In the present work, we have searched the existence of CDM-type cosmological model in anisotropic
Heckmann-Schucking space-time. The matter source that is responsible for the present acceleration
of the universe consist of cosmic fluid with , where is the equation of
state parameter. The Einstein’s field equations have been solved explicitly under some specific choice
of parameters that isotropizes the model under consideration. It has been found that the derived model
is in good agreement with recent SN Ia observations. Some physical aspects of the model has been discussed
Keywords: Heckmann-Schucking metric, dark energy, CDM model
CDM-type cosmological model and observational constraints
G. K. Goswami, Anil Kumar Yadav and Mandwi Mishra
Kalyan Post-Graduate College, Bhilai, C.G., India
Anand Engineering College, Keetham, Agra - 282007, India
Shri Shankaracharya Engineering College, Bhilai, C. G., India
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.  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 (CDM) cosmological model is the simplest
model of universe that describes the present acceleration of universe and fits with the present
day cosmological data . 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 which is mathematically
equivalent to vacuum energy with equation of state (EOS) parameter set equal to . It suffers from
two problems on theoretical front, concerning the cosmological constant . 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
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), , were proposed instead of constant
vacuum energy. The current cosmological data from large scale-structure , Supernovae
Legacy survey, Gold Sample of Hubble Space Telescope [9, 10] do not support the
possibility of . However, 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 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  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 , Yadav et al [17, 18], Akarsu and Kilinc , Yadav , Saha and Yadav  and Pradhan . In the present work, however, we present 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.
2 Field equations
. 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 and is the 4-velocity vector.
In co-moving co-ordinates
The Einstein field equations are
where , and stand for time derivatives of A, B, and C respectively.
The mass-energy conservation equation 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 that reveals . 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. which is easily obtained by putting in eq. (15). That is why has physical meaning.
Now we can assume
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 , 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 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 and i.e.
The equations of state for matter, and energies are as follows
where for matter in form of dust, for matter in form of radiation. There are certain more values of for matter in different forms during the course of evolution of the universe.
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 comprises of following components
Equations (20) and (31) yield
Equations (26) and (27) take the form
3 Dust filled universe
For dust filled universe, we have and
Then eq. (35) gives
where , and
3.1 Expression for Hubble’s Constant
Equations (31) and (35) yields
The behaviour of Hubble’s parameter versus redshift and scale function have been depicted in Figure 1. We notice from figure 1 that increases with redshift while it decreases with scale function. From the right panel of figure (1), it is also clear that for dominated universe, the Hubble’s parameter is either almost stationary or it is decreasing slowly with scale function i.e. time.
4 Some observational constraints
The luminosity distance which determines flux of the source is given by
where is the spatial co-ordinate distance of a source.
The Geodesic for metric (1) ensures that if in the beginning ; then ; .
So if a particle moves along x- direction, it continues to move along x- direction always.If we assume that line of sight of a vantage galaxy from us is along x-direction then path of photons traveling through it satisfies
From this we obtain
where we have used and from eqs. (21) and (31)
So, the luminosity distance is given by
4.1 Apparent Magnitude and Red Shift relation:
The absolute magnitude and apparent magnitude are related to the redshift by following relation
For low redshift, one can easily obtain the luminosity distance from eq. (41)
Combining equations (41), (42) and (43), one can easily obtain the expression for the apparent magnitude in terms of redshift parameter as follows
Figure 2 shows the behaviour of luminosity distance and apparent magnitude
with redshift for some certain values of , and .
In the present analysis, we use 60 data set of SN Ia for the low red shift as reported by Perlmutter et al. . In this case has been computed according to the following relation
Here, dof stands for degree of freedom. From the table 1, we note that the best fit values of with and the reduced value is 0.1257.
4.2 Age of the Universe
The present age of the universe is obtained as follows
where we have used and
The left panel of figure 3 shows the variation of time with redshift. It is also observed that the dominated universe gives the age of the universe as
From WMAP data, the empirical value of present age of universe is which is very close to present age of universe, estimated in the derived model.
5 Deceleration Parameter
The deceleration parameter is given by
From equation (34)
This equation clearly shows that without presence of term in the Einstein’s field
equation (4), one can’t imagine of accelerating universe.This equation also expresses the fact that anisotropy
raises the lower limit value of required for acceleration. This
may be seen in the following way.
For FRW model, acceleration requires
where as for anisotropic model
Combining equations (34), (35), (36) and (37), the expression for DP in terms of redshift is given by
Since in the derived model, the best fit values of , and are 0.29, 0.69 and 0.02 respectively hence we compute the present value of DP for derived CDM universe by putting in eq. (49). The present value of DP is given by
6 Some Physical Properties of the Model
6.1 The energy density in the universe
The energy density is given by
Here are the present energy density of various components. Taking,
Therefore, the present value of dust energy density and dark energy density are obtained as
6.2 Shear Scalar
The shear scalar is given by
In our model
From eq. (57), it is clear that shear scalar vanishes as .
6.3 Relative Anisotropy
The relative anisotropy is given by
This follows the same pattern as shear scalar. This means that relative anisotropy
decreases over scale factor i.e. time.
6.4 Evolution of the scale factor
We begin with the integral
Equations (37) and (59) lead to
The right panel of figure 3 shows the variation of time with scale function for the derived model.
7 Final remarks
In this paper, we have investigated the 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
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 CDM model fits better with the observational data.
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