The SN Ia observations [1, 2] suggest that the observable universe is undergoing an accelerated expansion. This remarkable discovery stands a major break through of the observational cosmology and indicates the presence of unknown fluid - dark energy (DE) that opposes the self attraction of the matter. This acceleration is realized with positive energy density and negative pressure. So, it violate the strong energy condition (SEC). The authors of ref. [3] confirmed that the violation of SEC gives a reverse gravitational effect that provides an elegant description of transition of universe from deceleration to cosmic acceleration. The cosmological constant cold dark matter ($\Lambda$CDM) cosmological model is the simplest model of universe that describes the present acceleration of universe and fits with the present day cosmological data [4]. It is based on the Einstein’s theory of general relativity with a spatially flat, isotropic and homogeneous space-time. The observed acceleration of universe has been explained by introducing a positive cosmological constant $\Lambda$ which is mathematically equivalent to vacuum energy with equation of state (EOS) parameter set equal to $-1$. It suffers from two problems on theoretical front, concerning the cosmological constant $\left(\Lambda \right)$. These problems are known as fine tunning and cosmic coincidence problems [5, 6]. In the contemporary cosmology, the source that derives the present acceleration of universe is still mystery and is discussed under the generic name DE. In the literature, the simplest candidate of dark energy is a positive $\Lambda$ besides some scalar field DE models, namely the phantom, quintessence and k-essence [6, 7]. In the physical cosmology, the dynamical form of DE with an effective equation of state (EOS), $\omega <-\frac{1}{3}$, were proposed instead of constant vacuum energy. The current cosmological data from large scale-structure [8], Supernovae Legacy survey, Gold Sample of Hubble Space Telescope [9, 10] do not support the possibility of $\omega <<-1$. However, $\omega =-1$ is a favorable candidate for DE that crossing the phantom divide line (PDL). Setare and Saridakis [11, 12] have studied the quintom model that described the nature of DE with $\omega$ across -1 and give the concrete theoretical justification for existence of quintom model

We notice that after publication of WMAP data, today there is considerable evidence in support of anisotropic model of universe. On the theoretical front, Misner [13] has investigated an anisotropic phase of universe, which turns into isotropic one. The authors of ref. [14, 15] have investigated the accelerating model of universe with anisotropic EOS parameter and have also shown that the present SN Ia data permits large anisotropy. Recently DE models with variable EOS parameter in anisotropic space-time have been studied by Yadav and Yadav [16], Yadav et al [17, 18], Akarsu and Kilinc [19], Yadav [20], Saha and Yadav [21] and Pradhan [22]. In the present work, however, we present $\Lambda$CDM-type cosmological model in spatially homogeneous and anisotropic Heckmann-Schucking space-time. The outline of paper is as follows: in section 2, the field equation and it’s solution are described. Section 3 deals with dust filled universe and Hubble’s parameter. Section 4 covers the study of observational parameters for the model under consideration. The deceleration parameter (DP) and certain physical properties of the universe are presented in section 5 and 6 respectively. Finally conclusions are summarized in section 7.

. We consider a general Heckmann-Schucking metric

where A, B and C are functions of time only. we consider energy momentum tensor for a perfect fluid i.e.

where $g_{ij}{u}^i{u}^j=1$ and $u^i$ is the 4-velocity vector.

In co-moving co-ordinates

The Einstein field equations are

Choosing co-moving coordinates,the field equations (4) in terms of line element (1) can be write down as

where $A_4$, $B_4$ and $C_4$ stand for time derivatives of A, B, and C respectively.
The mass-energy conservation equation $T_{;j}^{ij}=0$ gives

Subtracting eqs.(5) from (6),(6) from (7) and (7) from (6), we obtain

Subtracting(12) from (10), we get

This equation can be re-written in the following form

Integrating this equation, we get the following first integral

where L is constant of integration.
The exact solution of eq. (15), in general, is not possible however one can solve eq. (15) explicitly, by choosing $L=0$ that reveals $A^2=BC$. The present day observations suggest that the initial anisotropy dissipated out for large value of time and the directional scale factors have same values in all direction i.e. $A=B=C$ which is easily obtained by putting $L=0$ in eq. (15). That is why $L=0$ has physical meaning.
Now we can assume

where

Further integrating equation (11), we get the first integral

where K is an arbitrary constant of integration.
With help of equation(15), eq (9) simplifies as

Thus the Hubble’s parameter in this model is

Equations(5)-(8) are simplified as

Taking $K=0$, we obtain the Einstein’s field equation for spatially homogeneous and isotropic flat FRW model as

where A is expansion scale factor.
Thus equations(21)-(23) may be regarded as counterpart of FRW Equations in our anisotropic model. Equations(22) and (23) may be re-written as

We now assume that the cosmological constant $\Lambda$ and the term due to anisotropy also act like energies with densities and pressures as

It can be easily verified that energy conservation law (20) holds separately for ${\rho }_{\Lambda }$ and ${\rho }_{\sigma }$ i.e.

The equations of state for matter, $\sigma$ and $\Lambda$ energies are as follows

where ${\omega }_m=0$ for matter in form of dust, ${\omega }_m=\frac{1}{3}$ for matter in form of radiation. There are certain more values of ${\omega }_m$ for matter in different forms during the course of evolution of the universe.

Since

Therefore

Similarly

So,

Now we use the following relation between scale factor A and red shift z

The suffix(0) is meant for the value at present time.
The energy density $\rho$ comprises of following components

Equations (20) and (31) yield

Equations (26) and (27) take the form

For dust filled universe, we have $p_m=0$ and ${\rho }_m=\frac{3c^2{H}^2}{8\pi G}$
Then eq. (35) gives

where ${\Omega }_m=\frac{{\rho }_m}{{\rho }_c}=\frac{\left({\Omega }_m{)}_0{H}_0^2\right(1+z{)}^3}{}{H}^2$, ${\Omega }_{\Lambda }=\frac{{\rho }_{\Lambda }}{{\rho }_c}=\frac{({\Omega }_{\Lambda }{)}_0{H}_0^2}{H^2}$ and ${\Omega }_{\sigma }=\frac{{\rho }_{\sigma }}{{\rho }_c}=\frac{\left({\Omega }_{\sigma }{)}_0{H}_0^2\right(1+z{)}^6}{H^2}$

In this paper, we have investigated the $\Lambda$CDM-type cosmological model in Heckmann-Shucking space-time. Under some specific choice of parameters, the model under consideration isotropizes and have consistency with recent SN Ia observation. We have estimated some physical parameters at present epoch for derived model which is summarized as

Table: 2

We observe from the result displayed in table 2 that the derived model is observationally indistinguishable in the vicinity of present epoch of universe. Thus the $\Lambda$CDM model fits better with the observational data.