## Abstract

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
in detail.

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

Email: gk.goswam9i@gmail.com

Anand Engineering College, Keetham, Agra - 282007, India

Email: abanilyadav@yahoo.co.in

Shri Shankaracharya Engineering College, Bhilai, C. G., India

## 1 Introduction

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 (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 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 [8], 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 [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 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

(1) |

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

(2) |

where and is the 4-velocity vector.

In co-moving co-ordinates

(3) |

The Einstein field equations are

(4) |

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

(5) |

(6) |

(7) |

(8) |

where , and stand for time derivatives of A, B, and C respectively.

The mass-energy conservation equation gives

(9) |

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

(10) |

(11) |

(12) |

Subtracting(12) from (10), we get

(13) |

This equation can be re-written in the following form

(14) |

Integrating this equation, we get the following first integral

(15) |

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

(16) |

(17) |

where

(18) |

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

(19) |

where K is an arbitrary constant of integration.

With help of equation(15), eq (9) simplifies as

(20) |

Thus the Hubble’s parameter in this model is

(21) |

Equations(5)-(8) are simplified as

(22) |

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

(23) |

(24) |

(25) |

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

(26) |

(27) |

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

(28) |

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

(29) |

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.

(30) |

Since

Therefore

Similarly

So,

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

(31) |

The suffix(0) is meant for the value at present time.

The energy
density comprises of following components

(32) |

Equations (20) and (31) yield

(33) |

Equations (26) and (27) take the form

(34) |

(35) |

## 3 Dust filled universe

For dust filled universe, we have and

Then eq. (35) gives

(36) |

where , and

### 3.1 Expression for Hubble’s Constant

Equations (31) and (35) yields

(37) |

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

(38) |

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

(39) |

From this we obtain

(40) |

where we have used and from eqs. (21) and (31)

So, the luminosity distance is given by

(41) |

### 4.1 Apparent Magnitude and Red Shift relation:

The absolute magnitude and apparent magnitude are related to the redshift by following relation

(42) |

For low redshift, one can easily obtain the luminosity distance from eq. (41)

(43) |

Combining equations (41), (42) and (43), one can easily obtain the expression for the apparent magnitude in terms of redshift parameter as follows

(44) |

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. [2]. In this case has been computed according to the following relation

.29 | .69 | .02 | 7.41417 | 0.1257 |
---|---|---|---|---|

.98 | 0 | .02 | 8.9135 | 0.1511 |

0 | .98 | .02 | 7.8437 | 0.1329 |

Table: 1

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

(45) |

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

Since,

.

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

(46) |

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

(47) |

where as for anisotropic model

(48) |

Combining equations (34), (35), (36) and (37), the expression for DP in terms of redshift is given by

(49) |

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

(50) |

## 6 Some Physical Properties of the Model

### 6.1 The energy density in the universe

The energy density is given by

(51) |

where

(52) |

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

(53) |

(54) |

### 6.2 Shear Scalar

The shear scalar is given by

(55) |

where

(56) |

In our model

(57) |

From eq. (57), it is clear that shear scalar vanishes as .

### 6.3 Relative Anisotropy

The relative anisotropy is given by

(58) |

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

(59) |

Equations (37) and (59) lead to

(60) |

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

12.30891 Gyrs | |
---|---|

-0.505 | |

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 CDM model fits better with the observational data.

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