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 Reiss et al. (1998) and Perlmutter et al. (1997). 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 and 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. It is well known that \(f(R,T)\) gravity is considered to be a viable alternative to general relativity (GR) as it takes care of early inflation and late time acceleration of the universe by modifying the Hilbert–Einstein action of GR. The gravitational field equations of this theory are derived from the action

$$ S = \frac{1}{16\pi} \int f(R,T)\sqrt{ - g} d^{4}x + \int L_{m} \sqrt{ - g} d^{4}x $$
(1)

where \(L_{m}\) is the matter Lagrangian and the stress energy tensor of matter is

$$ T_{ij} = \frac{2}{\sqrt{ - g}} \frac{\delta \sqrt{ - g} L_{m}}{\delta g^{ij}} $$
(2)

and the trace by \(T = g^{ij} T_{ij}\) respectively. By assuming that \(L_{m}\) of matter depends only on the metric tensor components \(g_{ij}\), we have obtained the field equations of \(f(R,T)\) gravity as

$$\begin{aligned} &f_{R} ( R,T ) R_{ij} - \frac{1}{2}f ( R,T ) g_{ij} + ( g_{ij} \square - \nabla_{i} \nabla_{j} ) f_{R} ( R,T ) \\ &\quad = 8\pi T_{ij} - f_{T} ( R,T ) T_{ij} - f_{T} ( R,T ) \theta_{ij} \end{aligned}$$
(3)

where

$$\begin{aligned} \theta_{ij} = - 2 T_{ij} + g_{ij} L_{m} - 2 g^{lk}\frac{\partial^{2}L_{m}}{\partial g^{ij}\partial g^{lm}}, \qquad\square = \nabla^{k} \nabla_{k} \end{aligned}$$
(4)

here \(f_{R} = \frac{\partial f ( R,T )}{\partial R}\), \(f_{T} = \frac{\partial f ( R,T )}{\partial T}\) and \(\nabla^{i}\) is the covariant derivative.

It may be noted that when \(f ( R, T ) = f ( R )\), (3) yields the field equations of \(f(R)\) gravity.

The problem of perfect fluid described by an energy density \(\rho\), pressure \(p\) and four velocities \(u^{i}\) are complicated since there is no unique definition of the matter Lagrangian. However, here, we assume that stress energy tensor of matter is given by

$$ T_{ij} = \left( \rho + p \right) u_{i} u_{j} - p g_{ij} $$
(5)

and the matter Lagrangian can be taken as \(L_{m} = - p\) and we have

$$ u^{i} \nabla_{j} u_{i} = 0,\qquad u^{i} u_{i} = - 1 $$
(6)

Now with the use of (6) we obtain, for the variation of stress energy of perfect fluid, the expression

$$ \theta_{ij} = - 2T_{ij} - p g_{ij} $$
(7)

Generally, the field equations also depend through the tensor \(\theta_{ij}\), on the physical nature of the matter field. Hence in the case of \(f(R,T)\) gravity, depending on the nature of the matter source, we obtain several theoretical models corresponding to each choice of \(f(R,T)\).

Assuming

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

as a first choice where \(f(T)\) is an arbitrary function of the trace of stress energy tensor of matter, we get the gravitational field equations of \(f(R,T)\) gravity from (7) as

$$ R_{ij} - \frac{1}{2}g_{ij}R = 8 \pi T_{ij} + 2 f'(T) T_{ij} - 2 f'(T) \theta_{ij} + f(T) g_{ij} $$
(8)

where the prime denotes differentiation with respect to the argument. If the matter source is perfect fluid then the field equations of \(f ( R,T )\) gravity, in view of (7), become

$$ R_{ij} - \frac{1}{2} g_{ij} R = 8 \pi T_{ij} + 2f' ( T )T_{ij} + \bigl[ 2p f' ( T ) + f ( T ) \bigr] g_{ij} $$
(9)

Immediately after the big bang, during the phase 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. The interest in studying cosmic strings in the early universe stems from the fact that they provide one of the most promising scenarios for the generation of large scale structure in the Universe. Stachel (1980), Letelier (1983), Vilenkin (1987), Krori et al. (1990), Mahanta and Mukherjee (2001) and Battacharya and Baruah (2001) have investigated cosmological model with string source in general relativity.

It is well known that spatially homogeneous 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 (Letelier 1983) and Takabayasi (1976) strings in \(f(R,T)\) gravity. Recently, Kanakavalli and Ananda Rao (2016) and Kanakavalli et al. (2016) 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.

