LRS Bianchi Type-II String Cosmological Models in a Modified Theory of Gravitation

LRS Bianchi type-II string cosmological models in a modified theory of gravitation

T. Kanakavalli1, G. Ananda Rao 2 , D.R.K.Reddy3*

*[email protected]

1Department of Applied Mathematics, GITAM University, Visakhapatnam, India 2Department of Mathematics, ANITS, Visakhapatnam, India 3Department of Applied Mathematics, Andhra University, Visakhapatnam, India

Abstract:

This paper is devoted to the investigation of spatially homogenous anisotropic LRS Bianchi type-II cosmological models with string source in a modified theory of gravitation formulated by Harko et al. (Phys. Rev. D 84: 024020, 2011) which is universally known as f( R, T) gravity. Here R is the Ricci scalar and T is the trace of the energy momentum tensor. By solving the field equation we have presented massive string and Takabyasi or p- string models in this theory. However it is interesting to note that geometric string in this space – time does not exist in this theory. Physical and geometrical properties of the strings obtained are also discussed.

Key words: LRS Bianchi type II model, string models, f(R, T) gravity.

1. Introduction:

The most attractive subject of investigation that is attracting the attention of recent research workers in cosmology is the accelerated expansion of the universe. This is based on the super fold observations of Riess et al. (1998) and Perlmutter et al. (1999). It is believed that this expansion is driven by an exotic energy with large negative pressure which is usually called as “dark energy”. There are two ways of talking this late time acceleration of the universe. One way is to introduce dark energy component in the universe and study dark energy models and another way is to modify general relativity there have been several modifications of general relativity. Some of them are f(R) gravity (Caroll et al. 2004). f(R,T) gravity (Harko et al. 2011), scalar tensor theories of gravity proposed by Brans and Dicke (1961) and Saez – Ballester (1986). Here we are interested in f(R,T) gravity where R is the Ricci scalar and T is the trace of the energy momentum tensor. The gravitational field equations of this theory have been derived from the Hilbert – Einstein type variational principal whose action is given by

1 S  f (R,T )  L  g d 4 x   m  16G (1) where Lm is the matter Lagrangian the stress energy tensor of matter is 2   gL T  m ij  g ij  g (2)

Using gravitational units (G= C=1) the corresponding field equations f(R,T) gravity are obtained by varying the action (1) with respect to the metric gij as

1 f( R , T ) R- f ( R , T ) g + ( g -蜒 ) f ( R , T ) = 8p T - f ( R , T ) Q Rij ij ij i j R ij T ij 2 (3) Where

ab d f( R , T ) d f( R , T ) gd Tab f = , f = , Q = and = 蜒i R T ij ij W j (4) d R dT d g

i Here is the covariant derivative and Ti j . The usual energy momentum tensor derived from the lagrangian Lm. it can be seen that when f(R,T)= f(R), then Eq.(3) reduces to f(R) gravity field equations.

Now contracting eq.(3), we get

f( R , T ) R+ 3 f ( R , T ) - f ( R , T ) = 8p T - f ( R , T ) T - f ( R , T ) Q RW R T T (5) By specifying the functional forms of f as

f( R , T )= R + 2 f ( T )

f( R , T )= f( R) + f ( T ) 1 2 (6)

f( R , T )= f1( R) + f 2( R) f 3 ( T )

Harko et al. (2011) has obtained some modified gravity models.

In particular for the choice of f(R,T) given by

f( R , T )= R + 2 f ( T ) (7) f(R,T) gravity field equations are obtained as (Harko et al. 2011)

1 Rij- Rg ij =8p T ij + 2 fⅱ ( T ) T ij +( 2 pf ( T ) + f ( T )) g ij 2 (8)

Here the matter source is perfect field given by T=(r + p) u u - pg ij i j ij (9) Immediately after the big bang, during the phage transition in the early universe, spontaneous symmetry breaking gives rise to a random network of stable line like topological defects known as cosmic strings, domain walls and monopoles. Strings are line like structures with particles attached to them. They play a significant role in galaxy formation at the early stages of the universe. Satchel (1980), Letelier (1983), Vilenkin (1987), Krori et al. (1990), Mahanta and Mukherjee (2001) and Bhattacharya and Baruah (2001) have investigated cosmological model with string source in general relativity.

It is well known that spatially homogenous FRW models are best suited for the description of present day universe. But the models with anisotropic background are considered to be best suited for representing the early stages of the universe. Bianchi models with anisotropic background are the simplest models for this purpose. In recent years, investigation of Bianchi type models in modified theories of gravitation is giving importance.

