It is still an interesting area of research to discover its unknown phenomenon that has yet to observe to study the ultimate fate of the universe. But till today cosmologists cannot make a final and comprehensive conclusion about the origin and evolution of universe with strong evidence. So more and more investigations are required to discover and understand the unknown phenomenon of the universe and many mysterious particles which are to be observed to study the ultimate fate of the universe. The cosmologist or researchers developed the string theory to describe the universe, its early stages and the evolution during the time. So, the study on string cosmology is becoming very interesting area for the cosmologist, because of its significant role in the study of formation and evolution of the universe at the early stages and to understand about the future evolution. In the field of the general relativity the investigation of string was generally initiated by prominent authors, Stachel^{ }^{ }^{ }^{6, 7}).

The strings are crucial topological stable defects occurred due to the phase transition at the early days of the universe, when the temperature is lower than a specific temperature, known as critical temperature. The occurrence of strings inside the universe results in anisotropy within the space-time, though the strings aren't seen in the present epoch. Strings cause no damage to the cosmological models, but they can result in very interesting astrophysical effects. Because of the great position of strings in the description of the evolution of the early universe, nowadays, many prominent authors have significantly studied the string cosmology. Soon, after the big-bang, there was a breaking of symmetry during the time of phase transition and the cosmic temperature went down below some critical temperatures due to which the strings arose, as according to grand unified theories(Everett^{ }^{ }

Though the Einstein general relativity is one of the most acceptable theory in modern era to describe the universe, it is unable to explain some of the strong unknown facts about the universe such as accelerated expansion of universe, reason behind the expansion etc. So the several researchers are trying to solve and explain those aspects of the universe by the help of different modified theories of Einstein General theory of relativity such as Weyl’s theory, Brans-Dicke theory, f(R) gravity theory, f(R, T) theory, Lyra geometry, scalar tensor theory etc. Among these theories Lyra geometry is one of the most important modified theory. Inspired by the geometrization of gravitation, Weyl^{ }^{ }

Inspired by the above discussions, here we have studied the string cosmological model with particles connected to them in Bianchi type-III universe considering Lyra geometry. The work done in this paper and findings are somewhat distinct from the earlier findings. In the sec.2, Bianchi type-III metric is presented and the field equations in Lyra geometry are derived; In the sec.3, the determinate solutions of the field equations are determined by using some plausible conditions. Physical and geometrical properties of our model with the help of graph are discussed in sec.4; In sec.5 conclusions of the paper are given.

We consider the Bianchi type-III metric as

Here, a, b and c are the functions of `t' alone. For the above metric let

The field equations with gauge function and

Where,

Here,

The energy-momentum tensor for a cosmic string is taken as

Here,

If R is the average scale factor then volume is

The expansion scalar is given by

Hubble parameter is given by

The shear scalar is given by

And the mean anisotropy parameter is

Where,

The field

Here the overhead dots represent the order of differentiation w. r. t. time `t'.

Solving

Here

And using it, (18) can be written as,

Thus using relation (19) the field

We have 3 highly nonlinear independent differential

Here, we take the assumption that the shear scalar and expansion scalar are proportional to each other

Here

This is based on observations of velocity and red-shift relation for an extragalactic source which predicted that the Hubble expansion is 30 percent isotropic, which is supported by the works of Thorne^{ }^{ }^{ }

Secondly we adopt the assumption proposed by Berman^{ }

When h is negative then the model universe expand with acceleration and when q is positive then the model universe contract with deceleration. Although the present observations like CMBR and SNe Ia suggested the negative value of q but it can be remarkably state that they are not able to deny about the decelerating expansion (positive q) of universe. This is the most suitable condition to explore the physically meaningful solutions of the above field equations.

The scale factor R admits the solution-

Here

Using the

Without loss of generality we take h=1 and k=0 then (26), (27) becomes

Using (28) the metric (1) can be reduced to

This gives the geometry of the metric (1).

We obtained some of the important physical and geometrical parameters that are useful for the discussion on the evolution of the universe.

Using (28) in (22) we obtained

From (19) and (20) using (28) we obtained

From (30), (31) we obtained the

The gauge function

The spatial volume, scalar expansion, Hubble parameter, shear scalar and mean anisotropy parameter of the model are

The

From the expressions of energy density

For the model universe, the expression of particle density

In this model universe, at the initial epoch of time, the gauge function

The volume for this model increases as time increases. The expression of volume V as obtained in

From the expansion scalar and Hubble parameter for the model (29), at

In

From

In this article, we have attempted to present a new solution to the field equations obtained for Bianchi type-III universe in Lyra geometry by using the law of variation of Hubble’s parameter H which yields constant DP. This variational law for H in

The authors declare that there was no financial aid received and no conflict of interest associated with this research work.