1 Introduction

Several modifications of Einstein’s theory have been proposed from time to time. Noteworthy among them are the scalar-tensor theories of gravitation formulated by Brans and Dicke (1961), Nordtvedt (1970), Sen (1957), Sen and Dunn (1971) and Saez and Ballester (1986). In particular, Brans-Dicke and Saez-Ballester scalar-tensor theories are attracting more and more attention because of their applications in cosmology. In these theories gravity is mediated by a long range scalar field in addition to the usual tensor fields present in Einstein’s theory. Brans-Dicke scalar-tensor theory of gravitation introduces an additional scalar field ϕ interacting equally with all forms of matter (with the exception of electromagnetism) besides the metric g ij and a dimensionless coupling constant ω. In Saez-Ballester scalar-tensor theory of gravitation the metric is coupled with a dimensionless scalar field in a simple manner. This coupling gives a satisfactory description of the weak fields. One particularly interesting result of this theory is appearance of antigravity regime, which suggests a possible connection to the missing matter problem in non-flat FRW cosmologies. Canuto et al. (1977) have proposed a scale covariant theory of gravitation which is another viable alternative to general relativity (Wesson 1980; Will 1984). In this theory Einstein’s field equations are valid in gravitational units where as the physical quantities are measured in atomic units. The metric tensor in the two systems of units are related by a conformal transformation

$$ \overline{g_{ij}} = \phi^{2} \bigl(x ^{k} \bigr) g _{ij} $$
(1)

where Latin indices take the values 1, 2, 3, 4, bar denotes gravitational units and unbar denotes atomic units. The gauge scalar function ϕ (0<ϕ<∞) in its most general formulation is a function of all space-time coordinates. Thus, using the conformal transformation of the type given by Eq. (1) Canuto et al. (1977) transformed the usual Einstein’s equations into

$$ R _{ij} - \frac{1}{2} R g _{ij} + f _{ij} ( \phi) =-8\pi G (\phi ) T _{ij} + \Lambda(\phi) g _{ij} $$
(2)

where

$$ \phi^{2} f _{ij} =2\phi\phi_{i;j} - 4 \phi _{,i} \phi_{,j} - g _{ij} \bigl(\phi \phi_{;k} ^{k} - \phi^{, k} \phi _{, k} \bigr) $$
(3)

Here R ij is the Ricci tensor, R is the Ricci scalar, Λ is the cosmological ‘Constant’, G is the gravitational ‘constant’ and T ij is the energy momentum tensor of matter distribution. A semicolon denotes covariant derivative and comma denotes ordinary derivative with respect to x i. A particular feature of this theory is that no independent equation for ϕ exists. The possibility that have been considered for the gauge scalar function (Canuto et al. 1977) are

$$ \phi (t ) = \biggl( \frac{t}{t _{0}} \biggr) ^{\epsilon},\quad\epsilon = \pm1,\ \pm \frac{1}{2} $$
(4)

where t 0 is a constant. The form

$$ \phi\sim t ^{\frac{1}{2}} $$
(5)

is the one most favored to observations (Canuto and Goldman 1983).

The study of cosmological models in the modified theories of gravitation has been the active area of research for the last few decades. Singh and Rai (1983) presented a nice review of Brans-Dicke cosmological models while Singh and Agrawal (1991), Reddy and Rao (2001), Reddy et al. (2006) have studied some cosmological models in Saez-Ballester theory. Recently, Adhav (2012), Reddy and Shanthikumar (2013), Rao and Neelima (2013) have discussed some cosmological models in f(R,T) gravity (Harko et al. 2011), which is another modification of general relativity to explain the recent scenario of accelerated expansion of the universe (Riess et al. 1998; Perlmutter et al. 1999). Reddy et al. (2002), Reddy and Venkateswarlu (2004), Reddy et al. (2007), Adhav et al. (2008), Ram et al. (2009), Belinchon (2012), Singh and Anita (2007), Singh and Devi (2011), Pradhan et al. (2013), and Beesham (1986a, 1986b, 1986c) are some of the authors who have investigated several aspects and different isotropic and anisotropic cosmological models in the scale covariant theory of gravitation.

