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Mesonic perfect fluid solutions are found in general relativity with the aid of Einstein’s Rosen cylindrically symmetric space time. A static vacuum model and a non-static cosmological model corresponding to perfect fluid are investigated. The cosmological term Λ is found to be a decreasing function of time which is supported by the result found from recent type Ia Supernovae observations. The various physical and geometrical features of the model are discussed.

Theory of general relativity (Einstein 1916) has served as basis for the study of cosmological models of universe. The cosmological term Λ has been introduced in 1917 by Einstein to modify his own equation of general relativity. Now this Λ-term remains a focal point of interest in modern theories. In 1930s distinguished cosmologists, A. S. Eddington and Abbey Georges Lemaitre felt that introduction of Λ-term has attractive features in cosmology and models, so it should be discussed deeply. Moreover models with cosmological time-dependent term-Λ are becoming popular as they help to solve the cosmological constant problem in natural way. The generalized Einstein’s theory of gravitation with time-dependent G and Λ has been proposed by Lau [

To study the nature of scalar field without mass parameters interacting with perfect fluid in Einstein’s Rosen space time is a subject of interest due to its significant role in the description of the universe at the early stages of evolution. Patel [

Recently many authors like Tsagas and Maartens [

Very recently Adhav et al. [

In this paper we consider the cylindrically symmetric space time in mesonic perfect fluid with time-dependent Λ-term in general theory of relativity. A static vacuum model and a non-static cosmological model are presented and studied in detail.

We consider the nonstatic cylindrically symmetric Einstein Rosen metric

where

We denote the coordinates

The Einstein’s field equations with the cosmological term

where

and

are respectively the energy momentum tensors for the perfect fluid and massless scalar field. The massless scalar field satisfies the Klein-Gordan wave equation

Here

mesonic field and cosmological constant. Hereafter the semicolon (;) denotes covariant differentiation.

Using commoving coordinate system, the set of field Equation (2) for the metric (1) reduces to the following forms

and

The Klein-Gordon Equation (6) for the metric (1) yields

Equations (7)-(12) are highly nonlinear partial differential equations and hence it is very difficult to solve them, as there exists no standard method to derive their solution.

Here we consider two particular physical important cases:

1) static vacuum model and 2) non-static cosmological model.

Further to avoid the mathematical complexities, we consider scalar field

In this case we consider

Therefore, in this case the field Equations (7)-(12) reduces the following set of equations

The solutions of the field equations are given by

where

After a suitable choice of coordinates, Einstein-Rosen cylindrically symmetric metric (1) can be written as

Here we consider

and

The exact solution of this equation is given by

where

Now using the equation of state

we obtain the physical quantities

and

where

After a suitable choice of coordinates and constants, Einstein-Rosen cylindrically symmetric metric (1) becomes

Here we discuss three models corresponding to

Case-I: When

From Equation (26), we obtain

Therefore in this case the energy density and cosmological constant takes the form

Case-II: When

In this case the energy density and cosmological constant are equal i.e.

From Equation (26), we obtain

Case-III: When

In this case from Equation (26), we obtain the energy density and cosmological constant in the form

From Equations (32) and (36) we observe that the cosmological constant term Λ is a decreasing function of time whereas

Here we study Physical and Kinematical properties of the cosmological model given by Equation (29). For the model (29) the expressions for the spatial volume V, scalar expansion

The spatial volume v tend to zero as T tends to

We have studied Einstein Rosen cylindrically symmetric static vacuum model and non-static cosmological model with mesonic perfect fluid with time-dependent cosmological constant term Λ in general relativity. We

have discussed three physical models corresponding to values of γ, i.e.

cosmological model is nonsingular; contracting and deceleration parameter indicates inflation. The time-de- pendent cosmological term Λ is decreasing function of time and it approaches to small positive value at late time.

V. D. Elkar is thankful to the University Grants Commission, New Delhi, India for providing fellowship under F.I.P.

Vijay G. Mete,Vijay D. Elkar, (2016) Einstein Rosen Mesonic Perfect Fluid Cosmological Model with Time Dependent Λ-Term. International Journal of Astronomy and Astrophysics,06,99-104. doi: 10.4236/ijaa.2016.61007