The above discussion has motivated us to consider Bianchi type-II geometric, Takabayasi and massive (Reddy 2003a) strings in \(f(R,T)\) gravity which have not been investigated till date. 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 homogeneous and anisotropic LRS Bianchi type-II space–time is given by

$$\begin{aligned} ds^{2} &= - dt^{2} + X^{2}dx^{2} + Y^{2}dy^{2} + 2Y^{2}xdy dz \\ &\quad{}+ \bigl( Y^{2}x^{2} + X^{2} \bigr)dz^{2} \end{aligned}$$
(10)

where \(X\) and \(Y\) are functions of cosmic time \(t\) only.

We consider the energy momentum tensor for a cosmic string source (not the string filled with cosmic fluid so that \(p=0\) in the field equations (9))

$$ T_{ij} = \rho u_{i}u_{j} - \lambda x_{i}x_{j} $$
(11)

where \(\rho\) is the energy density of the string cloud, \(u^{i}\) is the four velocity, \(x^{i}\) is the string direction and \(\lambda\) is the string tension density. Also, we have

$$ u^{i}u^{j} = - x^{i}x_{j} = - 1,\qquad u^{i}x_{j} = 0 $$
(12)

and

$$ \rho = \rho_{p} + \lambda $$
(13)

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

$$ f(T) = \mu T $$
(14)

where \(\mu\) is a constant.

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

$$\begin{aligned} &\frac{\ddot{X}}{X} + \frac{\ddot{Y}}{Y} + \frac{\dot{X}\dot{Y}}{XY} + \frac{1}{4}\frac{Y^{2}}{X^{4}} = - \left( \lambda + \rho \right)\mu \end{aligned}$$
(15)
$$\begin{aligned} &2\frac{\ddot{X}}{X} + \frac{\dot{X}^{2}}{X^{2}} - \frac{3}{4}\frac{Y^{2}}{X^{4}} = - (8\pi + 3\mu )\lambda - \mu \rho \end{aligned}$$
(16)
$$\begin{aligned} &2\frac{\dot{X}\dot{Y}}{XY} + \frac{\dot{X}^{2}}{X^{2}} - \frac{1}{4}\frac{Y^{2}}{X^{4}} = - \left( 8\pi + 3\mu \right)\rho - \mu \lambda \end{aligned}$$
(17)

which reduced to the following two independent equations

$$\begin{aligned} &\frac{\ddot{Y}}{Y} - \frac{\ddot{X}}{X} - \frac{\dot{X}^{2}}{X^{2}} + \frac{\dot{X}\dot{Y}}{XY} + \frac{Y^{2}}{X^{4}} = \left( 8\pi + 2\mu \right)\lambda \end{aligned}$$
(18)
$$\begin{aligned} &\frac{\ddot{Y}}{Y} + \frac{\ddot{X}}{X} - \frac{\dot{X}^{2}}{X^{2}} - \frac{\dot{X}\dot{Y}}{XY} + \frac{1}{2}\frac{Y^{2}}{X^{4}} = \left( 8\pi + 2\mu \right)\rho \end{aligned}$$
(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

$$ V = a^{3}(t) = X^{2}Y $$
(20)

The generalized mean Hubble parameter \(H\) is defined as

$$ H = \frac{\dot{a}}{a} = \frac{1}{3} ( H_{1} + H_{2} + H_{3} ) $$
(21)

where

$$\begin{aligned} H_{1} = \frac{\dot{X}}{X},\qquad H_{2} = \frac{\dot{X}}{X},\qquad H_{3} = \frac{\dot{Y}}{Y} \end{aligned}$$
(22)

are the Hubble parameter in the directions of \(x\), \(y\) and \(z\) axes respectively.

The expansion scalar \(\theta\), shear scalar \(\sigma^{2}\) are given by

$$\begin{aligned} &\theta = 2\frac{\dot{X}}{X} + \frac{\dot{Y}}{Y} \end{aligned}$$
(23)
$$\begin{aligned} &\sigma^{2} = \frac{1}{2}\sigma^{ij}\sigma_{ij} = \frac{1}{3} \biggl( \frac{\dot{X}}{X} - \frac{\dot{Y}}{Y} \biggr)^{2} \end{aligned}$$
(24)

The mean anisotropic parameter is

$$ \Delta = \frac{1}{3}\sum_{i = 1}^{3} \biggl( \frac{H_{i} - H}{H} \biggr)^{2} $$
(25)

The deceleration parameter \(q\) is defined as

$$ q = \frac{d}{dt} \biggl( \frac{1}{H} \biggr) - 1 $$
(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 \(X\), \(Y\), \(\rho\) and \(\lambda\). Hence to obtain the deterministic solution we need two exact conditions.

  1. (i)

    The shear scalar \(\sigma\) is proportional to scalar expansion \(\theta\), which gives the relationship between the metric potentials given by

    $$ X = Y^{n} $$
    (27)

    where \(n \ne 1\) is a positive constant which takes care of the anisotropy of the space–time.