It is well known that viscosity plays an important role in cosmology (Singh and Devi 2011; Singh and Kale 2011; Setare and Sheyki 2010; Misner 1969). Also, bulk viscosity appears as the only dissipative phenomenon occurring in FRW models and has a significant role in getting accelerated expansion of the universe popularly known as inflationary space. Bulk viscosity contributes negative pressure term giving rise to an effective total negative pressure stimulating repulsive gravity. The repulsive gravity overcomes attractive gravity of matter and gives an impetus for rapid expansion of the universe hence cosmological models with bulk viscosity have gained importance in recent years. Barrow (1986), Pavon et al. (1991), Maartens et al. (1995), Lima et al. (1993), and Mohanthy and Pradhan (1992) are some of the authors who have investigated bulk viscous cosmological models in general relativity. Johri and Sudharsan (1989), Pimental (1994), Banerjee and Beesham (1996), Singh et al. (1997), Rao et al. (2007, 2012), Naidu et al. (2012) and Reddy et al. (2012) have studied bulk viscous and bulk viscous string cosmological models in Brans and Dicke (1961) and other modified theories of gravity. Very recently, Reddy et al. (2013a, 2013b) investigated Kaluza-Klein and LRS Bianchi type-II bulk viscous cosmic string models in the modified f(R,T) gravity while Naidu et al. (2013) have obtained bulk viscous FRW model in this modified gravity.

During the past two decades, string cosmological models have received considerable attention of research workers because of their importance in structure formation in the early stages of evolution of the universe. 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. It is well known that massive strings serve as seeds for the large structures like galaxies and cluster of galaxies in the universe. Letelier (1983), Stachel (1980), Vilenkin et al. (1987), Banerjee et al. (1990), Tripathy et al. (2009), Reddy (2003a, 2003), Katore and Rane (2006) and Sahoo (2008) are some of the authors who have investigated several important aspects of string cosmological models either in the frame work of general relativity or in modified theories of gravity.

Motivated by the above discussion and investigations in modified theories of gravity, we have studied, in this paper, LRS Bianchi type-II cosmological model in the scale covariant theory of gravity, proposed by Canuto et al. (1977), in the presence of cosmic strings and bulk viscosity. In Sect. 2, explicit field equations in scale covariant gravity for Bianchi type-II metric are obtained in the presence of bulk viscous fluid containing one dimensional cosmic strings. In Sect. 3 an exact cosmological model is presented by solving the field equations. Physical and kinematical properties of the model are discussed in Sect. 4. The last section contains some conclusions.

2 Metric and field equations

We consider spatially homogeneous and anisotropic LRS Bianchi type-II metric in the form

$$ ds ^{2} = - dt ^{2} + A ^{2} \bigl(dx ^{2} + dz ^{2} \bigr) + B ^{2} \ (dy+xz) ^{2} $$
(6)

We consider the energy momentum tensor for a bulk viscous fluid containing one dimensional cosmic strings as

$$ T _{ij} = ( \rho+ \overline{p} ) u _{i} u _{j} + \overline{p} g _{ij} - \lambda x _{i} x _{j} $$
(7)

and

$$ \overline{p} = p-3 \zeta H $$
(8)

where ρ is the rest energy density of the system, ζ(t) is the coefficient of bulk viscosity, 3ζH is usually known as bulk viscous pressure, H is Hubble’s parameter and λ is string tension density.

Also, \(u ^{i} = \delta_{4} ^{i}\) is a four-velocity vector which satisfies

$$ g _{ij} u ^{i} u_{j} =- x ^{i} x _{j} =-1\quad \mathrm{and}\quad u ^{i} x _{i} =0 $$
(9)

Here we also consider ρ, \(\overline{p}\) and λ are functions of time t only.