Adhav (2012) has obtained Bianchi type-I model in f(R,T) gravity while Reddy et al. (2012a, 2012b) have discussed Bianchi type-III and Kaluza- Klein cosmological models in thus theory. Reddy and Santhi Kumar (2013) have studied Bianchi type III dark energy model in f(R,T) gravity. Also, Reddy et al. (2013) have investigated Bianchi type II universe with cosmic strings and bulk viscosity in this modified theory of gravitation. Rao and Rao (2015) have discussed Bianchi type V string cosmological models which correspond to geometric (Letlier 1983) and Takabayasi (1976) strings in f(R,T) gravity. Recently, Valli and Rao (2016a, 2016b) have obtained geometric and p- strings and Bianchi type-III space-times, respectively, in this theory. Sahoo (2016) has discussed LRS Bianchi type-I string cosmological model in f(R,T) gravity using a constant deceleration parameter given by Berman (1983). However Bianchi type- II geometric and Takabayasi strings in this theory have not been discussed.

Our main purpose, in this paper, is to investigate geometric, Takabayasi and massive (Reddy 2003a) strings in Bianchi type- II space time in f(R,T) gravity. The work in this paper is organized as follows: in Sect. 2 we obtain the explicit form the field equations of f(R,T) gravity in the presence of massive string source with the help of Bianchi type – II metric. In Sect. 3 we present the solutions and the corresponding string models. Physical discussion of the models is also given. The last section contains some conclusions.

2. Metric and the field equations

Spatially homogenous and anisotropic LRS Bianchi type-II space-time is given by

2 2 2 2 2 2 2 2 2 2 2 ds  dt  X dx  Y dy  2Y xdy dz  Y x  X dz (10) where X and Y are the functions of the cosmic time t only. The energy momentum tensor for a cosmic string source is

T  u u  x x ij i j i j (11) where  is the energy density of the string cloud, ui is the four velocity, xi is the string direction and  is the string tension density. Also, we have

i j i i u u  x x j  1, u x j  0 (12) and

     p (13)

where  p is the rest energy density particle attached to the string. Letelier (1983) has pointed out that  may be positive or negative. Also we choose (Harko et al. 2011)

f (T)  T (14) where  is a constant.

Now using commoving coordinate system, the field equations (8) (replacing p by  in the view of eq. (11), for the metric with the help of (11) to (14), can be explicitly written as

X˙˙ Y˙˙ X˙Y˙ 1 Y 2    4     X Y XY 4 X (15)

X˙˙ X˙ 2 3 Y 2 2  2  4  8   X X 4 X (16)

X˙Y˙ X˙ 2 1 Y 2  2  4  8  2  XY X 4 X (17) which reduced to the following two independent equations

Y˙˙ X˙˙ X˙ 2 X˙Y˙ Y 2   2   4  8   Y X X XY X (18) Y˙˙ X˙˙ X˙ 2 X˙Y˙ 1 Y 2   2   4  8   Y X X XY 2 X (19) where an overhead dot indicates differentiation with respect to time t.

Now for the metric (1), we have the average scale factor a(t) and the spatial volume as

V  a3 (t)  X 2Y (20)

The generalized mean Hubble parameter H is defined as

a˙ 1 H   H1  H 2  H 3  a 3 (21)

X˙ X˙ Y˙ where H  , H  , H  (22) 1 X 2 X 3 Y are the Hubble parameter in the directions of x, y and z axes respectively.

The expansion scalar  , shear scalar  2 are given by

X˙ Y˙   2  (23) X Y

2 1 1  X˙ Y˙  2 ij   (24)     ij     2 3  X Y 

The mean anisotropic parameter is

2 1 3  H  H     i  (25) 3 i1  H 

The deceleration parameter q is defined as

d  1  q    1 dt  H  (26)

The behavior of the universe is determined by the sign of q. If q>0, we have decelerating universe and if q<0, we have accelerating universe. if q=0, we have uniform expansion of the universe.

The above physical and kinematic properties will be useful in solving the field equations. 3. Solutions of the field equations and the corresponding models

We can observe that the field equations (18) and (19) are two independent equations with four unknowns X ,Y,  and  . Hence to obtain the deterministic solution we need two exact conditions.

(i) The shear scalar  is proportional to scalar expansion q, which gives the relationship between the metric potentials given by

n X  Y (27) where n  1 is a positive constant which takes care of the anisotropy of the space-time. (ii) The equations of state

r= l (28)

r=(1 + w ) l (29)

r+ l = 0 (30)

Here solving the field equations with the conditions (27) and (28) we obtain geometric or Nambu string (Letelier 1983), conditions (27) and (29) correspond to Takabayasi string (1976) and conditions (27) and (30) yield the massive string or Reddy (2003a) string. We shall now discuss the above string models.

3.1 Geometric or Nambu string ( r= l ):

In this case, Eqs. (18) and (19) yield

X˙˙ X˙Y˙ 1 Y 2    0 (31) X XY 4 X 4

Using Eqs. (27) in Eqs. (31) we get

2 1 X˙ 2 1 4 X˙˙   R n  0 (32) n X 2 4

Now using X˙  f 2 and integrating Eqs. (32) we obtain

1 X˙  f 2   R2n2 (33) 4 which is imaginary. This leads to the conclusion that, in this case, strings do not exist. Hence in f(R,T) gravity Bianchi-II geometric or Nambu string do not survive.

