 investigated during the last
two decades in a non covariant way, (Chen \& Wu 1990); (Abdel 1990);
(Pavon 1991); (Carvalho et al 1992); (Kalligas et al 1992); (Lima \&
Maia 1994);  (Lima \& Carvalho 1994); (Lima \& Trodden 1996); (Arbab
\&  Abdel 1994); (Birkel \& Sarkar 1997); (Silveira \& Waga 1997);
(Starobinsky 1998); (Overduin \& Cooperstock 1998); (Vishwakarma
2000,2001); (Arbab 2001,2003,2004); (Cunha \& Santos 2004);
(Carneiro \& Lima 2005); (Fomin et al 2005); (Sola \& Stefancic
2005,2006); (Pradhan et al 2007); (Jamil \& Debnath 2011) and
(Mukhopadhyay 2011); in particular, in Fomin et al (2005) there are
several evolution relations for $\Lambda$ which many author have
used, also in Ref. Overduin \& Cooperstock (1998) appears a table
with these relations and the corresponding references where they
were considered. Anisotropic cosmological models, also has been
treated in this formalism from different points of view (Aroonkumar
1993,1994); (Arbab 1997); (Singh et al 1998); (Pradhan \& Kumar
2001); (Pradhan 2003,2007,2009); (Pradhan \& Pandey 2003,2006);
(Pradhan et al 2007,2008); (Carneiro 2005); (Esposito et al 2007);
(Bal \& Singh 2008); (Belinch\'on 2008); (Singh et al 2008), (Singh
et al 2013); (Shen 2013); (Tripathy 2013) and (Rahman \& Ansary
2013).

In this work we present an analysis in covariant way, using the
Lagrangian density of standard scalar field. The main idea arises by
proposing that as  in the cosmological constant case, the scalar
potential is identified as $ V(\phi)=2\Lambda$, with $\Lambda$ a
constant. So, we {\it extend} this idea and suggest that this
correspondence is valid even when this cosmological term has a
temporal dependence, i.e.,  $ V(\phi(t))=2\Lambda(t)$. We include a
barotropic equation state between the pressure and energy density of
the scalar field, $\rm p_\phi= \omega_\phi \rho_\phi$, quantities
that we shall define in the following lines. In order to built up
the analysis presented here, initially we solve the Klein-Gordon
equations, whose solution implies that the energy density of a
scalar field has a wide range of scaling behavior, $\rho_\phi \sim
A^{-m}$ with A the scale factor of the FRW model, (Ferreira \& Joyce
1998); (Liddle \& Sharrer 1998) and (Copeland et all 1998), that
emerges as a proportionality law between the energy density of the
scalar field and the energy density of the barotropic perfect fluid,
relation that is know as an "attractor solution", with the
proportionality constant $\rm m_\phi$ (Liddle \& Sharrer 1998), that
is, $\rm \rho_\phi=m_\phi \rho$.  However the nature of the matter
that corresponds to the scalar field is unknown since  this matter
is not detectable by the usual methods.

The research in  non covariant way goes  by
beginning with the relation when the cosmological term $ \Lambda$, often  treated as a constant, having a geometrical
interpretation. For explain this, the Einstein field equation is
written as
\begin{equation}
\rm G_{\mu \nu} + \Lambda g_{\mu \nu}=-8\pi G T_{\mu \nu},
\label{EFE}
\end{equation}
where $\rm G_{\mu \nu}$ is the usual Einstein tensor and $\rm T_{\mu
\nu}$ is the energy-momentum tensor of matter. When we take the
covariant divergence of this equation, the vanishing of the Einstein tensor  is
guaranteed by the Bianchi identities, then it is assumed that the
energy-momentum tensor satisfies the corresponding conservation law
$\rm \nabla^\nu T_{\mu \nu}=0$, and that the covariant divergence of
cosmological term must vanish, this implies that $\rm
\Lambda=constant$. Usually, this argument situates this cosmological
constant on the left-hand side of the field equations, given a
geometrical interpretation of the cosmological term.