Now using co moving coordinates and Eqs. (7)–(9) the field equations (2) and (3), for the metric (6), can be written, explicitly, as

$$\begin{aligned} & {\frac{\ddot{A}}{A} + \frac{\ddot{B}}{B} + \frac{\dot{A}}{A} \frac {\dot{B}}{B} + \frac{1}{4} \frac{B ^{2}}{A ^{4}} + \frac{\ddot{\phi}}{\phi} + \frac{\dot{\phi}}{ \phi} \biggl(3 \frac{\dot{A}}{A} + \frac{\dot{B}}{B} \biggr) -\biggl(\frac{\dot{\phi}}{ \phi} \biggr) ^{2} } \\ & { \quad {} =-8\pi G(\phi) \overline{p} } \end{aligned}$$
(10)
$$\begin{aligned} & {2\frac{\ddot{A}}{A} +\biggl(\frac{\dot{A}}{A} \biggr) ^{2} - \frac{3}{4} \frac {B ^{2}}{A ^{4}} + \frac{\ddot{\phi}}{\phi} + \frac{\dot{\phi}}{ \phi} \biggl(\frac{\dot{A}}{A} +3 \frac{\dot{B}}{B} \biggr) -\biggl(\frac{\dot{\phi}}{\phi} \biggr) ^{2} } \\ & { \quad {} =-8\pi G (\phi ) (\overline{p} - \lambda ) } \end{aligned}$$
(11)
$$\begin{aligned} & {2\frac{\dot{A}}{A} \frac{\dot{B}}{B} + \biggl(\frac{\dot{A}}{A} \biggr) ^{2} -\ \frac{1}{4} \frac{B ^{2}}{ A ^{4}} - \frac{\ddot{\phi}}{\phi} + \frac{\dot{\phi }}{\phi} \biggl(\frac{\dot{A}}{A} + \frac{\dot{B}}{B} \biggr) +3\biggl( \frac{\dot{\phi}}{\phi} \biggr) ^{2} } \\ & { \quad {} =8\pi G(\phi)\rho . } \end{aligned}$$
(12)

Also the conservation equation which is a consequence of the field equations (2) and (3) in this theory, is

$$ \dot{\rho} + (\rho+ \overline{p} ) u _{;k} ^{k} +\ \rho \frac{\dot{(G\phi)}}{ (G\phi)} +3 \overline{p} \frac{\dot{\phi}}{\phi} = 0 $$
(13)

and for the metric this becomes

$$ \dot{\rho} +(\rho+ \overline{p} ) \biggl(2 \frac{\dot{A}}{A} + \frac {\dot{B}}{B} \biggr) +\rho \biggl(\frac{\dot{G}}{G} + \frac{\dot{\phi}}{\phi} \biggr) +3 \overline{p} \frac{\dot{\phi}}{\phi} =0 $$
(14)

where an overhead dot denotes differentiation with respect to t.

The spatial volume for the metric (6) is defined by

$$ V= A ^{2} B= a ^{3} $$
(15)

where a(t) is the scale factor of the universe. the scalar expansion θ the shear scalar σ 2, the Hubble’s parameter H and mean anisotropy parameter A m in the model are defined by

$$ \theta= 2\frac{\dot{A}}{A} + \frac{\dot{B}}{B} $$
(16)
$$ \sigma^{2} = \frac{1}{3} \biggl(\frac{\dot{A}}{A} - \frac{\dot{B}}{B} \biggr) ^{2} $$
(17)
$$ H = \frac{1}{3} \theta=\frac{\dot{a}}{a} $$
(18)
$$\begin{aligned} 3A _{m} = \sum_{i=1} ^{3} \biggl(\frac{\Delta H _{i}}{H} \biggr) ^{2}\quad\mbox{where}\ \Delta H _{i} = H _{i} -H,\ i=1,2,3 \end{aligned}$$
(19)

3 Solutions and the model

The field equations (10)–(12) are a system of three independent equations in six unknowns A, B, \(\overline{p}\), ρ, ζ and λ. Also the field equations are highly non-linear in nature and therefore we use the following plausible physical conditions to find determinate solution.