3.2 Takabayasi or p-string ( r=(1 + w ) l )

In this particular case, integrating the field equations (18) and (19) with the help of Eq. (27), we obtain the metric potentials as

n  k  2n1 X  (2n 1) 2 t  k1  2n  2 

1  k  2n1 Y  (2n 1) 2 t (34)  k1  2n  2 

n2 where k  1   n(  2)(n 1)

 1     2 (35) k    2   n(  2)(n 1) and integration constants have been set equal to zero for simplicity.

Now we can write the metric (10) with the help of Eq. (34) as

2n 2 2n1 2n1 2 2  k  2  k  2 ds  dt  (2n 1) 2 t dx  (2n 1) 2 t dy   k1  2n  2   k1  2n  2  2  2 2n  2n1 2n1 2n1  k   k  2  k   2 2(2n 1) 2 t xdy dz  (2n 1) 2 t x  (2n 1) 2 t dZ  k1  2n  2   k1  2n  2   k1  2n  2    

where k1 and k2 are given by Eq. (35).

Physical discussion:

Eq. (36) represents LRS Bianchi type-II Takabayasi string cosmological model in f(R,T) gravity. The dynamical parameters of this model which play a significant role in the physical discussion of the model are Spatial volume

2n+ 1 2n- 1 (37) 轾 k2 V=犏(2 n - 1) t 臌 k1 +2 n - 2

The mean Hubble’s parameter

1V˙ 2 n + 1 H = = 3V 3(2 n- 1) t (38)

The scalar expansion is 2n + 1 q = (39) (2n- 1) t The shear scalar is

2 2 1骣n - 1 s = 琪 (40) 3t2 桫 2 n - 1 The average anisotropic parameter is

6(n - 1)2 D = (2n + 1)2 (41)

The deceleration parameter 3( 2n + 1) q = -1 + (2n - 1) (42)

It may be observe that the spatial volume of the universe increases with cosmic time. It can be seen that H,q , s2 , r and l diverse when t=0 and vanish t  . For n=1, we observe that

2 D =0 ands = 0 . This shows that for this value of n, the model is isotropic and shear free. The deceleration parameter q is negative for n <1 which shows that the universe accelerates. This is in agreement with the recent cosmological observations (Riess et al. 1998; Perlmutter et al. 1999). 3.3 Massive (Reddy) string: Reddy (2003a, 2003b) has obtained string models in Brans-Dicke (1961) and Saez-Ballester (1980) scalar-tensor theories of gravitation using the equation of state given by Eq. (30). Model obtained with this equation of state is usually known as Reddy string or massive string in literature. Now using Eqs. (27) and (30) the field equations (18) and (19) yield the solution

n  3  2n1 X  (2n 1) 2 t  4n  2n  2  1   2n1 3 (43) Y  (2n 1) 2 t  4n  2n  2  and integration constants have been set equal to zero for simplicity.

Now the metric (10) with the help of eq. (43) can be written as

2n 2 2 2n1 2n1 2n1 2 2  3  2  3  2  3  ds  dt  (2n 1) 2 t dx  (2n 1) 2 t dy  2(2n 1) 2 t xdy dz   4n  2n  2   4n  2n  2   4n  2n  2   2 2n  2n1 2n1  3  2  3   2 (2n 1) 2 t x  (2n 1) 2 t dZ  4n  2n  2   4n  2n  2    

The cosmological model given by Eq. (44) describes LRS Bianchi type-II Reddy string in f(R,T) gravity. The following dynamical parameters of the model will be useful for physical discussion of the model.

Spatial volume is

2n+ 1 2n- 1 轾 3 (45) V=犏(2 n - 1) t 犏 4n2 + 2 n - 2 臌 ( )

The mean Hubble’s parameter

1V˙ 2 n + 1 H = = 3V 3(2 n- 1) t (46)

The scalar expansion is 2n + 1 q = (47) (2n- 1) t Shear scalar is

2 2 1骣n - 1 s = 琪 (48) 3t2 桫 2 n - 1 The average anisotropic parameter is given by 6(n - 1)2 D = (2n + 1)2 (49) The deceleration parameter is 3( 2n + 1) q = -1 + (2n - 1) (50)

The energy density in the string is 2(n + 1) r = 3(1- 2n ) t 2 (51)

The tension density in the string is

2(n + 1) l = 3(2n- 1) t 2 (52)

It may be observed that the physical and kinematical parameters of this model are similar to the model in Sect.3.2 and hence we have the same physical behavior. However, in this case the 1 density is positive for n  . 2 4. Conclusion

It is well known that string cosmological models play a vital role in the discussion of early stages of evolution of the universe. Hence, in this paper, we have investigated LRS Bianchi type- II cosmological models in the presence of massive string source in f(R,T) gravity proposed by Harko et al. (2011). To obtain determinate solutions of the field equations of this theory we have used the three equations of state for strings which correspond to geometric, Takabayasi and Reddy strings. It is interesting to note that the LRS Bianchi type- II string in this theory, does not survive. However, we have presented the Takabyasi strings and Reddy strings in this particular space- time in this modified theory. We have also studied the physical behavior of the models presented.

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