However, if the cosmological term is moved to  the right-hand side
of the field equation, the interpretation of $\rm \Lambda$ change as
part of the matter content in the following sense. The field
equations now are written as
\begin{equation}
\rm G_{\mu \nu} =-8\pi G \tilde T_{\mu \nu}, \qquad  \tilde T_{\mu
\nu}= T_{\mu \nu}+ \frac{\Lambda}{8\pi G} g_{\mu \nu}. \label{EFE-N}
\end{equation}
Once this is done, there is  no a priori reason why this
cosmological term should not vary, considering that it is  the effective
energy-momentum tensor $\rm \tilde T_{\mu \nu}$ that satisfies the
conservation law
\begin{equation} \rm \nabla^\nu \tilde T_{\mu \nu}=0,\label{C-l}
\end{equation}.

The set of equation (\ref{EFE-N}, \ref{C-l}) and one state equation,
are the fundamental tools to do the research, considering a variable
cosmological term in non covariant way, because there is no a
Lagrangian density that reproduce these field equations
(\ref{EFE-N}, \ref{C-l}).

In the present treatment we take into  account the corresponding
Lagrangian density with a scalar field
\begin{equation}
 \rm {\cal L}[g,\phi]=\sqrt{-g}\left(R-\frac{1}{2}g^{\mu \nu}\nabla_\mu\phi\nabla_\nu\phi
 +V(\phi)\right)+\sqrt{-g} {\cal L}_{matter}\label{lagra}
\end{equation}
where R is the Ricci scalar, ${\cal L}_{matter}$ correspond to a
barotropic perfect fluid, $\rm p=\omega \rho$, $\rho$ is the energy
density and p is the pressure of the fluid in co-moving frame and
$\omega$ is the barotropic constant.

The corresponding variation of  (\ref{lagra}), with respect to the metric and
the scalar field gives the Einstein  and Klein-Gordon field equations
\begin{eqnarray}
&&\rm
R_{\alpha\beta}-\frac{1}{2}g_{\alpha\beta}R=-\frac{1}{2}\left(\nabla_\alpha\phi\nabla_\beta\phi-\frac{1}{2}g_{\alpha\beta}
g_{\mu\nu}\nabla_\mu\phi\nabla_\nu\phi\right)+\frac{1}{2}g_{\alpha\beta}V(\phi)-8\pi GT_{\alpha\beta}, \label{camrel}\\
&&\rm \square\phi-\frac{\partial V}{\partial\phi}=0.\label{klein}
\end{eqnarray}
From  (\ref{camrel}) it  can be deduced  that the energy-momentum
tensor associated with the scalar field is
\begin{equation}
\rm
8 \pi G T^{(\phi)}_{\alpha\beta}=\frac{1}{2}\left(\nabla_\alpha\phi\nabla_\beta\phi-\frac{1}{2}g_{\alpha\beta}
g_{\mu\nu}\nabla_\mu\phi\nabla_\nu\phi\right)-\frac{1}{2}g_{\alpha\beta}V(\phi)
\end{equation}\\
and the corresponding tensor for a barotropic perfect fluid  becomes
$$\rm T_{\alpha \beta}= \left(p+\rho\right) u_\alpha u_\beta +
g_{\alpha \beta} p$$
here $\rm u_\alpha$ is the four-velocity in the
comoving  frame. The line element to be considered in this work is the FRW
\begin{equation}
\rm ds^2=-N(t)^2 dt^2 +A(t)^2 \left[\frac{dr^2}{1-\kappa r^2}
+r^2(d\theta^2+sin^2\theta d\phi^2) \right]=-d\tau^2 + A(\tau)^2
\left[\frac{dr^2}{1-\kappa r^2} +r^2(d\theta^2+sin^2\theta d\phi^2)
\right], \label{frw}
\end{equation}
where we identify the time transformation $\rm N(t) dt=d\tau$, this
transformation will be used in the whole work,