parameter and deceleration parameter are obtained as

H = a2 ,

(62)

q = −1 .

(63)

5. DISCUSSION

In this paper cosmological models have been obtained in the context of BransDicke theory by considering two expressions for cosmological “constant”. The

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G.P. Singh, A.Y. Kale, J. Tripathi

12

cosmological solutions are obtained for flat and non-flat FRW space time. In both cases, k = 0 and k ≠ 0 energy density, pressure, cosmological “constant” are decreasing with evolution of the universe. These results are in fair agreement with the observations. In case of non-flat models a dynamic deceleration parameter has been obtained. In section 3.1.1. subcase-I model with power law solution had been discussed. Considering observational value of deceleration parameter Fig.1 shows rapid growth in scale factor R(t) with respect to cosmic time, while Fig. 2 indicates that scalar field φ(t) is increasing with evolution of the universe. It can be easily seen from Fig. (3) and (4) respectively that energy density is decreasing and dynamic cosmological term is always negative for all values of “a” which satisfy observational limits of deceleration parameter. Further, section 3.1.2. subcase-II is not interesting due to the fact that energy density ρ and pressure p takes uniform values for all time which are not consistent with observational results. Hence in this case exact solutions are not presented. The case-II deals with cosmological models where Λ ∝ R − n . In this case behaviour of scale factor R(t), scalar field φ(t), energy density ρ and cosmological term Λ behave similar to the previous case with different rate during evolution of the universe. Acknowledgements. Authors would like to thank Inter-University Centre for Astronomy and Astrophysics, Pune, for providing facilities where part of this work was completed.

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