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In this paper we have constructed five dimensional Bianchi type-I cosmological model generated by a cloud of string with particles attached to them in Lyra manifold. Out of the two different cases, we obtained one case leads to the five dimensional vacuum universe in Lyra manifold while the other case yields a string cosmological model in Lyra manifold. Some physical and geometrical properties of the models are briefly discussed.

Now a day cosmologists are interested to study cosmic strings in the framework of general theory of relativity as well as in alternative theory. Since the discovery of general theory of relativity by Einstein, there has been numerous modification of it. Lyra [

The concept of string theory was developed to describe events at the early stages of the evolution of the universe. So strings are important in the early stages of evolution of the universe before the particle creation. The present day observations do not rule out the possible existence of large scale networks of string in the early universe. Gauge theories with spontaneous symmetry breaking in elementary particle physics have given rise to anopontensive study of cosmic strings. It appears that after big bang the universe may have experienced a number of phase transitions Linde [

The unification of gravitational forces with other forces in nature is not possible in the usual four dimensional space times. So higher dimensional theory might be useful at very early stages of the evolution of the universe. In fact, as time evolves, the standard dimensions expand while the extra dimensions shrink to the Planckian dimension, which is beyond our ability to detect with the currently available experimental facilities Chatterjee et al. [

The five dimensional Bianchi type-I metric can be written as

where A, B, C and D are function of cosmic time “t” only. Here the extra co-ordinate is taken to be space like.

The Einstein’s field equations based on Lyra manifold is proposed by Sen [

where is the displacement vector and other symbols have their usual meanings as in the Riemannian geometry. The displacement vector is taken in the form

The energy momentum tensor for a cosmic string is taken as

where is the rest energy density of cloud of strings with particle attached to them, is the tension density of strings, is the rest energy density of the particles, is the five velocity of particles and

the direction of the string satisfies

(6)

The field Equation (2) together with Equations (3)-(6) for the metric (1) yield the following equations

where dash denotes the differentiation with respect to “t”.

In this section we find physically meaningful solutions of the field Equations (7)-(11) by taking some simplifying assumptions.

As there is no independent equation for the gauge function, so here we consider

is the most suitable form to fit the observations.

Let

and

where n and are arbitrary constants. By making use of Equations (12)-(14) in the field Equations (7)-(11) we get

From Equation (19) we get n = 1/3 or n = 1/6 For n = 1/3 Using n = 1/3 in Equation (18) we obtained n_{1} = 0 or n_{1} = 1/3 For n = 1/6 Using n = 1/6 in Equation (18) we obtained n_{1} = 1/2 or n_{1} = 1/6 Equations (13) and (14) show that the universe expand indefinitely as t increases if n > 0 and the extra dimension contract to a Planckian length as if. Hence to get a physically realistic string cosmological model, we take n = 1/3 and n_{1} = 0.

The geometry of the model described by the metric

Now using n = 1/3 and n_{1} = 0 in Equations (15) and (16) we get

This shows that string does not survive for isotropic model in Lyra geometry.

The scalar expansion is obtained as

At initial epoch t = 0, and.

The shear scalar is obtained as

The spatial volume of the universe is obtained as

V = kt (25)

The deceleration parameter (q) is obtained as

q = 0 (26)

It is observed that the deceleration parameter “q” is identically equal to zero, earlier discussed by Venkateswarlu and Pavan Kumar [

In this case we take

where are arbitrary constants.

Now from Equations (7) and (8), by the use of Equations (12) and (27) we find

The rest energy density () is given by Equation (28). It is observed that at initial epoch i.e. t = 0, , and satisfies the reality condition when

The string tension density () given by Equation (29) tends to zero as t tends to infinity. The particle density () given by Equation (30) satisfy the reality condition when

We observe that, the anisotropic three space will expand as when are all positive and the extra dimension will contract as if. By making use of the solutions given by Equations (27)- (29) in the field Equations (9)-(11), we conclude that

The geometry of the model described by the metric

The scalar expansion (), the shear (), the spatial volume (V) and the deceleration parameter (q) for the model (31) are obtained as

V = t (34)

q = 0 (35)

(Venkateswarlu and Pavan Kumar [

In this paper we constructed five dimensional Bianchi type-I cosmological models in the framework of Lyra geometry in the context of cosmic strings. From Case-I it is observed that, the isotropic Bianchi type-I higher dimensional cosmic strings do not survive. Hence we obtained higher dimensional vacuum isotropic Bianchi type-I string cosmological model in Lyra geometry. Further we observed that, the three spatial co-ordinates expand indefinitely as while the extra dimension remains constant. It is also interesting to note that, the deceleration parameter (q) is identically equal to zero. Although the solutions obtained are special in nature.

From Case-II we obtained five dimensional anisotropic Bianchi type-I string cosmological model. At initial epoch i.e. t = 0, the rest energy density, tension density, particle density, scalar expansion and shear become infinity. Therefore we can say that, the model admit initial singularity. The spatial volume (V) becomes zero at t = 0 and this shows the expansion of the model with time. It is also interesting to note that, the deceleration parameter (q) is identically equal to zero.

The authors are extremely grateful to the referees for their valuable suggestions.