  2. (ii)

    The equations of state

    $$\begin{aligned} &\rho = \lambda \end{aligned}$$
    (28)
    $$\begin{aligned} &\rho = (1 + \omega )\lambda \end{aligned}$$
    (29)
    $$\begin{aligned} &\rho + \lambda = 0 \end{aligned}$$
    (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 (\(\rho = \lambda\))

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

$$ \frac{\ddot{X}}{X} - \frac{\dot{X}\dot{Y}}{XY} - \frac{1}{4}\frac{Y^{2}}{X^{4}} = 0 $$
(31)

Using (27) in (31) we get

$$ \frac{\ddot{X}}{X} - \frac{1}{n}\frac{\dot{X}^{2}}{X^{2}} - \frac{1}{4}X^{\frac{2}{n} - 4} = 0 $$
(32)

Now using \(\dot{X} = f(X)\) and integrating (32) we obtain

$$ \dot{X}^{2} = f^{2} = - \frac{1}{4}X^{\frac{2(1 - n)}{n}} $$
(33)

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 (\(\rho = (1 + \omega)\lambda\))

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

$$\begin{aligned} &\!\begin{aligned} &X = \biggl[ (2n - 1)\sqrt{\frac{k_{2}}{k_{1} + 2n - 2}} t \biggr]^{\frac{n}{2n - 1}} \\ &Y = \biggl[ (2n - 1)\sqrt{\frac{k_{2}}{k_{1} + 2n - 2}} t \biggr]^{\frac{1}{2n - 1}} \end{aligned} \end{aligned}$$
(34)

where

$$\begin{aligned} &\!\begin{aligned} &k_{1} = \frac{\omega n^{2}}{\omega - n(\omega + 2)(n - 1)} \\ & k_{2} = \frac{ ( \omega + \frac{1}{2} )}{\omega - n(\omega + 2)(n - 1)} \end{aligned} \end{aligned}$$
(35)

and integration constants have been set equal to zero for simplicity.

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

$$\begin{aligned} ds^{2} =& - dt^{2} + \biggl[ (2n - 1)\sqrt{\frac{k_{2}}{k_{1} + 2n - 2}} t \biggr]^{\frac{2n}{2n - 1}}dx^{2} \\ &{}+ \biggl[ (2n - 1)\sqrt{ \frac{k_{2}}{k_{1} + 2n - 2}} t \biggr]^{\frac{2}{2n - 1}}dy^{2} \\ &{}+2 \biggl[ (2n - 1)\sqrt{\frac{k_{2}}{k_{1} + 2n - 2}} t \biggr]^{\frac{2}{2n - 1}}xdy dz \\ &{}+ \biggl( \biggl[ (2n - 1)\sqrt{\frac{k_{2}}{k_{1} + 2n - 2}} t \biggr]^{\frac{2}{2n - 1}}x^{2} \\ &{}+ \biggl[ (2n - 1)\sqrt{\frac{k_{2}}{k_{1} + 2n - 2}} t \biggr]^{\frac{2n}{2n - 1}} \biggr)dZ^{2} \end{aligned}$$
(36)

where \(k_{1}\) and \(k_{2}\) are given by (35).

3.2.1 Physical discussion

Equation (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

$$ V = \biggl[ (2n - 1) \sqrt{\frac{k_{2}}{k_{1} + 2n - 2}} t \biggr]^{\frac{2n + 1}{2n - 1}} $$
(37)

The mean Hubble’s parameter

$$ H = \frac{1}{3}\frac{\dot{V}}{V} = \frac{2n + 1}{3(2n - 1)t} $$
(38)

The scalar expansion is

$$ \theta = \frac{2n + 1}{(2n - 1)t} $$
(39)

The shear scalar is

$$ \sigma^{2} = \frac{1}{3t^{2}} \biggl( \frac{n - 1}{2n - 1} \biggr)^{2} $$
(40)

The average anisotropic parameter is

$$ \Delta = \frac{6(n - 1)^{2}}{(2n + 1)^{2}} $$
(41)

The deceleration parameter

$$ q = \frac{4 ( n - 1 )}{(2n + 1)} $$
(42)

The energy density and tension in the string is

$$\begin{aligned} &\!\begin{aligned} &\rho = \frac{(1 + \omega ) ( 2k_{2} ( 1 - n ) + k_{1} + 2n - 2 )}{3(8\pi + 2\mu )(1 - 2n)t^{2}} \\ &\lambda = \frac{ ( 2k_{2} ( 1 - n ) + k_{1} + 2n - 2 )}{3(8\pi + 2\mu )(1 - 2n)t^{2}} \end{aligned} \end{aligned}$$
(43)

It may be observe that the spatial volume of the universe increases with cosmic time. It can be seen that \(H\), \(\theta\), \(\sigma^{2}\), \(\rho\) and \(\lambda\) diverse when \(t=0\) and vanish \(t \to \infty\). For \(n= 1\), we observe that \(\Delta = 0\) and \(\sigma^{2} = 0\). This shows that for this value of \(n\), the model is isotropic and shear free and the deceleration parameter \(q\) is negative for \(0< n <1\). This establishes the late time acceleration of the universe. This is in agreement with the recent cosmological observations (Reiss et al. 1998; Perlmutter et al. 1997).