  1. (i)

    The shear scalar σ 2 is proportional to scalar expansion θ so that we can take (Collins et al. 1980)

    $$ A =B ^{m} $$
    (20)

    where m≠1 is a constant and it takes care of the anisotropic nature of the model.

  2. (ii)

    Variation of Hubble’s parameter proposed by Bermann (1983) that yields constant deceleration parameter models of the universe defined by

    $$ q =-a \frac{\ddot{a}}{\dot{a} ^{2}} = \mbox{constant} $$
    (21)

    which admits the solution

    $$ a = (ct+d) ^{\frac{1}{1+q}} $$
    (22)

    where c≠0 and d are constants of integration. This equation implies that the condition for accelerated expansion of the universe is 1+q>0.

  3. (iii)

    For a barotropic fluid, the combined effect of the proper pressure and the bulk viscous pressure can be expressed as

    $$ \overline{p} = p-3 \varsigma H = \varepsilon\rho $$
    (23)

    where

    $$ \varepsilon= \varepsilon_{0} -\alpha\quad (0 \leq\varepsilon _{0} \leq 1 ),\qquad p=\varepsilon_{0} \rho $$
    (24)

    and α and ε 0 are constants.

Now using Eqs. (20)–(23),the field equations yield the expressions for the metric coefficients as

$$ {A} = (ct+d) ^{\frac{3m}{ (2m+1 ) (1+q)}} $$
(25)
$$ {B}= (ct+d) ^{\frac{3}{(2m+1 ) (1+q)}} $$
(26)

where the constants c≠0 and d are constants of integration.

Hence the metric (6) through a proper choice of coordinates and constants (i.e. c=1, d=0) can be written as

$$\begin{aligned} ds ^{2} =& - dt ^{2} + t ^{\frac{6m}{ (2m+1 ) (1+q)}} \bigl(dx ^{2} + dz ^{2} \bigr) \\ &{} + t ^{\frac{6}{ (2m+1 ) (1+q )}} (dy+xz) ^{2} \end{aligned}$$
(27)

4 Some physical properties of the model

The model given by Eq. (27) represents LRS Bianchi type-II bulk viscous string cosmological model in the scale covariant theory proposed by Canuto et al. (1977) with the following physical and kinematical properties:

The physical and kinematical quantities of observational interest in cosmology are the energy density ρ, bulk viscous pressure \(\overline{p}\), pressure p, Hubble’s parameter H, coefficient of bulk viscosity ζ, scalar field ϕ, string tension density λ, spatial volume V 3, scalar expansion θ, shear scalar σ 2, and the mean anisotropy parameter A m which, for the model (25), are given by

