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 [1] proposed a modification of Riemannian geometry by introducing a gauge function in to the structure less manifold that bears close resemblances to Weyl’s [2] geometry. Subsequent investigations were done by several authors Sen [3], Sen and Dunn [4], Halford [5], Bhamra [6], Beesham [7], Soleng [8], Pradhan et al. [9], Agarwal et al. [10] in scalar tensor theory and cosmology within the framework of Lyra geometry. Singh and Singh [11-13] and Singh and Desickan [14] have studied Bianchi type-I, III, Kantowski-Sachs cosmological models with time dependent displacement field in the cosmological theory based on Lyra’s manifold. Soleng [8] has pointed out that the cosmologies based on Lyra’s geometry based with constant gauge vector will either introduce creation field and be equal to Holy’s creation field cosmology or contain a special vacuum field which together with the gauge vector term may be considered as a cosmological term.

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 [15]. These phase transitions can produce vacuum domain structures such as domain walls, string and monopoles Kibble [16], Zel’dovich [17] of all these cosmological structures, cosmic strings have excited perhaps the most interest. Cosmic strings may act as gravitational lenses Vilenkin [18,19] and may give rise to density perturbations leading to formation of galaxies. Kibble [20] mentioned that the presence of strings in the early universe can be explained using grand unified theories. These are two major fields that the string theory ought to illuminate some day; particle physics and cosmology. Particle physics addresses the microscopic extreme while cosmology the microscopic extreme. Relativists and particle theorists have both identified the important problem of reconciling quantum theory with general relativity. The prospect of achieving this attracts both of them to string theory.

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. [21]. This fact has attracted many researchers to investigate the problems in the field of higher dimensions Appelquist et al. [22], Chodos and Detweller [23] constructed massive string cosmological model in higher dimensional homogeneous space time in general relativity. Krori et al. [24] constructed a Bianchi type-I string cosmological model in higher dimensional space time and obtained that matter and string coexist throughout the evolution of the universe. Venkateswarlu and Pavan Kumar [25] constructed higher dimensional string cosmological model in scale covariant theory of gravitation. Rahaman et al. [26] obtained exact solutions of the field equations for five dimensional space time in Lyra manifold when the source of gravitation is a massive string. Moanty and Samanta [27] constructed five dimensional axially symmetry string cosmological models with bulk viscous fluid In general theory of relativity. Recently Khadekar and Shelote [28] studied five dimensional FRW cosmological models with quark and strange quark matter. In this paper we constructed five dimensional Bianchi type-I string cosmological models in Lyra manifold.

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 [3] and Sen and Dunn [4] in normal gauge may be written as

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.

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.