3.3 Massive (Reddy) string

Reddy (2003a, 2003b) has obtained string models in Brans and Dicke (1961) and Saez and Ballester (1986) scalar-tensor theories of gravitation using the equation of state given by (30). Model obtained with this equation of state is usually known as Reddy string or massive string in literature.

Now using (27) and (30) the field equations (18) and (19) yield the solution

$$\begin{aligned} &\!\begin{aligned} & X = \biggl[ (2n - 1)\sqrt{\frac{3}{4 ( n^{2} + 2n - 2 )}} t \biggr]^{\frac{n}{2n - 1}} \\ &Y = \biggl[ (2n - 1)\sqrt{\frac{3}{4 ( n^{2} + 2n - 2 )}} t \biggr]^{\frac{1}{2n - 1}} \end{aligned} \end{aligned}$$
(44)

and integration constants have been set equal to zero for simplicity.

Now the metric (10) with the help of (44) can be written as

$$\begin{aligned} ds^{2} =& - dt^{2} + \biggl[ (2n - 1)\sqrt{ \frac{3}{4 ( n^{2} + 2n - 2 )}} t \biggr]^{\frac{2n}{2n - 1}}dx^{2} \\ &{} + \biggl[ (2n - 1) \sqrt{\frac{3}{4 ( n^{2} + 2n - 2 )}} t \biggr]^{\frac{2}{2n - 1}}dy^{2} \\ &{}+ 2 \biggl[ (2n - 1)\sqrt{\frac{3}{4 ( n^{2} + 2n - 2 )}} t \biggr]^{\frac{2}{2n - 1}}xdy dz \\ &{}+\biggl( \biggl[ (2n - 1)\sqrt{\frac{3}{4 ( n^{2} + 2n - 2 )}} t \biggr]^{\frac{2}{2n - 1}}x^{2} \\ &{}+ \biggl[ (2n - 1)\sqrt{\frac{3}{4 ( n^{2} + 2n - 2 )}} t \biggr]^{\frac{2n}{2n - 1}} \biggr)dZ^{2} \end{aligned}$$
(45)

3.3.1 Physical discussion

The cosmological model given by (45) 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

$$ V = \biggl[ (2n - 1)t\sqrt{\frac{3}{4 ( n^{2} + 2n - 2 )}} \biggr]^{\frac{2n + 1}{2n - 1}} $$
(46)

The mean Hubble’s parameter

$$ H = \frac{1}{3}\frac{\dot{V}}{V} = \frac{2n + 1}{3(2n - 1)t} $$
(47)

The scalar expansion is

$$ \theta = \frac{2n + 1}{(2n - 1)t} $$
(48)

Shear scalar is

$$ \sigma^{2} = \frac{1}{3t^{2}} \biggl( \frac{n - 1}{2n - 1} \biggr)^{2} $$
(49)

The average anisotropic parameter is given by

$$ \Delta = \frac{6(n - 1)^{2}}{(2n + 1)^{2}} $$
(50)

The deceleration parameter is

$$ q = \frac{4 ( n - 1 )}{(2n + 1)} $$
(51)

The energy density in the string is

$$ \rho = \frac{2 ( n + 1 )}{3(8\pi + 2\mu )(1 - 2n)t^{2}} $$
(52)

The tension density in the string is

$$ \lambda = \frac{2 ( n + 1 )}{3(8\pi + 2\mu )(2n - 1)t^{2}} $$
(53)

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 density is positive for \(n < \frac{1}{2}\).

It is well known that the issue of accelerated expansion of the universe can be explained by taking into account the modified theories of gravity such as \(f(R,T)\) gravity. In \(f(R,T)\) gravity cosmic acceleration may result not only due to geometric contribution to the matter but also depends on matter contents of the universe. The coupling between matter and geometry in \(f(R,T)\) gravity yields non-geodesic motion of test particle and an extra acceleration is always present. We have investigated in this paper the geometric, Takabayasi and massive (Reddy) strings in this theory which explains the cosmic acceleration.

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). Our work in this paper will be helpful to study the structure formation of the universe in \(f(R,T)\) gravity which is a viable alternative to general relativity. 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. It is observed that our model confirms the late time acceleration of the universe.