$$\begin{aligned} 8\pi G (\phi ) \rho =& \biggl[ \frac{1}{t ^{2}} \biggl\{ \frac{9m (m+2 )}{ (2m+1 ) ^{2} (q+1 ) ^{2}} + \frac{3\epsilon (m+1 )}{ (2m+1 ) (q+1 )} \\ &{} + 2\epsilon (\epsilon+1 ) \biggr\} \biggr] - \frac{1}{4} t ^{\frac{-6(2m-1)}{ (2m+1 ) (q+1)}} \end{aligned}$$
(28)
$$\begin{aligned} 8\pi G (\phi ) \overline{p} =& \biggl[ \frac{\varepsilon }{t ^{2}} \biggl\{ \frac{9m (m+2 )}{ (2m+1 ) ^{2} (q+1 ) ^{2}} + \frac{3\epsilon (m+1 )}{ (2m+1 ) (q+1 )} \\ &{} +2\epsilon (\epsilon+1 ) \biggr\} \biggr] - \frac{\varepsilon}{4} t ^{\frac{-6(2m-1)}{ (2m+1 ) (q+1)}} \end{aligned}$$
(29)
$$\begin{aligned} 8\pi G (\phi ) p =& \biggl[ \frac{\varepsilon_{0}}{t ^{2}} \biggl\{ \frac{9m (m+2 )}{ (2m+1 ) ^{2} (q+1 ) ^{2}} + \frac{3\epsilon (m+1 )}{ (2m+1 ) (q+1 )} \\ &{} +2\epsilon (\epsilon+1 ) \biggr\} \biggr] - \frac{\varepsilon_{0}}{4} t ^{\frac{-6(2m-1)}{ (2m+1 ) (q+1)}} \end{aligned}$$
(30)
$$ H= \frac{1}{(q+1)t} $$
(31)
$$\begin{aligned} 8\pi G(\phi)\zeta =&\frac{ (\varepsilon_{0} -\varepsilon )}{3} (q+1) \biggl[ \frac{1}{t} \biggl\{ \frac{9m (m+2 )}{ (2m+1 ) ^{2} (q+1 ) ^{2}} \\ &{} + \frac{3\epsilon (m+1 )}{ (2m+1 ) (q+1 )} +2\epsilon ( \epsilon+1 ) \biggr\} \biggr] \\ &{} - \frac{1}{4} t ^{\frac{-(12m-7)}{ (2m+1 ) (q+1)}} \end{aligned}$$
(32)
$$ \phi (t ) =\biggl(\frac{t}{t _{0}} \biggr) ^{\epsilon} $$
(33)
$$\begin{aligned} 8\pi G (\phi ) \lambda =& \biggl[ \frac{1}{t ^{2}} \biggl\{ \frac{9(2 m ^{2} -m-1)}{ (2m+1 ) ^{2} (q+1 ) ^{2}} \\ &{} + \frac{6\epsilon (m-1 )}{ (2m+1 ) (q+1 )} \biggr\} \biggr] - t ^{\frac{-6(2m-1)}{ (2m+1 ) (q+1)}} \end{aligned}$$
(34)
$$ V ^{3} =t ^{\frac{3}{q+1}} $$
(35)
$$ \theta= \frac{3}{ (1+q ) t} $$
(36)
$$ \sigma^{2} = \frac{3(m-1) ^{2}}{[ (2m+1 ) (q+1 ) t] ^{2}} $$
(37)
$$ A _{m} = \frac{4}{3} $$
(38)

From the above results, we can observe that the model (25) has no initial singularity i.e. at t=0. The spatial volume of the model increases as t increases showing the expansion of the universe since 1+q>0. It can also be seen that all the physical and kinematical quantities except ϕ diverge at the initial epoch, i.e. at t=0 and they all tend to zero as t becomes infinitely large. Also, since A m , \(\frac{\sigma^{2}}{ \theta^{2}} \neq0\), the model is anisotropic through out the evolution of the universe and it will help to understand the structure of the universe and the formation of galaxies at the early stages of evolution of the universe in scale covariant cosmology.

5 Conclusions

Scalar field cosmology is of immense importance in the study of the early universe and particularly in the investigation of inflation(during which the universe undergoes a period of accelerated expansion). It is well known that the present day universe is well represented by the spatially homogeneous isotropic FRW model. However experiments reveal that there is certain amount of anisotropy in our universe. Hence, in this paper, we present a spatially homogeneous and anisotropic LRS Bianchi type-II model in scale covariant theory of gravitation proposed by Canuto et al. (1977) when the source of energy momentum tensor is a viscous fluid containing one dimensional cosmic strings. We have used the special law of variation for Hubble’s parameter proposed by Bermann (1983) to obtain a determinate solution of the highly non linear field equations. A barotropic cosmic fluid is considered for this study. A general equation of state for the energy density is assumed. We have also assumed that the scalar expansion of the space-time is proportional to shear scalar. The model presented will help to discuss the role of bulk viscosity in getting an inflationary model and to understand structure formation in the early stages of evolution of the universe.